Group Field Theory in dimension four minus epsilon

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.