Some Two-Loop Corrections to the Finite Temperature Effective Potential in the Electroweak Theory

Perturbation theory at finite temperature suffers from well-known infrared problems. In the standard model, as a result, one cannot calculate the effective potential for arbitrarily small values of $\phi$, the Higgs expectation value. Because the Higgs field is now known not to be extremely light, it is necessary to determine whether perturbation theory is a reliable guide to properties of the weak phase transition. In this note, we evaluate the most singular contributions to the potential at two loops as well as the leading strong interaction contributions. Above the critical temperature, the strong interaction corrections are reasonably small, while the weak corrections are about 10\%, even for rather small values of the Higgs field. At the critical temperature, the weak corrections have a more substantial effect, rendering the transition significantly more first order, but not significantly changing the upper bound on the Higgs mass required for baryogenesis.