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 $\phi^4$ theory using a systematic $1/N$ expansion. Our approach automatically avoids problems associated with infrared divergences in ordinary perturbation theory in $\hbar$. 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 $T_1$, 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 $T_2$, 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.