PHYSICAL REVIEW D |
VOLUME 9, NUMBER 12 |
15 JUNE 1974 |

### Symmetry behavior at finite temperature*

#### L. Dolan and R. Jackiw

*Laboratory for Nuclear Science, and Department of Physics,*

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

(Received 18 March 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.

#### I. INTRODUCTION

By drawing an analogy with the Meissner effect, Kirzhnits and Linde^{1}
have suggested that spontaneous symmetry violation in relativistic field theory
will disappear above a critical temperature. They gave qualitative arguments to
support this contention in a theory with global symmetry (*not* a gauge
theory) and obtained an order-of-magnitude expression for the critical
temperature in terms of the parameters of the theory. This problem was next
examined by Weinberg, who, in a preliminary investigation,^{2} derived a
numerical value for the critical temperature in the Kirzhnits-Linde model. He
then began to develop a complete analysis of spontaneous symmetry violation
and/or persistence at finite temperature, with special emphasis on gauge
theories with local symmetries.
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

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