  The Ward-Takahashi identities are considered as the generalization of the
Noether currents available to quantum field theory and include quantum
fluctuation effects. Usually, they take the form of relations between
correlation functions, which ultimately correspond to the relation between
coupling constants of the theory. For this reason, they play a central role in
the construction of renormalized theory, providing strong relations between
counter-terms. Since last years, they have been intensively considered in the
construction of approximate solutions for nonperturbative renormalization group
of tensorial group field theories. The construction of these identities is
based on the formal invariance of the partition function under a unitary
transformation, and Ward's identities result from a first-order expansion
around the identity. Due to the group structure of the transformation under
consideration, it is expected that a first-order expansion is indeed
sufficient. We show in this article that this does not seem to be the case for
a complex tensor theory model, with a kinetic term involving a Laplacian.
