Triviality proof for mean-field $\varphi_4^4$-theories

TL;DR

This paper proves the triviality of mean-field $ ext{φ}_4^4$-theories by extending previous results to $O(N)$ models and replacing the infrared cutoff with a physical mass, advancing the understanding of scalar quantum field theories.

Contribution

It extends the triviality proof of mean-field scalar theories to $O(N)$ models and removes the infrared cutoff, using rigorous non-perturbative analysis of Wilson renormalization group equations.

Findings

Proved triviality of mean-field $ ext{φ}_4^4$-theories with infinite bare coupling.

Extended the analysis to $O(N)$ vector models.

Replaced the infrared cutoff with a physical mass in the analysis.

Abstract

The differential equations of the Wilson renormalization group are a powerful tool to study the Schwinger functions of Euclidean quantum field theory. In particular renormalization theory can be based entirely on inductively bounding their perturbatively expanded solutions. Recently the solutions of these equations for scalar field theory have been analysed rigorously without recourse to perturbation theory, at the cost of restricting to the mean-field approximation. In particular it was shown there that one-component $\varphi^4_4$-theory is trivial if the bare coupling constant of the UV regularized theory is not large. This paper presents progress w.r.t. Kopper's previous paper on asymptotically free solutions of the mean-field scalar flow equations: 1. The upper bound on the bare coupling is sent to infinity and the proof is extended to $O(N)$ vector models. 2. The unphysical infrared cutoff used for technical simplicity is replaced by a physical mass.