Radiatively induced first-order phase transitions the necessity of the renormalization group

We advocate a (Wilson) renormalization-group (RG) treatment of finite-temperature first-order phase transitions, in particular those driven by radiative corrections such as occur in the standard model, and other spontaneously broken gauge theories. We introduce the scale-dependent coarse-grained free energy S_Λ[φ] which we explicitly calculate, using the Wilson RG and a (4 - ε)-expansion, for a scalar toy model that shares many features of the gauged case. As argued by Langer and others, the dynamics of the phase transition are described by S_Λ[φ] with 1/Λ of order the bubble wall thickness, and not by the usual (RG-improved) finite-temperature effective action which is reproduced by S_Λ[φ] for Λ → 0. We argue that for weakly first-order transitions (such as that in the he (4 - ε)-expansion is necessary to control an inevitable growth of the effective scale-dependent coupling towards the strong-coupling regime, and that diagrammatic resummation techniques are unlikely to be appropriate.