Finite-temperature bubble nucleation with shifting scale hierarchies

TL;DR

This paper develops a state-of-the-art framework to compute finite-temperature bubble nucleation rates in theories with shifting scale hierarchies, focusing on supercooled phase transitions in a scale-invariant SU(2)cSM model. It critically assesses the derivative expansion used in thermal EFTs for nucleation and demonstrates that gauge-field fluctuations induce divergences beyond leading orders, necessitating direct evaluation of fluctuation determinants. By combining a 3D high-temperature EFT with exact functional determinants (via Gel'fand–Yaglom methods) for gauge- and scalar-sector fluctuations, the authors achieve gauge- and renormalization-scale-invariant nucleation rates at NLO, including two-loop soft logarithms. Numerical results show substantial improvements over derivative-expansion approaches and reveal large shifts in percolation temperatures and bubble properties, with important implications for gravitational-wave predictions in beyond-the-Standard-Model scenarios. The work thus provides a robust, scalable framework for predicting GW signals from supercooled phase transitions in models with multi-scale dynamics and motivates extending these methods to broader theories and higher-loop accuracy.

Abstract

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 various 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 leading two 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 large explicit logarithms at two loops, utilising the exact renormalisation group structure of the EFT. 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.

Figures (7)

• Figure 1: Field-dependent masses evaluated on the bounce solution for a benchmark point with $g_{\hbox{\tiny\rm{$X$}}} = 0.8$ and $M_{\hbox{\tiny\rm{$X$}}} = 10^4$ GeV. Here, $m_{{\hbox{\tiny\rm{$X$}}},3}$, $m_{{\hbox{\tiny\rm{$X_0$}}},3}$ corresponds to the spatial and temporal gauge modes, respectively, $m_3$ is the scalar field bare mass. The prime denotes derivatives with respect to the field $v_3$. The vertical grey line denotes the tail of bounce solution where the gauge modes become lighter than nucleating field.
• Figure 2: Different approaches for computing $\Gamma_{\hbox{\tiny\rm{$T$}}}/T^4$ for a benchmark point with $g_{\hbox{\tiny\rm{$X$}}} = 0.8$ and $M_{\hbox{\tiny\rm{$X$}}} = 10^4$ GeV. Bands illustrate the sensitivity of different approaches to the choice of 4D RG scale at $\bar{\mu}_{{\hbox{\tiny\rm{4d}}}} = \pi T$ (solid) and $\bar{\mu}_{{\hbox{\tiny\rm{4d}}}} = 7 T$ (dotted). The \ref{['it:NLO-det-T4']} curve is evaluated at $\bar{\mu}_{{\hbox{\tiny\rm{4d}}}} = 7T$. The 3D scale is set to $\bar{\mu}_{{\hbox{\tiny\rm{3d}}}}=T$.
• Figure 3: Predictions for the percolation temperature for different approximations to the NLO nucleation rate. Left panel: \ref{['it:NLO-det']} results for $T_{\rm p}$, right panel: relative difference between \ref{['it:NLO-det']} and \ref{['it:NLO-grad']} results for $T_{\rm p}$.
• Figure 4: Comparison of different approximations to the nucleation rate and the corresponding percolation temperature. Left panel: relative difference between \ref{['it:NLO-det']} and \ref{['it:NLO-det-T4']}, right panel: relative difference between \ref{['it:NLO-det-T4']} and \ref{['it:NLO-grad']}.
• Figure 5: Percolation temperature as a function of $g_{\hbox{\tiny\rm{$X$}}}$ for fixed $M_{\hbox{\tiny\rm{$X$}}} = 10^{4.05}$ GeV, obtained with different approximations to the scalar determinant in the prefactor of the nucleation rate. Here, $\mathcal{I}_\phi = \bigl(S_{\rm 3}^{{\hbox{\tiny\rm{EFT,LO}}}}[v_{3,b}]/(2\pi)\bigr)^{3/2}$ is the Jacobian factor obtained by the removal of scalar zero modes.