The (φ 4)3 + 1 theory with infinitesimal bare coupling constants

Abstract

We study the (φ4)3 + 1 theory by means of a variational method improved with a BCS-type vacuum state. We examine the theory with both negative and positive infinitesimal bare coupling constants, where the theory has been suggested to exist nontrivially and stably in the infinite ultraviolet cutoff limit. When the cutoff is sent to infinity, we find the instability of the vacuum energy at the end point value of the variational parameter in the case of the negative bare coupling constant. For the positive bare coupling constant, we can renormalize the vacuum energy without using the extremal condition with respect to the variational mass parameter. We do not find an instability for the whole range of parameters including the end point. We still have a possibility that the theory with this bare coupling constant is nontrivial and stable.