Convergence of hydrodynamic modes: insights from kinetic theory and holography

TL;DR

This paper investigates the factors determining the convergence limits of hydrodynamic dispersion relations using kinetic theory and holography, revealing new features like branch cuts and pole collisions affecting convergence.

Contribution

It introduces a kinetic theory approach to analyze hydrodynamic convergence, highlighting differences from holographic models and providing a practical method to identify singularities.

Findings

Convergence radius in shear channel set by pole-branch point collision.

In the sound channel, convergence determined by pole-pole collision.

Provides a prescription using the Implicit Function Theorem to locate singularities.

Abstract

We study the mechanisms setting the radius of convergence of hydrodynamic dispersion relations in kinetic theory in the relaxation time approximation. This introduces a qualitatively new feature with respect to holography: a nonhydrodynamic sector represented by a branch cut in the retarded Green's function. In contrast with existing holographic examples, we find that the radius of convergence in the shear channel is set by a collision of the hydrodynamic pole with a branch point. In the sound channel it is set by a pole-pole collision on a non-principal sheet of the Green's function. More generally, we examine the consequences of the Implicit Function Theorem in hydrodynamics and give a prescription to determine a set of points that necessarily includes all complex singularities of the dispersion relation. This may be used as a practical tool to assist in determining the radius of convergence of hydrodynamic dispersion relations.