Resolving the unitary gauge puzzle of thermal phase transitions

Broken gauge symmetries are typically restored at high temperature, and the critical temperature ${\mathit{T}}_{\mathit{c}}$ can be found to leading order by a simple, one-loop calculation in a renormalizable gauge. However, the one-loop calculation in unitary gauge yields a different result than in typical renormalizable gauges, and it has long been a puzzle how to calculate thermal quantities in unitary gauge. We show in the Abelian Higgs model that, for temperature T small compared to ${\mathit{T}}_{\mathit{c}}$, the loop expansion in unitary gauge is an expansion in ${\mathit{T}}^2$/${\mathit{T}}_{\mathit{c}}^2$. Thus, all orders in the loop expansion are relevant to the leading-order determination of the critical temperature. By explicit two-loop calculation, we verify that gauge-invariant quantities such as the effective scalar mass and the free energy of the vacuum agree with renormalizable gauge results to the corresponding order in ${\mathit{T}}^2$/${\mathit{T}}_{\mathit{c}}^2$. To this order, the equivalence of unitary and renormalizable gauges may be succinctly summarized by the corresponding high-temperature effective Lagrangians, which differ only by a nonlinear field redefinition.