Schwinger-Keldysh effective action for hydrodynamics with approximate symmetries

TL;DR

This paper develops a systematic method using spurious symmetry transformations within the Schwinger-Keldysh effective action framework to analyze how approximate symmetries, like chiral symmetry in QCD, influence hydrodynamic behavior and relaxation processes.

Contribution

It introduces a novel approach to constrain symmetry-breaking effects in non-equilibrium hydrodynamics with approximate symmetries, applying it to chiral symmetry in QCD at finite temperature and density.

Findings

Relaxation term in axial Ward-Takahashi identity is second order in quark mass.

In broken phase, relaxation is subleading compared to pion mass term.

In restored phase, relaxation becomes the dominant axial charge relaxation mechanism.

Abstract

We study the hydrodynamic theories with approximate symmetries in the recently developed effective action approach on the Schwinger-Keldysh (SK) contour. We employ the method of spurious symmetry transformation for small explicit symmetry-breaking parameters to systematically constrain symmetry-breaking effects in the non-equilibrium effective action for hydrodynamics. We apply our method to the hydrodynamic theory of chiral symmetry in Quantum Chromodynamics (QCD) at finite temperature and density and its explicit breaking by quark masses. We show that the spurious symmetry and the Kubo-Martin-Schwinger (KMS) relation dictate that the Ward-Takahashi identity for the axial symmetry, i.e., the partial conservation of axial vector current (PCAC) relation, contains a relaxational term proportional to the axial chemical potential, whose kinetic coefficient is at least of the second order in the quark mass. In the phase where the chiral symmetry is spontaneously broken, and the pseudo-Nambu-Goldstone pions appear as hydrodynamic variables, this relaxation effect is subleading compared to the conventional pion mass term in the PCAC relation, which is of the first order in the quark mass. On the other hand, in the chiral symmetry-restored phase, we show that our relaxation term, which is of the second order in the quark mass, becomes the leading contribution to the axial charge relaxation. Therefore, the leading axial charge relaxation mechanism is parametrically different in the quark mass across a chiral phase transition.