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

TL;DR

This paper advances the constructive analysis of rank four tensor field theories with quartic melonic interactions, establishing Borel summability and setting the stage for studying more complex models.

Contribution

It extends the constructive approach to a new superrenormalizable tensor model with quartic interactions and demonstrates Borel summability beyond previous models.

Findings

Controlled the $T^4_4$ tensor model using multiscale loop vertex expansion.

Proved Borel summability of the model in the coupling constant.

Prepared groundwork for analyzing renormalizable tensor models like $T^4_5$.

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.