The O(2) model in polar coordinates at nonzero temperature

TL;DR

This paper investigates the behavior of the O(2) model at finite temperature using polar coordinates and the CJT formalism, revealing insights into symmetry restoration, Goldstone theorem validity, and mass degeneracy issues.

Contribution

It provides a detailed analysis of the O(2) model at nonzero temperature in polar coordinates, including the effects of explicit symmetry breaking and the limitations of the Goldstone theorem.

Findings

Angular mass becomes tachyonic above 300 MeV with explicit breaking.

Goldstone theorem holds below critical temperature when symmetry breaking is infinitesimal.

Radial and angular masses never degenerate at any temperature.

Abstract

We study the restoration of spontaneously broken symmetry at nonzero temperature in the framework of the O(2) model using polar coordinates. We apply the CJT formalism to calculate the masses and the condensate in the double-bubble approximation, both with and without a term that explicitly breaks the O(2) symmetry. We find that, in the case with explicitly broken symmetry, the mass of the angular degree of freedom becomes tachyonic above a temperature of about 300 MeV. Taking the term that explicitly breaks the symmetry to be infinitesimally small, we find that the Goldstone theorem is respected below the critical temperature. However, this limit cannot be performed for temperatures above the phase transition. We find that, no matter whether we break the symmetry explicitly or not, there is no region of temperature in which the radial and the angular degree of freedom become degenerate in mass. These results hold also when the mass of the radial mode is sent to infinity.