Finite Temperature Scalar Potential from a 1/N Expansion

We compute the leading and next–to–leading corrections to the finite temperature scalar potential for a 3+1 dimensional φ4 theory using a systematic 1/N expansion. Our approach automatically avoids problems associated with infrared divergences in ordinary perturbation theory in h̄. The leading order result does not admit a first order phase transition. The subleading result shows that the exact theory can admit at best only a very weak first order phase transition. For N = 4 and weak scalar coupling we find that T1, the temperature at which tunneling from the origin may begin in the case of a first order transition, must be less than about 0.5 percent larger than T2, the temperature at which the origin changes from being a local minimum to being a local maximum. We compare our results to the effective potential found from a sum of daisy graphs.