Group field theory in dimension 4-ε

Building on an analogy with ordinary scalar field theories, an e-expansion for rank-3 tensorial group field theories with gauge invariance condition is introduced. This allows us 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 phi(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 phi(6) coupling constants and one phi(4) super-renormalizable interaction spoils this interesting property. However, the existence of a nontrivial fixed point is established in dimension 4 - epsilon, hence suggesting that the SU(2) theory might be asymptotically safe. To pave the way to future nonperturbative calculations, the present perturbative results are discussed in the framework of the effective average action.