Symmetry behavior at finite temperature

Spontaneous symmetry breaking at finite temperature is studied. We show that for the class of theories discussed, symmetry is restored above a critical temperature βc^-1. We determine βc by a functional-diagrammatic evaluation of the effective potential and the effective mass. A formula for βc 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.