Constructive tensor field theory: The $T^{4}_{4}$ model

Abstract: We continue our constructive study of tensor field theory through the next natural model, namely the rank four tensor theory with quartic melonic interactions and propagator inverse of the Laplacian on $U(1)^4$. This superrenormalizable tensor field theory has a power counting quite similar to ordinary $\phi^4_3$. We control the model via a multiscale loop vertex expansion which has to be pushed quite beyond the one of the $T^4_3$ model and we establish its Borel summability in the coupling constant. This paper is also a step to prepare the constructive treatment of just renormalizable models, such as the $T^4_5$ model with quartic melonic interactions.