by the $\star$-product in the position space can induce us to fear some problems for the renormalization. In fact it was discovered [@ncft3] that after computing the Feynman rules for such theory and deriving the loop amplitude we find that the non-local interaction terms in the action induce an oscillatory factors (involving loop momenta) in the Feynman integrals. Studying the structure of Feynman diagrams for the action, Filk [@one-loop-nc3] has showed that are present two types of loop diagram the planar and non-planar diagrams. The planar diagrams do not have this oscillatory factors coming from the non-local interaction terms, and therefore the corresponding integrals are the same as in usual quantum field theory. On the other hand, all non-planar diagrams have the oscillatory factors involving loop momenta. Due to this terms the renormalization of quantum field theories on the non-commutative $\mathbb{R}^n$ is not achieved and these models show a phenomenon called UV/IR-mixing [@UV-mix]. Chepelev and Roiban [@UV-mix1] analyses UV/IR-mixing to all orders, the conclusion of the power-counting theorem is that field theories on non-commutative $\mathbb{R}^n$ are not renormalizable if the divergence of their commutative counterparts are higher than logarithmic.

A great step towards the non-commutative field theory was made when 