  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.
