Hydrodynamic gradient expansion in gauge theory plasmas

TL;DR

The leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrod dynamic excitation corresponding to a 'nonhydrodynamic' quasinormal mode on the gravity side is identified.

Abstract

We utilize the fluid-gravity duality to investigate the large order behavior of hydrodynamic gradient expansion of the dynamics of a gauge theory plasma system. This corresponds to the inclusion of dissipative terms and transport coefficients of very high order. Using the dual gravity description, we calculate numerically the form of the stress tensor for a boost-invariant flow in a hydrodynamic expansion up to terms with 240 derivatives. We observe a factorial growth of gradient contributions at large orders, which indicates a zero radius of convergence of the hydrodynamic series. Furthermore, we identify the leading singularity in the Borel transform of the hydrodynamic energy density with the lowest nonhydrodynamic excitation corresponding to a `nonhydrodynamic' quasinormal mode on the gravity side.

Figures (2)

• Figure 1: Behavior of the coefficients of hydrodynamic series for the energy density as a function of the order. At large enough order the coefficients start exhibiting factorial growth. The radius of convergence of the Borel transformed series is estimated to be $6.37$, in rough agreement with \ref{['eq.closestpole']}: $2/3\times6.37 = 4.25$.
• Figure 2: Real and imaginary parts of poles $\zeta_{0}$ of the symmetric Padé approximant of the Borel transform of the energy density \ref{['eq.enden']}. From the plot we removed numerically spurious poles. The pole closest to the origin governs the convergence radius of the Borel transformed series and gives rise to the lowest quasinormal frequency. The poles from the encircled region (magenta) lead to the powerlike preexponential factor in \ref{['eq.qnmcontrib']}.