  In this paper we consider the complete momentum-independent quartic order
truncation for the effective average action of a real Abelian rank 3 tensorial
group field theory. This complete truncation includes non-melonic as well as
double-trace interactions. In the usual functional renormalization group
perspective, the inclusion of more operators that belong to the underlying
theory space corresponds to an improvement of the truncation of the effective
average action. We show that the inclusion of non-melonic and double-trace
operators in the truncation brings subtleties. In particular, we discuss the
assignment of scaling dimensions to the non-melonic sector and how the
inclusion of double-trace operators considerably changes the results for
critical exponents when they are not included. We argue that this is not a
particular problem of the present model by comparing the results with a pure
tensor model. We discuss how these issues should be investigated in future
work.
