Ward identity violation for melonic $T^4$-truncation

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.