# Spectral functions and critical dynamics of the $O(4)$ model from classical-statistical lattice simulations

## TL;DR

This study uses classical-statistical lattice simulations to analyze spectral functions and critical dynamics of the relativistic $O(4)$ model, revealing complex behaviors near phase transitions and critical points.

## Contribution

It provides the first detailed real-time spectral function analysis of the $O(4)$ model across different phases and near criticality using lattice simulations.

## Key findings

- Spectral functions show relativistic quasi-particle peaks in low and high temperature phases.
- Additional structures appear in spectral functions during the crossover region.
- Spectral functions exhibit an infrared power law near the critical point, indicating critical dynamics.

## Abstract

We calculate spectral functions of the relativistic $O(4)$ model from real-time lattice simulations in classical-statistical field theory. While in the low and high temperature phase of the model, the spectral functions of longitudinal $(\sigma)$ and transverse $(\pi)$ modes are well described by relativistic quasi-particle peaks, we find a highly non-trivial behavior of the spectral functions in the cross over region, where additional structures appear. Similarly, we observe a significant broadening of the quasi-particle peaks, when the amount explicit $O(4)$ symmetry breaking is reduced. We further demonstrate that in the vicinity of the $O(4)$ critical point, the spectral functions develop an infrared power law associated with the critical dynamics, and comment on the extraction of the dynamical critical exponent $z$ from our simulations.