|PHYSICAL REVIEW D||VOLUME 9, NUMBER 12||15 JUNE 1974|
Spontaneous symmetry breaking at finite temperature is studied. We show that for the class of theories discussed, symmetry is restored above a critical temperature. We determine this by a functional-diagrammatic evaluation of the effective potential and the effective mass. A formula 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.
It was Weinberg who suggested to us that the diagrammatic-functional method for evaluating effective potentials in field theory, which had recently been developed,3-5 might be profitably employed to study temperature effects. We report here the results of our investigation. Weinberg has also presented an analysis of the problem.6 He uses diagrammatic methods to determine a temperature-dependent mass, as well as operator techniques to compute a temperature-dependent potential. We give a functional-diagrammatic evaluation of these quantities, from which the critical temperature can be deduced. All physical results are in agreement and confirm the qualitative observations of Kirzhnits and Linde.1
We examine a field theory at nonzero temperature, or equivalently the ensemble of finite-temperature Green's functions, defined by