Resurgence for the non-conformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics

TL;DR

This paper explores the resurgence phenomena in non-conformal Bjorken flow with quantum statistics, constructing transseries solutions and analyzing their Borel plane singularities to understand the flow's nonperturbative structure.

Contribution

It develops a conjecture on the resurgent relations in non-conformal hydrodynamics with quantum statistics and verifies it through numerical evaluation of Stokes constants.

Findings

Resurgent relations are conjectured based on Borel plane singularities.

Numerical checks support the conjecture for various initial conditions and particle masses.

Comments on transseries structure and special cases like massless particles are provided.

Abstract

We consider resurgence for the nonconformal Bjorken flow with Fermi-Dirac and Bose-Einstein statistics on the extended relaxation-time approximation. We firstly consider full formal transseries expanded around the equilibrium and then construct the resurgent relation by looking to the structure of Borel transformed ODEs. We form a conjecture of the resurgent relation based on the considerations that Stokes constants constituting of the resurgent relation originate only from singularities of dissipative variables on the Borel plane and that the other variables such as temperature and chemical potential become Borel nonsummable through nonlinear terms with the dissipative variables. We numerically check the conjecture for fundamental variables by explicitly evaluating values of the dominant Stokes constant depending on initial conditions and a particle mass. We also make comments on some issues related to transseries structure and resurgence such as the case of broken $U(1)$ symmetry, the massless case, generalized relaxation-time, and attractor solution.