# Longitudinal Sound and Diffusion in Holographic Massive Gravity

## TL;DR

This paper investigates the longitudinal quasi-normal modes in a holographic massive gravity model, confirming hydrodynamic predictions for sound modes but finding unexpected discrepancies in the diffusion mode, and exploring higher frequency dynamics.

## Contribution

It provides a detailed numerical and analytical study of hydrodynamic and non-hydrodynamic longitudinal modes in holographic massive gravity, highlighting unexpected behavior in the diffusion constant.

## Key findings

- Sound mode speed and attenuation match hydrodynamic predictions.
- Diffusion constant from quasi-normal modes disagrees with hydrodynamics.
- Higher quasi-normal modes reveal dynamics beyond the hydrodynamic limit.

## Abstract

We consider a simple class of holographic massive gravity models for which the dual field theories break translational invariance spontaneously. We study, in detail, the longitudinal sector of the quasi-normal modes at zero charge density. We identify three hydrodynamic modes in this sector: a pair of sound modes and one diffusion mode. We numerically compute the dispersion relations of the hydrodynamic modes. The obtained speed and the attenuation of the sound modes are in agreement with the hydrodynamic predictions. On the contrary, we surprisingly find disagreement in the case of the diffusive mode; its diffusion constant extracted from the quasi-normal mode data does not agree with the expectations from hydrodynamics. We confirm our numerical results using analytic tools in the decoupling limit and we comment on some possible reasons behind the disagreement. Finally, we extend the analysis of the collective longitudinal modes beyond the hydrodynamic limit by displaying the dynamics of the higher quasi-normal modes at large frequencies and momenta.