arXiv : Hydrodynamic series and hydrodynamization of expanding plasma in kinetic theory

Heller, Michal P.; Svensson, Viktor; Kurkela, Aleksi; Spaliński, Michal

doi:10.1103/PhysRevD.97.091503

Article
Report number	arXiv:1609.04803 ; CERN-TH-2016-199
Title	Hydrodynamic series and hydrodynamization of expanding plasma in kinetic theory
Author(s)	Heller, Michal P. (Potsdam, Max Planck Inst. ; Perimeter Inst. Theor. Phys. ; Warsaw, Inst. Nucl. Studies) ; Kurkela, Aleksi (CERN ; Stavanger U.) ; Spaliński, Michal (Warsaw, Inst. Nucl. Studies ; Bialystok U.) ; Svensson, Viktor (Warsaw, Inst. Nucl. Studies ; Potsdam, Max Planck Inst.)
Publication	2018-06-01
Imprint	15 Sep 2016
Number of pages	6
Note	Comments: 6 pages, 3 figures, coefficients of the gradient expansion included in the submission 6 pages, 2 figures, v2: author added, major rewrite, mysterious off real axis singularities in the Borel plane explained (!), see also arXiv:1802.08225 [nucl-th] by Heller and Svensson; v3: references added, minor improvements in the text, first 426 terms from Eq. (8) included in the submission; v4: title changed, matches published version
In:	Phys. Rev. D 97 (2018) 091503
DOI	10.1103/PhysRevD.97.091503
Subject category	physics.flu-dyn ; Other Fields of Physics ; hep-th ; Particle Physics - Theory ; hep-ph ; Particle Physics - Phenomenology ; nucl-th ; Nuclear Physics - Theory
Abstract	We explore the transition to hydrodynamics in a weakly-coupled model of quark-gluon plasma given by kinetic theory in the relaxation time approximation with conformal symmetry. We demonstrate that the gradient expansion in this model has a vanishing radius of convergence due to the presence of a transient (nonhydrodynamic) mode, in a way similar to results obtained earlier in strongly-coupled gauge theories. This suggests that the mechanism by which hydrodynamic behaviour emerges is the same, which we further corroborate by a novel comparison between solutions of different weakly and strongly coupled models. However, in contrast with other known cases, we find that not all the singularities of the analytic continuation of the Borel transform of the gradient expansion correspond to transient excitations of the microscopic system: some of them reflect analytic properties of the kinetic equation when the proper time is continued to complex values.
Copyright/License	arXiv nonexclusive-distrib. 1.0 Publication: © 2018-2025 authors

$Poles of the Pad{\'e} approximant to the Borel transform of the gradient expansion. We have checked that the structure of singularities remains stable as the number of terms kept in the series is varied. We have also checked that the residues of the pictured poles lie well above what was set as the numerical accuracy, i.e. that they are not numerical artefacts. The depicted singularities are discretizations of branch cuts with branch points at values of $\xi$ given in Eqs.~\eqref{eq.leadsing} and~\eqref{eq.omegacomplex}, as well as at $\xi_{0} + \xi_{\pm}$ and $2\, \xi_{\pm}$ (see also Ref.~\cite{Casalderrey-Solana:2017zyh} for a related statement based on a smaller data set from the previous version of the present manuscript).$ $Poles in the Pade approximant to the Borel transform of the gradient expansion. Note that although the expression is not Borel-summable due to a singularity on the positive real line, its resummation is expected to give a well-defined answer (up to a choice of initial conditions) due to the phenomenon of resurgence~\cite{Heller:2015dha}.$ Show more plots

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