Functional renormalization group approach for tensorial group field theory: a rank-6 model with closure constraint

Abstract

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.

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

“…As announced, a new term appears with respect to the truncated version (48), which contains a dependence on η and then move the critical line. The flow equation for mass may be obtained from (119) setting p = 0 on both sides.…”

Tensor Field Theory

“…The next steps have consisted in extending the method to TGFTs, both in rank 3 with a linear kinetic term [74] and in rank 6 with a quadratic kinetic term [56] and again at the level of quartic melonic truncations, essentially confirming the same qualitative behavior of asymptotic freedom in the ultraviolet regime with a fixed point in the infrared. The decompactification limit where the group U(1) is replaced by R has then been performed explicitly in [75], confirming again the existence of a promising transition to a condensed phase in the infrared regime.…”

“…The current period centers around a more systematic investigation of their properties and phase structure, generalizing many standard field theoretic tools such as the parametric representation [53], renormalization group equations of the Polchinski [54] and Wetterich type [55,56], Ward identities combined with Schwinger-Dyson equations [57] and Connes-Kreimer algebras [58].…”

Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow

“…TGFTs are celebrated for their nonlocal behavior in the interactions and the difficulties related to combinatorics. Thus new computation tools are needed to address the question of the renormalization group for TGFT [1]- [12]. First insights have been gained by nonperturbative Wetterich equation and, in particular by an investigation of the leading order melonic interactions, with a new method called effective vertex expansion (EVE) [1]- [5].…”