Relaxation time approximation revisited and non-analytical structure in retarded correlators

TL;DR

This paper rigorously justifies the relaxation time approximation (RTA) in kinetic theory, identifies its limitations with energy-dependent relaxation times, proposes an improved approach, and analyzes the non-analytical structures in retarded correlators for different interaction regimes.

Contribution

It provides a mathematical justification for RTA, proposes a method to restore collision invariance, and characterizes non-analytical structures in retarded correlators for hard and soft interactions.

Findings

RTA with energy-independent relaxation time is justified for hard interactions.

Hydrodynamic poles are the dominant modes in the retarded correlators for hard interactions.

Gapless branch-cuts appear in soft interactions due to the spectrum of the collision operator.

Abstract

In this paper, we give a rigorous mathematical justification for the relaxation time approximation (RTA) model. We find that only the RTA with an energy-independent relaxation time can be justified in the case of hard interactions. Accordingly, we propose an alternative approach to restore the collision invariance lacking in traditional RTA. Besides, we provide a general statement on the non-analytical structures in the retarded correlators within the kinetic description. For hard interactions, hydrodynamic poles are the long-lived modes. Whereas for soft interactions, commonly encountered in relativistic kinetic theory, the gapless eigenvalue spectrum of linearized collision operator leads to gapless branch-cuts. We note that particle mass and inhomogeneous perturbations would complicate the above-mentioned non-analytical structures.