One-loop corrections to the metastable vacuum decay

TL;DR

We address the problem of computing the one‑loop prefactor in false vacuum decay for a self‑interacting scalar field in $3+1$ dimensions using a numerical functional‑determinant method. Fluctuations around the $O(4)$‑symmetric bounce are decomposed into radial partial waves, with the determinant evaluated via a radial ODE approach and careful treatment of the negative and translational zero modes; renormalization is carried out in the $\overline{MS}$ scheme and the finite nonperturbative part ${\cal D}^{\overline{(3)}}$ is computed numerically. The perturbative parts $A^{(1)}$ and $A^{(2)}$ are obtained analytically, and the full one‑loop effective action combines these with ${\cal D}^{\overline{(3)}}$. Numerically, quantum corrections generally suppress the decay rate and are modest away from the thin‑wall limit, while near thin‑wall the gradient expansion captures the dominant contributions. The approach is general and extensible to four dimensions, providing a practical tool for quantum corrections to tunnelling rates in realistic field theories.

Abstract

We evaluate the one-loop prefactor in the false vacuum decay rate in a theory of a self interacting scalar field in 3+1 dimensions. We use a numerical method, established some time ago, which is based on a well-known theorem on functional determinants. The proper handling of zero modes and of renormalization is discussed. The numerical results in particular show that quantum corrections become smaller away from the thin-wall case. In the thin-wall limit the numerical results are found to join into those obtained by a gradient expansion.

Figures (6)

• Figure 1: Potential $U(\Phi)$ in dimensionless form Eq. (\ref{['pot']}). The curves are labelled with the value of $\alpha$.
• Figure 2: Bounce profiles for different $\alpha$.
• Figure 3: Classical action $\tilde{S}_{cl}$ versus $\alpha$ (left) and the ratio $\tilde{S}_{cl}/\tilde{S}^{tw}_{cl}$ for $\alpha > 0.5$ (right).
• Figure 4: The ratio $G(\alpha,\beta)=S^{eff}_{1-loop}/ S_{cl}$ for $\beta=1$.
• Figure 5: Our results for the effective action $S^{eff}_{\rm 1-loop}$ (squares) together with the perturbative part $S^{eff}_{\rm 1-loop, p}$ (dotted line) and the leading parts of the gradient expansion $S^{eff}_{{\rm grad}, 0+2}$ (dashed line, $\alpha=0.45-0.95$). All shown quantities are multiplied by the factor $(1-\alpha)^3$.