A positive definite formulation of vacuum decay with reduced symmetry

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.