Tensor models admit the large N limit dominated by the graphs called melons. The melons are caracterized by the Gurau number = 0 and the amplitude of the Feynman graphs are proportional to N − . Other leading order contributions i.e.> 0 called pseudomelons can be taken into account in the renormalization program. The following paper deals with the renormalization group for a U (1)-tensorial group field theory model taking into account these two sectors (melon and pseudo-melon). It generalizes a recent work [arXiv:1803.09902], in which only the melonic sector have been studied. Using the power counting theorem the divergent graphs of the model are identified. Also, the effective vertex expansion is used to generate in detail the combinatorial analysis of these two leading order sectors. We obtained the structure equations, that help to improve the truncation in the Wetterich equation. The set of Ward-Takahashi identities is derived and their compactibility along the flow provides a non-trivial constraints in the approximation shemes. In the symmetric phase the Wetterich flow equation is given and the numerical solution is studied. 1 vincent.lahoche@cea.fr 2 dine.ousmanesamary@cipma.uac.bj An important notion for tensorial Feynman diagrams is the notion of faces, whose we recall in the following definition:Definition 2 A face is defined as a maximal and bicolored connected subset of lines, necessarily including the color 0. We distinguish two cases:• The closed or internal faces, when the bicolored connected set correspond to a cycle.3 Strictly speaking the term "quantum" is abusive, we should talk about statistical model, or quantum field theory in the euclidean time.

We develop the functional renormalization group formalism for a tensorial group field theory with closure constraint, in the case of a just renormalizable model over U (1) ⊗6 , with quartic interactions. The method allows us to obtain a closed but non-autonomous system of differential equations which describe the renormalization group flow of the couplings beyond perturbation theory. The explicit dependence of the beta functions on the running scale is due to the existence of an external scale in the model, the radius of S 1 U (1). We study the occurrence of fixed points and their critical properties in two different approximate regimes, corresponding to the deep UV and deep IR. Besides confirming the asymptotic freedom of the model, we find also a nontrivial fixed point, with one relevant direction. Our results are qualitatively similar to those found previously for a rank-3 model without closure constraint, and it is thus tempting to speculate that the presence of a Wilson-Fisher-like fixed point is a general feature of asymptotically free tensorial group field theories.

Renormalization group methods are an essential ingredient in the study of nonperturbative problems of quantum field theory. This paper deal with the symmetry constraints on the renormalization group flow for quartic melonic tensorial group field theories. Using the unitary invariance of the interactions, we provide a set of Ward-Takahashi identities which leads to relations between correlation functions. There are numerous reasons to consider such Ward identities in the functional renormalization group. Their compatibility along the flow provides a non-trivial constraint on the reliability of the approximation schemes used in the non-perturbative regime, especially on the truncation and the choice of the regulator. We establish the so called structure equations in the melonic sector and in the symmetric phase. As an example we consider the T 4 5 TGFT model without gauge constraint. The Wetterich flow equation is given and the way to improve the truncation on the effective action is also scrutinized.