  This manuscript aims at giving our new advance on the functional
renormalization group applied to tensorial group field theory. It is based on a
series of our three papers [arXiv:1803.09902], [arXiv:1809.00247] and
[arXiv:1809.06081]. We consider the polynomial Abelian $U(1)^d$ models without
closure constraint, especially we discuss the case of quartic melonic
interaction. We present a new approach to solve the exact Wetterich flow
equation called the effective vertex expansion method, and investigate the
resulting flow equations, especially about the existence of non-Gaussian fixed
points for their connection with phase transitions. To complete this method, we
consider a non-trivial constraint arising from the Ward-Takahashi identities,
and discuss the disappearance of the global non-trivial fixed points taking
into account this constraint. Finally, we argue in favor of an alternative
scenario involving a phase transition into the reduced phase space given by the
Ward constraint.
