previous fields into we get a positive action with minimum in zero: $$\begin{aligned}
&S(\psi,Y_A,Y_B)&= +\frac{1}{(1+\Omega^2)^2}\int d^4z\Bigg\{D\Big(\left[Y_{A\mu},Y_{A\nu}\right]_\star\left[Y_{A}^\mu,Y_{A}^\nu\right]_\star 
+ \left[Y_{B\mu},Y_{B\nu}\right]_\star\left[Y_{B}^\mu,Y_{B}^\nu\right]_\star\Big) \nonumber \\
& &+ \left(\psi\star\bar{\psi}+\mu\cos\alpha(\psi+\bar{\psi})+ CY_A^\mu\star Y_{A\mu} +\mu \textbf{I}^\mu Y_{A\mu}\sin\alpha \right)^2  \nonumber \\
& &+ \left(\bar{\psi}\star\psi +\mu\cos\alpha(\psi+\bar{\psi})+ CY_B^\mu\star Y_{B\mu} +\mu \textbf{I}^\mu Y_{B\mu}\sin\alpha  \right)^2 \nonumber \\ \nonumber \\
& &+2(1+\Omega^2)\left((Y_{B\mu}-Y_{A\mu})\mu\cos\alpha +\psi\star Y_{B\mu}-Y_{A\mu}\star\psi\right)\nonumber \\
& &\star\left((Y_{A}^\mu-Y_{B}^\mu)\mu\cos\alpha+\bar{\psi}\star Y_{A}^\mu -Y_{B}^\mu\star\bar{\psi}\right)\Bigg\}(z)  +\mathcal{O}(\chi_1)
\label{Sf} \end{aligned}$$ Where for simplicity we put: $$\begin{aligned}
C=\frac{1+\Omega^2}{4\Omega^2},& D=\frac{\left(1-\Omega^2\right)^2}{2}-\frac{\left(1-\Omega^2\right)^4}{6\left(1+\Omega^2\right)^2} , & \frac{\chi_{-1}}{\chi_0}=\mu^2\end{aligned}$$

Discretization by Moyal base
----------------------------

The following treatment is mainly taken from [@moyal-triple; @bondia2] as introduction to the Moyal base which will be used later. We can define on the algebra $\mathbb{R}_\Theta^2$ a natural basis of eigenfunctions $f_{mn}$ of the harmonic oscillator, where $m, n \in \mathbb{N}$. This base satisfy the $\star$-multiplication rule: $$(f_{mn} \star f_{kl} )(x) = \delta_{nk} f_{ml} (x) \label{star-rule}$$ and this useful property: $$\begin{aligned}
\int d^2 x f_{mn}(x)&=& \delta_{mn} \int d^2x  f_0 =2\pi\theta\delta_{mn}\end{aligned}$$ The previous multiplication rule associates the $\star$-product between $f_{ml}$ with the ordinary matrix product: In this base we can write any elements of $ \mathbb{R}_\Theta^2$ but we have to require the rapid decay [@bondia2] of the sequences of coefficients $\{a_{mn}\}$: $$\sum^\infty_{m,n=0} a_{mn} f_{mn}(x) \in \mathbb{R}_\Theta^2, \ \textrm{if}, \ \sum^\infty_{m,n=0} \left((2m+1)^{2k} (2n+1)^{2k} |a_{mn}|^2 \right)^\frac{1}{2} < \infty \ 