Linear Structure, Convergence and Resurgence of Hydrodynamic Transseries

Abstract

We construct an exact formulation for the solution of Bjorken expanding relativistic viscous hydrodynamics with a constant relaxation time and derive its late time transseries structure. The two fundamental modes of the linear auxiliary equation separate the hydrodynamic contribution from the exponentially suppressed transient mode. We show that this single transient mode generates the entire hierarchy of nonperturbative sectors through nonlinear mode mixing, while its decay explains the loss of initial condition dependence and the emergence of late time hydrodynamic attractor behavior. The hydrodynamic gradient expansion is factorially divergent for generic values of the transport parameter but converges at the discrete values $a=N^2$, where the auxiliary asymptotic expansions terminate. We further determine the exact Stokes constant of the auxiliary linear equation and establish its relation to the logarithmic Borel singularity coefficient. The vanishing of the Stokes constant at these exceptional values establishes a direct connection between auxiliary series termination, hydrodynamic convergence, and Stokes decoupling between the hydrodynamic and the nonhydrodynamic transient sector.