  Models with radiative symmetry breaking typically feature strongly
supercooled first-order phase transitions, which result in an observable
stochastic gravitational wave background. In this work, we analyse the role of
higher order thermal corrections for these transitions, applying
high-temperature dimensional reduction to a theory with dimensional
transmutation. In particular, we study to what extent high-temperature
effective field theories (3D EFT) can be used. We find that despite significant
supercooling down from the critical temperature, the high-temperature expansion
for the bubble nucleation rate can be applied using the 3D EFT framework, and
we point out challenges in the EFT description. We compare our findings to
previous studies, and find that the next-to-leading order corrections obtained
in this work have a significant effect on the predictions for GW observables,
motivating a further exploration of higher order thermal effects.
