Towards Accurate Gravitational Wave Predictions: Gauge-Invariant Nucleation in the Electroweak Phase Transition

TL;DR

This work tackles the longstanding issue of gauge dependence in predicting gravitational waves from a first-order electroweak phase transition within a SMEFT framework. By employing a three-dimensional EFT and generalizing Nielsen identities to finite temperature, the authors show that gauge-invariant nucleation rates and phase-transition parameters can be achieved, especially under the ultrasoft power counting $\lambda \sim g^3$. They demonstrate that the gauge dependence is substantially reduced at soft and ultrasoft scales, with ultrasoft results offering the most reliable predictions for GW signatures. The framework enables more precise connections between beyond-SM physics scales, PT dynamics, and observable GW spectra, and points to valuable cross-checks with non-perturbative methods and lattice studies. Overall, the gauge-invariant SMEFT-3D EFT approach provides a principled route to robustly predicting GW signals from early Universe phase transitions.

Abstract

The vacuum decay in the early Universe should be gauge-invariant. In this work, we study the gauge dependence of the vacuum decay occurring through a first-order phase transition and the associated gravitational wave production. We investigate the gauge dependence of the bubble nucleation and phase transition parameters within the framework of the Standard model effective field theory in three dimension. By considering the power-counting and utilizing the Nielsen identity at finite temperature, we show that, depending on the power-counting scheme favored by the new physics scale, the perturbative computation methodology allow we get the gauge-independent nucleation rates and phase transition, this enables more accurate predictions of gravitational wave signatures.

Figures (17)

• Figure 1: The effective potential at $\Lambda = 600$ GeV for $T=T_c\,, T_n$ with different gauge parameters in the soft scale (left) and ultrasoft scale (right).
• Figure 2: The two graphs that contribute to $C$ at one-loop order
• Figure 3: The diagrams needed for calculating the $Z$-factor.The dashed lines represent scalar propagators, the wavy lines represent gauge field propagators, and the dotted lines represent ghost field propagators.
• Figure 4: Left: the deviation $\Delta$ as function of $\Lambda$, where the Red, Green, and Blue curves respectively correspond to $\xi =0$, $\xi =0.5$, $\xi =1$. Right: the deviation $\delta$ as a function of $\Lambda$, the red curve represents the contribution from Eq. (\ref{['Nielsen_1b']}), which includes the terms $D$ and $\tilde{D}$, while the blue curve corresponds to the contribution without $D$ and $\tilde{D}$, like Eq. (\ref{['like2']}), different line styles denote different gauge parameters.
• Figure 5: The effective action $\mathcal{B}$ as function of temperature T at the soft scale and ultrasoft scale with $\Lambda$ =570 GeV.