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

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