Pedagogical comments about nonperturbative Ward-constrained melonic renormalization group flow

This paper, in addition to our recent works, intends to explore the behavior of the Wetterich flow equations in the portion of the theory space spanned by non-branching melons constrained with Ward-identities. We focus on a rank-5 just-renormalizable tensorial group field theory and consider a non-trivial extension of the local potential approximation namely effective vertex expansion for just-renormalizable quartic melonic interactions, disregarding effects coming from disconnected interactions. Investigating the dynamically constrained flow, we show explicitly that results weakly rely on the number of quartic interactions involved in the classical action. In particular, the predictions for the fully connected model are essentially the same as for the single colored model. Finally, closing the flow equations using Ward identities without additional assumptions to compute integrals involved in the effective vertex expansion, we do not find reliable fixed point in the unconstrained theory space connected with the Gaussian region.