Dynamical Manifestation of the Goldstone Phenomenon at 1-loop

TL;DR

This paper calculates the damping rate of Goldstone modes in a thermal medium with explicit symmetry breaking, revealing how damping behaves near zero explicit breaking and highlighting the Goldstone phenomenon's dynamical aspects.

Contribution

It provides a one-loop analysis of Goldstone mode damping in a thermal medium, including effects of explicit symmetry breaking, and shows the non-analytic dependence of damping on the breaking parameter.

Findings

Damping rate depends non-analytically on explicit symmetry breaking parameter h.

Damping rate vanishes as h approaches zero, indicating no restoring force.

The analysis demonstrates the dynamical manifestation of the Goldstone phenomenon.

Abstract

We have calculated the damping rate $\Gamma (|{\bf k}|)$ for classical on-shell Goldstone modes of the O(2) symmetric scalar fields propagating in a thermal medium of the broken symmetry phase taking into account the effect of the explicit symmetry breaking. The result of the one-loop analysis can be expanded around $\Gamma (0)$, which depends non-analytically on the parameter of the explicit symmetry breaking, h. $\Gamma (0)$ vanishes when $h\to 0$, demonstrating in this way the absence of the restoring force, when the equilibrium direction of the symmetry breaking is modulated homogeneously.