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 licit. We determine xc by a functional-diagrammatic evaluation of the effective potential and the effective mass. A formula for do 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 eritical point 1 or 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' 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 andjor persistence at finite temperature, with special emphasis on gauge theories with local symmetries.

It was Weinberg who suggested to us that the diagrammatic-functional methods 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

VII. CONCLUSION

The restoration of a spontaneously broken symmetry above a critical temperature is a phenomenon whose aspects in field theory have been exhibited in this investigation. An interesting distinction emerges between symmetries broken dynamically and those broken explicitly by scalar fields: The former remain broken at high temperature, the latter can be restored.

We have computed the critical temperature in terms of the renormalized parameters of the theory for a variety of models. Also in the target limit of an O(N)-invariant spinless theory, we obtained a parameter-independent description of the behavior near the critical point. Renormalization, which defined the parameters, was performed at zero temperature. This is by no means necessary - an alternate procedure is to renormalize at finite temperature; a convenient point is the critical temperature. In that case by bc is no longer calculable, but other parameters of the theory are expressed in terms of bc, and no information is lost.21 Finite-temperature field theories are examples of dynamical systems with long-range (infrared) modifications. It is possible that they present an analog to models with infrared trapping of various excitations.

The leading N summation can also be used at zero temperature to obtain information about the effective potential and about symmetry violation in the O(N)-invariant Bose field theory, (3.21). The zero-temperature version of our gap equation (3.39) is

where

This is not especially interesting, since it merely sums the loops of the mass renormalization. However, it is easy to see that

is given in the leading N approximation, by exactly the same series of loops. Hence

Renormalization is trivial; we find

where we have set

and have rescaled the field

Clearly (7.3b) has a symmetry-breaking solution for m2<0:

A detailed study of (7.3b) is planned.22