Theεexpansion and the electroweak phase transition

Standard perturbative (or mean field theory) techniques are not adequate for studying the finite-temperature electroweak phase transition in some cases of interest to scenarios for electroweak baryogenesis. We instead study the properties of this transition using the renormalization group and the $\epsilon$ expansion. This expansion, based on dimensional continuation from $3 \mathrm{to} 4-\epsilon$ spatial dimensions, provides a systematic approximation for computing the effects of (near-)critical fluctuations. The $\epsilon$ expansion is known to predict a first-order transition in Higgs theories, even for heavy Higgs boson masses. The validity of this conclusion in the standard model is examined in detail. A variety of physical quantities are computed at leading and next-to-leading order in $\epsilon$. For moderately light Higgs boson masses (below 100 GeV), the $\epsilon$ expansion suggests that the transition is more strongly first order than is predicted by the conventional analysis based on the one-loop (ring-improved) effective potential. Nevertheless, the rate of baryon nonconservation after the transition is found to be larger than that given by the one-loop effective potential calculation. Detailed next-to-leading order calculations of some sample quantities suggests that the $\epsilon$ expansion is reasonably well behaved for Higgs boson masses below 100-200 GeV. We also compare the $\epsilon$ expansion with large-$N$ results (where $N$ is the number of scalar fields) and find that the $\epsilon$ expansion is less well behaved in this limit.