The tunneling decay rate in QFT beyond the thin wall approximation

The tunneling decay rate per unit volume in Quantum Field Theory (QFT), at order $\mathrm{\hslash }$, is given by $\mathrm{\Gamma }\mathrm{/}V=A{e}^{-B}$, where $B$ is the Euclidean action evaluated at the so-called bounce, and $A$ is proportional to the determinant of a second-order differential operator. The dominant contribution comes from the exponential factor. To estimate $\mathrm{\Gamma }\mathrm{/}V$, one must determine the bounce configuration, which satisfies a highly nonlinear equation. A common approach in the literature is the thin-wall approximation. In this work, we extend the formalism to cases where the thin-wall approximation is not valid. We employ a simple variational method to estimate both the bounce and the decay rate, and we find good agreement between our results and full numerical calculations.