Gauge-independent treatment of electroweak phase transition

TL;DR

The paper tackles the gauge-dependence problem in computing bubble nucleation during the electroweak phase transition within the Standard Model Effective Field Theory (SMEFT). By employing a three-dimensional EFT obtained via dimensional reduction with power counting $\lambda \sim g^3$, the authors apply Nielsen identities to prove that the nucleation exponent can be decomposed as $\mathcal{B}=\mathcal{B}_0+\mathcal{B}_1$ and remains gauge-invariant up to two-loop order. This leads to a gauge-independent nucleation rate $\Gamma$ and more reliable phase-transition parameters, such as the nucleation temperature $T_n$ and the strength parameter $\alpha$, with $T_n$ typically lowered and $\alpha$ enhanced compared to traditional treatments. The resulting gravitational-wave predictions become less sensitive to gauge choice, with stronger signals for smaller new-physics scales $\Lambda$ and potential detectability by future observatories like LISA, Taiji, and TianQin. Overall, the work strengthens the predictive power of perturbative SMEFT analyses of the electroweak phase transition and its gravitational-wave signatures.

Abstract

We provide the first certificate of the gauge-independent bubble nucleation at the electroweak phase transition with the standard model effective field theory. Taking advantage of the thermal effective field theory framework, with the power counting $λ\sim g^3$, we rigorously demonstrate the gauge independence of the bubble nucleation rate up to two-loop order. Furthermore, we analyze the influence of relevant phase transition parameters on the gauge parameter and investigate its implications for gravitational waves generated by the electroweak phase transitions.

Figures (7)

• Figure 1: The effective action $\mathcal{B}$ as function of temperature $T$ with $\Lambda=570$ GeV and $590$ GeV. The solid line denotes the results of tradition method, the dash line denotes the results of the gauge-independent method, and the color denotes the different values of $\xi$ and NP scales $\Lambda$.
• Figure 2: The behaviors of nucleation temperature $T_n$(top), the parameter $\alpha$ (middle) and $\beta/H$ (bottom) as functions of $\Lambda$. The solid line denotes the results with traditional method, the dash line denotes the gauge-independent results, the color denotes the different values of $\xi$.
• Figure 3: The effect of gauge parameter on GW prediction at $\Lambda=570$ and $590$ GeV. The color region dentes the sensitivity of detectors, including Taiji Hu:2017mdeRuan:2018tsw, Tianqin TianQin:2015yphZhou:2023rop, LISA LISA:2017pwjBaker:2019nia, BBO Crowder:2005nrCorbin:2005nyHarry:2006fi and DECIGO Seto:2001qfKawamura:2006upYagi:2011wgIsoyama:2018rjb.
• Figure 4: Contribution of $A^a_0,B_0$ to the two-loop effective potential. Solid line denote $A^a_0,B_0$, dashed line denote scalar, wave line denote gauge boson.
• Figure 5: The two graphs that contribute to $C$ at one-loop order.