  The nontrivial fixed point discovered for $\phi^4$-marginal couplings in
tensorial group field theories have been showed to be incompatible with
Ward-Takahashi identities. In this 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's identities. In this paper we
focus on a rank-4 Abelian $\phi^6$-just renormalizable tensorial group field
theory and describe the renormalization group flow over the sub-theory space
where Ward constraint is satisfied along the flow, by using an improved version
of the effective vertex expansion. We show that this model exhibit a nontrivial
fixed points in this constrained subspace. Finally, the well-known
asymptotically freedom of this model is highlighted.

High Energy Physics - Theory