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We introduce a new family of tensorial field theories by coupling differentfields in a non-trivial way, with a view towards the investigation of thecoupling between matter and gravity in the quantum regime. As a first step, weconsider the simple case with two tensors of the same rank coupled together,with Dirac like kinetic kernel. We focus especially on rank-$3$ tensors, whichlead to a power counting just-renormalizable model, and interpret Feynmangraphs as Ising configurations on random lattices. We investigate therenormalization group flow for this model, using two different andcomplementary tools for approximations, namely, the effective vertex expansionmethod and finite-dimensional truncations for the flowing action. Due to thecomplicated structure of the resulting flow equations, we divided the work intotwo parts. In this first part we only investigate the fundamental aspects onthe construction of the model and the different ways to get tractablerenormalization group equations, while their numerical analysis will beaddressed in a companion paper.

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