We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop γ-and β-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV. Pacs numbers: 11.10.Gh, 04.60.-m We recall some definitions (see [41][54][45]):
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 study the polynomial Abelian or U (1) d Tensorial Group Field Theories equipped with a gauge invariance condition in any dimension d. We prove the just renormalizability at all orders of perturbation of the ϕ 4 6 and ϕ 6 5 random tensor models. We also deduce that the ϕ 4 5 tensor model is super-renormalizable.
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