Absolute Lower Bound on the Bounce Action

Abstract

The decay rate of a false vacuum is determined by the minimal action solution of the tunnelling field: bounce. In this Letter, we focus on models with scalar fields which have a canonical kinetic term in $N(>2)$ dimensional Euclidean space, and derive an absolute lower bound on the bounce action. In the case of four-dimensional space, we show the bounce action is generically larger than $24/\lambda_{\rm cr}$, where $\lambda_{\rm cr} \equiv {\rm max} [ -4V(\phi) /|\phi|^4] $ with the false vacuum being at $\phi=0$ and $V(0)=0$. We derive this bound on the bounce action \textit{without solving the equation of motion explicitly}. Our bound is derived by a quite simple discussion, and it provides useful information even if it is difficult to obtain the explicit form of the bounce solution. Our bound offers a sufficient condition for the stability of a false vacuum, and it is useful as a quick check on the vacuum stability for given models. Our bound can be applied to a broad class of scalar potential with any number of scalar fields. We also discuss a necessary condition for the bounce action taking a value close to this lower bound.