Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

TL;DR

This paper introduces a new tensorial field theory coupling matter and gravity, focusing on rank-3 tensors, and studies its renormalization group flow using approximation methods, interpreting Feynman graphs as Ising models on random lattices.

Contribution

It presents a novel coupled tensorial field theory framework and explores its renormalization group behavior, linking quantum gravity models with statistical physics.

Findings

Model is power counting just-renormalizable.

Feynman graphs interpreted as Ising configurations on random lattices.

Renormalization group flow equations constructed and analyzed.

Abstract

We introduce a new family of tensorial field theories by coupling different fields in a non-trivial way, with a view towards the investigation of the coupling between matter and gravity in the quantum regime. As a first step, we consider the simple case with two tensors of the same rank coupled together, with Dirac like kinetic kernel. We focus especially on rank-$3$ tensors, which lead to a power counting just-renormalizable model, and interpret Feynman graphs as Ising configurations on random lattices. We investigate the renormalization group flow for this model, using two different and complementary tools for approximations, namely, the effective vertex expansion method and finite-dimensional truncations for the flowing action. Due to the complicated structure of the resulting flow equations, we divided the work into two parts. In this first part we only investigate the fundamental aspects on the construction of the model and the different ways to get tractable renormalization group equations, while their numerical analysis will be addressed in a companion paper.