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The $ε$-expansion 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 $ε$-expansion. This expansion, based on dimensional continuation from 3 to $4{-}ε$ spatial dimensions, provides a systematic approximation for computing the effects of (near)-critical fluctuations. The $ε$-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 $ε$. For moderately light Higgs masses (below 100~GeV), the $ε$-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 non-conservation after the transition is found to be {\em larger\/} than that given by the one-loop effective potential calculation. Detailed next-to-leading order calculations of some sample quantities suggests that the $\eps$-expansion is reasonably well behaved for Higgs masses below 100--200 GeV. We also compare the $\eps$-expansion with large-$N$ results (where $N$ is the number of scalar fields) and find that the $\eps$-expansion is less well behaved in this limit.