Abstract
The nontrivial fixed point discovered for ${\ensuremath{\phi}}^{4}$-marginal couplings in tensorial group field theories has been shown to be incompatible with Ward-Takahashi identities. In a previous analysis, we have stated that the case of models with interactions of order greater than four could probably lead to a fixed point compatible with local Ward identities. In this paper, we focus on a rank-4 Abelian ${\ensuremath{\phi}}^{6}$-just-renormalizable tensorial group field theory and describe the renormalization group flow over the subtheory space where the Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion. We show that this model exhibits a nontrivial fixed point in this constrained subspace. Finally, the well-known asymptotically freedom of this model is highlighted.
Highlights
Group field theories (GFTs) are a type of nonlocal field theories defined on d copies of a group manifold
We focus on a rank-4 Abelian φ6-just-renormalizable tensorial group field theory and describe the renormalization group flow over the subtheory space where the Ward constraint is satisfied along the flow, by using an improved version of the effective vertex expansion
Some important developments were given in various directions to think about the question of quantum gravity, such as random geometry, canonical quantum gravity, and the covariant approach with spin-foam models, which have together converged toward the definition of GFTs [8,9,10,11,12,13,14,15,16,17]
Summary
Group field theories (GFTs) are a type of nonlocal field theories defined on d copies of a group manifold. The renormalization group (RG) allows one to build effective field theories from an elementary scale and to understand dynamical phase transitions in statistical and quantum systems (see [43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80], and references therein) In this respect, it was considered as a very important tool to understand the condensation mechanism in GFT. The aim of this paper is to determine if such a fixed point remains compatible with Ward identities, by describing the flow at all orders in the coupling and the parameter ε and by using the EVE approach to solve the nonperturbative RG equation. We improve our analysis by implementing the so-called Ward-constraint melonic flow and conclude that one new nontrivial fixed point can be found on this subspace
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