Constructive Tensor Field Theory: The T44 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 ϕ34. We control the model via a multiscale loop vertex expansion which has to be pushed quite beyond the one of the T34 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 T54 model with quartic melonic interactions.