Functional renormalization group analysis of rank-3 tensorial group field theory: The full quartic invariant truncation

Abstract

In this paper, we consider the complete momentum-independent quartic order truncation for the effective average action of a real Abelian rank-3 tensorial group field theory. This complete truncation includes nonmelonic as well as double-trace interactions. In the usual functional renormalization group perspective, the inclusion of more operators that belong to the underlying theory space corresponds to an improvement of the truncation of the effective average action. We show that the inclusion of nonmelonic and double-trace operators in the truncation brings subtleties. In particular, we discuss the assignment of scaling dimensions to the nonmelonic sector and how the inclusion of double-trace operators considerably changes the results for critical exponents with respect to those obtained when they are not included. We argue that this is not a particular problem of the present model by comparing the results with a pure tensor model. We discuss how these issues should be investigated in future work.

Towards background independent quantum gravity with tensor models

“…Actually, requiring that the system allows a 1/N expansion is not enough to uniquely fix the scaling dimensions: Typically, demanding the existence of a welldefined and non-trivial large-N limit only results in upper bounds for the scaling dimensions of couplings at least if one works in truncations. It is still consistent with the 1/N expansion to only saturate the upper bound for some couplings but not for all, see, e.g., [29,39]. A choice of canonical dimension below the upper bound leads to a decoupling of the corresponding interaction in the system of beta functions.…”

Renormalization group flow of coupled tensorial group field theories: Towards the Ising model on random lattices

“…In a first time, we can interpret our result as a pathology of the vertex expansion, and as a hint about the role of higher interactions to correctly understand the physical theory space and the fixed point structure. This was already pointed out in [77] for disconnected interactions, and confirmed the necessity of finding methods allowing to explore more sophisticated theory spaces. There is no version of the EVE for disconnected interactions, but this result should be viewed as a strong motivation for future investigations.…”

“…The same diagrams appears in two case. For the Ward identity (77), the contraction involves a variation of the propagator with respect to the momentum, and the equation express the variation of the quantity π (i,1) 2 with respect to |p|. In the same way, the flow equation (63) describes the evolution of π (i,1) 2 with the change of scale k, and the contractions involves the variation of the propagator with respect to k. In both case, this is the non-trivial variation of the propagator which generate the change; and there are no reason to discard the Ward identity, especially because as it is easy to cheek, the two variations, with respect to p and k are not independent (see equation (56)) 8 .…”

Ward-constrained melonic renormalization group flow for the rank-four ϕ6 tensorial group field theory

“…In this section we propose to take into account the constraint (108) along the flow following the strategy described in [76]. As explained by the authors, the difficulty come from the definition (68) or (69), inherited from the structure of the melonic diagrams. This definition is too rigid to be conserved on the global phase space without introducing hard singularities.…”

“…Truncation consist as a systematic projection into a reduced dimensional phase space along which the flow equations may be solved analytically or numerically. This methods has been performed for a very large class of models [61]- [68]. Interestingly, all of them reveal the occurrence of nonperturbative fixed point in the symmetric phase, that is, as long as the vanishing mean field remains a good vacuum around which we can expand the flow equation.…”