Vacuum Decay Rate in D-dimensional Electroweak theories

TL;DR

The paper addresses vacuum decay in D-dimensional electroweak theories by introducing a unified framework that combines a WKB expansion with dimensional regularization to compute quantum fluctuations from scalar, fermion, and gauge sectors. The method expresses the decay rate as $\gamma=\mathcal{A}e^{-\mathcal{B}}$, where $\mathcal{B}$ is the bounce action and $\mathcal{A}$ is the fluctuation prefactor obtained from a determinant-based treatment of all fields, organized via $O(D)$-symmetric radial modes and Gelfand–Yaglom-type relations $\ln \mathcal{A}_{X}=\sum_{\nu}\text{deg}_X(\nu;D)\ln(\Psi_\nu^X/\hat{\Psi}_\nu^X)$. The authors provide explicit degeneracy formulas and reduced-mode equations for Higgs, fermions, and gauge bosons, and implement a renormalization framework in the $\overline{\text{MS}}$ scheme to enable arbitrary-order, dimensionally regulated calculations. The framework is demonstrated in $D=4$ SMEFT and in the finite-temperature, dimensionally reduced $D=3$ SMEFT, yielding large, dimensionless decay-rate scales (e.g., $\log_{10}(\gamma\times\mathrm{Gyr}\times\mathrm{Gpc}^3)=183$ in 4D and nontrivial two-loop results in 3D). Overall, this method provides a robust, scalable tool for vacuum-stability analyses and phase-transition phenomenology across dimensions.

Abstract

We present a systematic framework for calculating the vacuum decay rate in D-dimensional electroweak theories, providing a unified treatment of quantum fluctuations for scalar, fermion, and gauge boson fields via a combined WKB expansion and dimensional regularization. This method ensures rapid convergence even at large angular momenta. Application to a $D=4$ standard model effective field theory gives $\log_{10}(γ\times \text{Gyr} \times \text{Gpc}^3)=183$. In the finite-temperature, dimensionally reduced standard model effective field theory with $D=3$, the values are $\log_{10}(Γ_T \times \text{Gyr} \times \text{Gpc}^3)=42.4$ at tree level and $116.1$ at two-loop order. The approach offers a general and efficient tool for analyzing vacuum stability and decay across dimensions.

Figures (4)

• Figure 1: Residuals of the WKB approximation for W boson($\times10^6$). X$-n$th represents the result obtained by subtracting its WKB approximation of X field up to order $n$. The result is not only scaled but also multiplied by $\bar{\nu}$ due to $\ln R^X_\nu\sim \delta C_1\bar{\nu}^{-1}+\dots$. And the factor by which the T mode part is multiplied is $\bar{\nu}+1$.
• Figure 2: Comparison of previous and new methods for the Da mode of W boson. Here, the previously method employs the 3-dimensional version of the method described in Ref. endoFalseVacuumDecay2017.
• Figure 3: Convergence of the equations($\nu=3$).
• Figure 4: Residuals of the WKB approximation for Higgs($\times10^4$) and top quarks($\times10^6$).