Nonperturbative renormalization group beyond melonic sector: The Effective Vertex Expansion method for group fields theories

Abstract: Tensor models admit the large $N$ limit, dominated by the graphs called melons. The melons are characterized by the Gurau number $\varpi=0$ and the amplitude of the Feynman graphs are proportional to $N^{-\varpi}$. Other leading order contributions i.e. $\varpi> 0$ called pseudo-melons 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 (Lahoche and Ousmane Samary, arXiv:1803.09902), in which only the melonic sector has 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, which help to improve the truncation in the Wetterich equation. The set of Ward-Takahashi identities is derived and their compatibility along the flow provides a nontrivial constraint in the approximation schemes. In the symmetric phase, the Wetterich flow equation is given, and the numerical solution is studied.