Symmetry behavior at finite temperature

TLDR

Spontaneous symmetry breaking at finite temperature is investigated. The authors determine the critical temperature by diagrammatically evaluating the effective potential and mass, derive an approximate gap equation for near‑critical behavior, and ensure gauge invariance in gauge theories. They find that symmetry is restored above a critical temperature, provide a renormalized‑parameter formula for this temperature that is accurate at weak coupling, and argue that no critical point exists when symmetry is dynamically violated.

Abstract

Spontaneous symmetry breaking at finite temperature is studied. We show that for the class of theories discussed, symmetry is restored above a critical temperature ${{\ensuremath{\beta}}_{c}}^{\ensuremath{-}1}$. We determine ${\ensuremath{\beta}}_{c}$ by a functional-diagrammatic evaluation of the effective potential and the effective mass. A formula for ${\ensuremath{\beta}}_{c}$ is obtained in terms of the renormalized parameters of the theory. By examining a large subset of graphs, we show that the formula is accurate for weak coupling. An approximate gap equation is derived whose solutions describe the theory near the critical point. For gauge theories, special attention is given to ensure gauge invariance of physical quantities. When symmetry is violated dynamically, it is argued that no critical point exists.