A positive definite formulation of vacuum decay with reduced symmetry

TL;DR

This paper develops a positive definite formulation for vacuum decay calculations with reduced spherical symmetry, extending the tunneling potential method to impurity-catalyzed decay scenarios.

Contribution

It introduces a generalized, positive definite action for vacuum decay with $O(3)$ symmetry, broadening the applicability of tunneling potential methods beyond $O(4)$ symmetry.

Findings

The new action reduces to the tunneling potential in $O(4)$ symmetric cases.

Analytic examples demonstrate the method with $O(3)$ symmetry and arbitrary wall thickness.

The formulation explicitly accounts for impurity effects on vacuum decay.

Abstract

The Euclidean bounce for vacuum decay enjoys an $O(4)$ symmetry that is lost in the presence of impurities than can catalyze the decay. We present a formulation for the calculation of the tunneling decay action, that is explicitly positive definite, for impurities whose effects are spherically symmetric so that the bounce symmetry is reduced to $O(3)$. The action constructed can be regarded as a generalization of the tunneling potential method, which implicitly assumed $O(4)$ symmetry. We show that the action obtained reduces to the tunneling potential for $O(4)$-symmetric cases and provide analytic examples with $O(3)$ symmetry and arbitrary wall thickness.