Beyond the thin-wall approximation : precise numerical computation of prefactors in false vacuum decay

TL;DR

This work delivers a practical numerical framework to compute the false vacuum decay rate in four-dimensional scalar field theories, including the quantum fluctuation (one-loop) prefactor, without relying on the thin-wall approximation. It leverages the Gelfand–Yaglom method for determinants, extended with an angular momentum cutoff to handle higher dimensions, and combines low-$l$ numerical results with high-$l$ WKB analytics, together with MS-bar renormalization. A key contribution is a simple, entirely asymptotic-bounce-based formula for the l=1 zero-mode prefactor, and the approach is shown to reproduce known thin-wall results in the appropriate limit while remaining broadly applicable. The method is generalizable to multi-field settings, other dimensions, and finite-temperature cases, offering a versatile tool for semiclassical analyses of metastable decay.

Abstract

We present a general numerical method for computing precisely the false vacuum decay rate, including the prefactor due to quantum fluctuations about the classical bounce solution, in a self-interacting scalar field theory modeling the process of nucleation in four dimensional spacetime. This technique does not rely on the thin-wall approximation. The method is based on the Gelfand-Yaglom approach to determinants of differential operators, suitably extended to higher dimensions using angular momentum cutoff regularization. A related approach has been discussed recently by Baacke and Lavrelashvili, but we implement the regularization and renormalization in a different manner, and compare directly with analytic computations made in the thin-wall approximation. We also derive a simple new formula for the zero mode contribution to the fluctuation prefactor, expressed entirely in terms of the asymptotic behavior of the classical bounce solution.

Figures (7)

• Figure 1: Field potential $U(\phi)$ showing the true and false vacua, $\phi_+$ and $\phi_-$, respectively.
• Figure 2: Plots of the rescaled potential, $U(\Phi)=\frac{1}{2}\Phi^2-\frac{1}{2}\Phi^3+\frac{\alpha}{8}\Phi^4$, for $\alpha=0.6, 0.7, 0.8 , 0.9, 0.99.$ As $\alpha$ approaches 1, the vacua become degenerate.
• Figure 3: Plots of the bounce solution $\Phi_{\rm cl}(r)$ for various values of $\alpha$: $\alpha= 0.5$, $0.9$, $0.95$, $0.96$, $0.97$, $0.98$, $0.99$, with the plateau ending farther to the right for increasing $\alpha$. Observe that as $\alpha\to 1$, the sharp falloff in $\Phi_{\rm cl}$ occurs at $r\sim \frac{1}{1-\alpha}$, and $\Phi_{\rm cl}$ can be approximated by a step function.
• Figure 4: Plots of the fluctuation potential $U^{\prime\prime}(\Phi_{\rm cl}(r))$ for various values of $\alpha$ : $\alpha= 0.5$, $0.9$, $0.95$, $0.96$, $0.97$, $0.98$, $0.99$, with the binding well of the potential appearing farther to the right for increasing $\alpha$. Observe that as $\alpha\to 1$, the potential $U^{\prime\prime}(\Phi_{\rm cl}(r))$ is localized at $r\sim \frac{1}{1-\alpha}$, and is approximated well by the analytic form in (\ref{['thinwall']}).
• Figure 6: The $l$ dependence of $\ln T_{(l)}(\infty)$ for three different values of $\alpha$: $\alpha=0.01$ (top curve), $\alpha=0.7$ (middle curve), and $\alpha=0.9$ (bottom curve). The dots show the values of $\ln T_{(l)}(\infty)$ evaluated numerically using (\ref{['tequation']}), while the solid lines show the WKB prediction (\ref{['largel']}) of the leading large $l$ behavior. Notice the excellent agreement for $l\geq 20$.