Nonperturbative renormalization group beyond the 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 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.

Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow

“…k ( 0 ) are both renormalized and effective field strength and mass. From the definition (8), and with some calculation (see [4]), we obtain the following statement: Proposition 2. In the UV domain Λ k 1 and in the symmetric phase, the leading order flow equations for essential and marginal local couplings are given by:…”

“…But the propagator has other source of variability. In particular, it is not invariant with respect to the unitary symmetry of the classical interactions (4). Focusing on an infinitesimal transformation : δ 1 := (id + , id, · · · , id) acting nontrivially only on the color 1 for some infinitesimal anti-Hermitian transformations , the transformation rule for the propagator follows the Lie bracket:…”

“…The basic strategy of the EVE is to close the hierarchical system coming from (9) using the analytic properties of the effective vertex functions 2 and the rigid structure of the melonic diagrams. More precisely, the EVE express all the melonic effective vertices Γ (n) having negative flow dimension (that is for n > 4) in terms of effective vertices with positive or null flow dimension, that is Γ (2) and Γ (4) , and their flow is entirely driven by just-renormalizable couplings. As recalled, in this way we keep the entirety of the melonic sector and the full momentum dependence of the effective vertices.…”

“…In the same way, one expect that the melonic effective vertex Γ (6),i melo p1, p2, p3, p4, p5, p6 is completely determined by a reduced effective vertex π (i) 3 : Z 3 → R hooked to a boundary configuration such as: (18) and so one for Γ (n),i k, p1,··· , pn , involving the reduced vertex π (i) n : Z n → R. In the last expression, perm denote the permutation of the external edges like in (17). The reduced vertices π (i) 2 can be formally resumed as a geometric series [2]- [4]:…”

“…This was concerned π 3 defined by (22), and the derivative of the 4-point vertex (29). We checked in [4]- [5] the compatibility of this approximation with the second Ward identity (35). This is a strong constraint, in favor of the reliability of our approximation.…”