Phase transition temperatures at next-to-leading order

Broken gauge symmetries are typically restored at high temperature, and the leading-order result for the critical temperature ${\mathit{T}}_{\mathit{c}}$ has been known for many years. I find a simple expression for the next-to-leading-order correction to ${\mathit{T}}_{\mathit{c}}$, which is order ${\mathit{eT}}_{\mathit{c}}$ where e is the gauge coupling. The result is a simple consequence of recent work on summing ring diagrams at high temperature in gauge theories. The result is valid when the Higgs self-coupling λ is the same order as ${\mathit{e}}^2$, and it does not address the case of strongly first-order phase transitions, which arise when λ≪${\mathit{e}}^2$.