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.
Highlights
We introduce and motivate a new class of models that we will study throughout this work, by mixing different types of tensorial group field theories, that we call motley tensorial group field theories (MTGFT)
The truncated effective action (139) provides a parametrization of the theory space going beyond the connected melonic sector that we considered in the previous sections with the effective vertex expansion (EVE) method
This section is devoted to a first look at the numerical investigations of the flow equations obtained in the previous sections. Before coming to this analysis, we briefly summarize the results of the previous sections: (i) We defined a nonconventional TGFT, mixing two complex tensors interacting together
Summary
The construction of a fundamental theory, which describes the dynamics of spacetime at the quantum level, requires a consistent account for the superposition of (pre)geometric fluctuations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The existence of a well-defined power counting, in particular allows us to start a renormalization program, which has been strongly developed since the five last years as a promising way to investigate the continuum limit of discrete quantum gravity models. This success does not concern only the tensor models, but a new class of field theories derived from them and called tensorial field theories (TFTs) [51,52,53,54,55,56,57,58,59]. We provide three Appendixes A–C on which the power counting theorem, the proof of important propositions, and the computation of the sums which are used throughout the paper are given, respectively
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