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 $S_{\text{La}}[\varphi ]$ which we explicitly calculate, using the Wilson RG and a $(4-\text{ep})$-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 $S_{\text{La}}[\varphi ]$ with $\text{La}$ of order the bubble wall thickness, and {\it not} by the usual (RG-improved) finite-temperature effective action which is reproduced by $S_{\text{La}}[\varphi ]$ for $\text{La}\to 0$. We argue that for weakly first-order transitions (such as that in the standard model) the $(4-\text{ep})$-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.