Group Field Theory in dimension four minus epsilon

TL;DR

This paper introduces an epsilon expansion for four-dimensional tensorial group field theories, revealing asymptotic freedom in one case and potential asymptotic safety in another, with implications for quantum gravity models.

Contribution

It develops an epsilon expansion framework for rank-3 tensorial group field theories, analyzing their renormalization properties and fixed points across dimensions.

Findings

The SU(2) x U(1) model is asymptotically free.

A non-trivial fixed point exists in four minus epsilon dimensions.

Potential for asymptotic safety in the SU(2) case is suggested.

Abstract

Building on an analogy with ordinary scalar field theories, an epsilon 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) X U(1) and the dimension three SU(2). In the first situation, there is a unique marginal 4-valent 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 6-valent coupling constants and one 4-valent super-renormalizable interaction spoils this interesting property. However, the existence of a non-trivial fixed point is established in dimension four minus epsilon, 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.