Finite-temperature bubble nucleation with shifting scale hierarchies

A bstract Focusing on supercooled phase transitions in models with classical scale symmetry, we formulate a state-of-the art framework for computing the bubble nucleation rate, accounting for the presence of varying energy scales. In particular, we examine the limitations of derivative expansions in constructing a thermal effective field theory for bubble nucleation. We show that for gauge-field fluctuations, derivative expansions diverge after the two leading orders due to the strong variation in gauge-field masses between the high- and low-temperature phases. By directly computing these contributions using the fluctuation determinant, we capture these effects while also accounting for the dominant contribution at two-loop. Finally, we demonstrate how this approach significantly improves nucleation rate calculations compared to leading-order results, providing a more robust framework for predicting gravitational-wave signals from supercooled phase transitions in models such as the SU(2)cSM.