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 $24/{\lambda }_{\mathrm{cr}}$, where ${\lambda }_{\mathrm{cr}}\equiv \max [-4V(\varphi )/|\varphi {|}^4]$ with the false vacuum being at $\varphi =0$ 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.

Comments: 7 pages, 4 figure; v2 version published, discussions and references are added