Group field theory in dimension4−ϵ

Abstract

Building on an analogy with ordinary scalar field theories, an ε-expansion for rank-3 tensorial group field theories with gauge invariance condition is introduced. This allows to continuously interpolate between the dimension four group SU(2) × U(1) and the dimension three SU(2). In the first situation, there is a unique marginal ϕ 4 coupling constant, but in contrast to ordinary scalar field theory this model is asymptotically free. In the SU(2) case, the presence of two marginally relevant ϕ 6 coupling constants and one ϕ 4 super-renormalizable interaction spoils this interesting property. However, the existence of a non-trivial fixed point is established in dimension 4 − ε, hence suggesting that the SU(2) theory might be asymptotically safe. To pave the way to future non-perturbative calculations, the present perturbative results are discussed in the framework of the effective average action.

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

“…However, we have to keep in mind that = 0 increase the strength of the inessential couplings; as a result, if such a formal fixed point survive in the limit → 1/3, when the φ 6 couplings become marginal, it may be viewed as an UV attractive fixed point ensuring UV completion of the RG flow. The same phenomena has been pointed out in [72]. However, the aim of this paper is to determine if such a fixed point remains compatible with Ward identities, by describing the flow at all order in the coupling and the parameter and by using the EVE approach to solve the nonperturbative RG equation.…”

“…The same kind of equations has been obtained for all φ 6 -models studied in the literature, and extended discussions may be found in [64]- [66], [69], [72]. A qualitative phase portrait may be easily deduced from the system (102).…”

“…The motivation to be interested in such a model is the following: Note that the existence of a non-Gaussian fixed point can be understood from the following heuristic argument. As showed in [72], the TGFT Feynman amplitudes can be analytically continued with respect to the group dimension D: U(1) → U(1) D . The power counting then indicate that there are two just-renormalizable models in rank 4: The φ 6 melonic model that we will describe in section 2 and which is asymptotically free; and the formal quartic melonic model in dimension D = 4/3, which is also asymptotically free.…”

Towards background independent quantum gravity with tensor models

“…Plugging the value for [ḡ 2 4,2 ] and [ḡ 2,i 6,1 ] (which can be completely fixed as well) in (43) and (44) gives [ḡ 3 6,3 ] = −6r. This shows that it is not possible to assign canonical dimensions for such multitrace interactions lower than the upper bounds as a strategy to decouple them from the other beta functions.…”

Tensor Field Theory

“…In [69] the ε expansion of Wilson-Fisher was adapted to this rank-three SU(2) TGFT and a non-trivial fixed point was found in dimension 4 − ε, suggesting that this theory might be asymptotically safe.…”