Triviality proof for mean-field $\varphi_4^4$-theories
Pierre Wang, Christoph Kopper
TL;DR
The paper provides a rigorous nonperturbative analysis of triviality for mean-field φ^4_4 theories by employing Wilson renormalization group flow equations. It first establishes the flow for O(N) vector models and reduces it to a finite set of mean-field equations in terms of scale-dependent functions f_n(μ), proving local analyticity and uniqueness of mean-field solutions. The key result is the existence and uniqueness of smooth trivial solutions whose UV limits vanish, extended to arbitrarily large bare couplings and validated in the large-N regime. The analysis is then extended to a theory with a physical infrared mass, deriving a modified flow with a mass-dependent organizing function H(μ) and showing that triviality persists. Overall, the work strengthens nonperturbative triviality results for scalar field theories in four dimensions and demonstrates the robustness of the mean-field approach under both massless and massive settings.
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.