H.Grosse and R.Wulkenhaar [@phi4-non], found a non-commutative $\varphi^4$-theory renormalizable action which develops additional marginal coupling, corresponding to an harmonic oscillator potential for the real-valued free field $\varphi$ on $\mathbb{R}^4_\Theta$ : $$S[\varphi]=\int d^4x \left(\frac{1}{2} \varphi \star (-\Delta+\Omega^2 \tilde{x}^2 + \mu^2)\star\varphi + \frac{\lambda}{4}\varphi \star \varphi \star \varphi \star \varphi\right)(x) \nonumber$$ Where $x = 2\Theta^{-1} \cdot x, \ \lambda \in \mathbb{R} \ \Omega \in [0,1]$, and $\mu$ is a real parameter. Using the Moyal matrix base, which turns the $\star$-product into a standard (infinite) matrix product, H.Grosse and R.Wulkenhaar were able to prove the perturbative renormalizability of the theory [@matrix-renorm]. Afterward, R.Wulkenhaar et al. [@nc-renorm] found an alternative simpler normalization proof using multi-scale analysis in matrix base, showing the equivalence of various renormalization schemes. A last, but useful, renormalization proof was formulated using Symanzik type hyperbolic polynomials [@nc-renorm2].

The non-commutative model treated in this work is a sort of extension, via spectral action principle, of the scalar W-G model, in which we are interested to formulate a Yang-Mills theory in renormalizable way on Moyal space. We can expect that usual Yang-Mills theory on Moyal space without modifications of the action by something similar to an oscillator potential, to be not renormalizable [@UV-mix]. Additionally, 