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Referring to recent works concerning the functional renormalization group fortensorial group fields theories [arXiv:1803.09902] and [arXiv:1809.00247], thispaper gives in-depth explanation for, the ambiguity around the search of fixedpoints in the Wetterich flow equation and the Ward-Takahashi identities. Weconsider the $U(1)$-tensors models and discuss the non-existence of phasetransition taking into account the Ward-identities as new constraint along theflow. We prove that the quartic melonic tensorial group fields theories withoutclosure constraint are devoid with the fixed points and therefore the phasetransitions.

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