Constructive Tensorial Group Field Theory II: The $U(1)-T^4_4$ Model

TL;DR

This paper advances the non-perturbative construction of tensorial group field theories by proving analyticity and Borel summability for a specific super-renormalizable model with ultraviolet divergences, using a multiscale loop vertex expansion.

Contribution

It introduces a multiscale loop vertex expansion to establish analyticity and Borel summability for the $U(1)-T^4_4$ tensorial group field theory, extending constructive techniques to non-local, renormalizable models.

Findings

Proved analyticity of the model in a suitable domain.

Established Borel summability of the perturbative series.

Extended the loop vertex expansion technique for non-local theories.

Abstract

In this paper we continue our program of non-pertubative constructions of tensorial group field theories (TGFT). We prove analyticity and Borel summability in a suitable domain of the coupling constant of the simplest super-renormalizable TGFT which contains some ultraviolet divergencies, namely the color-symmetric quartic melonic rank-four model with Abelian $U(1)$ gauge invariance, nicknamed $U(1)-T^4_4$. We use a multiscale loop vertex expansion. It is an extension of the loop vertex expansion (the basic constructive technique for non-local theories) which is required for theories that involve non-trivial renormalization.