  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.
