Progress in Solving the Nonperturbative Renormalization Group for Tensorial Group Field Theory

Abstract

This manuscript aims at giving new advances on the functional renormalization group applied to the tensorial group field theory. It is based on a series of our three papers [arXiv:1803.09902], [arXiv:1809.00247] and [arXiv:1809.06081]. We consider the polynomial Abelian U (1) d models without closure constraint. More specifically, we discuss the case of the quartic melonic interaction. We present a new approach, namely the effective vertex expansion method, to solve the exact Wetterich flow equation, and investigate the resulting flow equations, specially regarding the existence of non-Gaussian fixed points for their connection with phase transitions. To complete this method, we consider a non-trivial constraint arising from the Ward-Takahashi identities, and discuss the disappearance of the global non-trivial fixed points taking into account this constraint. Finally, we argue in favor of an alternative scenario involving a first order phase transition into the reduced phase space given by the Ward constraint.

Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow

“…It can be easily checked that the structure equations (19) and (22) satisfy the second Ward identity (35) see [2]- [4] and also [22]- [25]. In the same way the first Ward identity (34) has been checked to be compatible with the equation (19) and the melonic closed equation for the 2-point function.…”

“…These ones being called assorted functions. Moreover, due to the momentum conservation along the boundaries of faces, Γ (2) k must be diagonal:…”

“…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]:…”

“…The two orientations of the external effective vertices being took into account in the definition of π (i) 2,pp . Expanding the exact flow equation (9), and keeping only the relevant contraction for large k, one get the following relevant contributions forΓ (2) k andΓ…”

“…whereẊ := k∂X/∂k. The computation require the explicit expression of Γ (2) k . In the melonic sector, the self energy obey to a closed equation, reputed difficult to solve.…”