Critical phenomena in field theories at finite temperature

Abstract

It is shown how renormalization-group techniques which have been successfully applied to critical phenomena in condensed-matter systems can be adapted to the study of phase transitions in field theories at finite temperature. Explicit calculations are described for the simplest case of a self-interacting scalar field. A systematic method of estimating the transition temperature to all orders of renormalized perturbation theory is given, and the two-loop contribution is found. It appears, however, that there is an additional, non-perturbative contribution which is not determined in the present work. An improved approximation to the effective potential is obtained, in which the renormalization group is used to resum the infrared singularities of perturbation theory. This shows that the transition is of second order, disproving recent claims to the contrary. Gauge theories are discussed qualitatively. While application of the techniques described in this paper will probably show that the transitions in some gauge theories are of first order, it is argued that the order of these transitions can probably not be determined reliably using methods currently available.