# Kubo formulae for the shear and bulk viscosity relaxation times and the scalar field theory shear $\tau_\pi$ calculation

## TL;DR

This paper derives Kubo formulas for shear and bulk relaxation times using quantum field theory, and computes the shear relaxation time in massless bb theory via diagrammatic methods, linking it to thermal width.

## Contribution

It introduces Kubo formulas for relaxation times and calculates shear relaxation time in bb theory using a diagrammatic approach, connecting microscopic dynamics to hydrodynamic parameters.

## Key findings

- Kubo formulas for shear and bulk relaxation times derived from stress-energy tensor response functions.
- Shear relaxation time computed in massless bb theory using integral equations.
- Shear relaxation time inversely related to thermal width, consistent with theoretical expectations.

## Abstract

In this paper we provide a quantum field theoretical study on the shear and bulk relaxation times. First, we find Kubo formulas for the shear and the bulk relaxation times, respectively. They are found by examining response functions of the stress-energy tensor. We use general properties of correlation functions and the gravitational Ward identity to parametrize analytical structures of the Green functions describing both sound and diffusion mode. We find that the hydrodynamic limits of the real parts of the respective energy-momentum tensor correlation functions provide us with the method of computing both the shear and bulk viscosity relaxation times. Next, we calculate the shear viscosity relaxation time using the diagrammatic approach in the Keldysh basis for the massless $\lambda\phi^4$ theory. We derive a respective integral equation which enables us to compute $\eta\tau_\pi$ and then we extract the shear relaxation time. The relaxation time is shown to be inversely related to the thermal width as it should be.