  Tensor models admit the large $N$ limit, dominated by the graphs called
melons. The melons are characterized by the Gurau number $\varpi=0$ and the
amplitude of the Feynman graphs are proportional to $N^{-\varpi}$. Other
leading order contributions i.e. $\varpi> 0$ called pseudo-melons can be taken
into account in the renormalization program. The following paper deals with the
renormalization group for a $U(1)$-tensorial group field theory model taking
into account these two sectors (melon and pseudo-melon). It generalizes a
recent work (Lahoche and Ousmane Samary, arXiv:1803.09902), in which only the
melonic sector has been studied. Using the power counting theorem the divergent
graphs of the model are identified. Also, the effective vertex expansion is
used to generate in detail the combinatorial analysis of these two leading
order sectors. We obtained the structure equations, which help to improve the
truncation in the Wetterich equation. The set of Ward-Takahashi identities is
derived and their compatibility along the flow provides a nontrivial constraint
in the approximation schemes. In the symmetric phase, the Wetterich flow
equation is given, and the numerical solution is studied.
