  We build constructively the simplest tensor field theory which requires some
renormalization, namely the rank three tensor theory with quartic interactions
and propagator inverse of the Laplacian on $U(1)^3$. This superrenormalizable
tensor field theory has a power counting almost similar to ordinary $\phi^4_2$.
Our construction uses the multiscale loop vertex expansion (MLVE) recently
introduced in the context of an analogous vector model. However to prove
analyticity and Borel summability of this model requires new estimates on the
intermediate field integration, which is now of matrix rather than of scalar
type.
