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

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.