Triviality vs perturbation theory: an analysis for mean-field $\varphi^4$-theory in four dimensions

TL;DR

This paper links mean-field trivial solutions of four-dimensional $ ext{O}(N)$ $ ext{phi}^4$ theory to perturbation theory, proving Borel-summability and asymptotic equivalence with non-perturbative solutions when an UV cutoff is used.

Contribution

It establishes a rigorous connection between trivial mean-field solutions and perturbation theory, including Borel-summability and asymptotic behavior in four-dimensional $ ext{O}(N)$ $ ext{phi}^4$ theory.

Findings

Proves local Borel-summability of mean-field perturbation theory with UV cutoff.

Shows perturbative solutions are asymptotic to non-perturbative solutions.

Defines a renormalized coupling constant within the mean-field framework.

Abstract

We have constructed the mean-field trivial solution of the $\varphi^4$ theory $O(N)$ model in four dimensions in two previous papers using the flow equations of the renormalization group. Here we establish a relation between the trivial solutions we constructed and perturbation theory. We show that if an UV-cutoff is maintained, we can define a renormalized coupling constant $g$ and obtain the perturbative solutions of the mean-field flow equations at each order in perturbation theory. We prove the local Borel-summability of the renormalized mean-field perturbation theory in the presence of an UV cutoff and show that it is asymptotic to the non-perturbative solution.