  Referring to recent works concerning the functional renormalization group for
tensorial group fields theories [arXiv:1803.09902] and [arXiv:1809.00247], this
paper gives in-depth explanation for, the ambiguity around the search of fixed
points in the Wetterich flow equation and the Ward-Takahashi identities. We
consider the $U(1)$-tensors models and discuss the non-existence of phase
transition taking into account the Ward-identities as new constraint along the
flow. We prove that the quartic melonic tensorial group fields theories without
closure constraint are devoid with the fixed points and therefore the phase
transitions.
