On the correlation functions in stable first-order relativistic hydrodynamics

TL;DR

This paper analyzes the correlation functions in stable first-order relativistic conformal hydrodynamics, developing methods to understand their analytical structure and long-time behavior, confirming results with effective field theory calculations.

Contribution

It introduces a method to numerically and analytically analyze the branch cut structure of response functions in relativistic hydrodynamics, extending understanding beyond linear response.

Findings

Identified branch cuts and threshold singularities in response functions.

Derived the long-time tail of correlation functions.

Confirmed analytical results with effective field theory calculations.

Abstract

First-order relativistic conformal hydrodynamics in a general (hydrodynamic) frame is characterized by a shear viscosity coefficient and two UV-regulator parameters. Within a certain range of these parameters, the equilibrium is stable and propagation is causal. In this work we study the correlation functions of fluctuations in this theory. We first compute hydrodynamic correlation functions in the linear response regime. Then we use the linear response results to explore the analytical structure of response functions beyond the linear response. A method is developed to numerically calculate the branch cut structure from the well-known Landau equations. We apply our method to the shear channel and find the branch cuts of a certain response function, without computing the response function itself. We then solve the Landau equations analytically and find the threshold singularities of the same response function. Using these results, we achieve the leading singularity in momentum space, by which, we find the long-time tail of the correlation function. The results turn out to be in complete agreement with the loop calculations in effective field theory.