Triviality in a Non-Perturbative Second-Order Mean-Field Theory for $ϕ^4_4$

Abstract

We introduce a second-order mean-field description of the four-dimensional Euclidean $ϕ^4$ model within the Wilson--Polchinski renormalization-group framework. The construction is based on the connected amputated Schwinger functions evaluated at the symmetric momentum configurations $ (p,-p,\ldots,p,-p)$, which are decomposed into a momentum-independent component, a component quadratic in $p$, and a higher-order remainder. The first two components are chosen to satisfy a closed nonlinear hierarchy. We prove the existence of solutions to this hierarchy for arbitrary positive bare coupling and establish their convergence to the Gaussian fixed point as the ultraviolet cutoff is removed. In particular, both the momentum-independent and the quadratic momentum sectors are asymptotically trivial.