be consider a non-perturbative approach using a discretized matrix model of the action obtained using a Moyal base. In this setting the action reduces to $$\begin{aligned}
&&S(\mathcal{D}_A)=  \nonumber \\
&&+\frac{1}{(1+\Omega^2)^2} \int d^4z \Bigg\{\left(\frac{(1-\Omega^2)^2}{2}-\frac{(1-\Omega^2)^4}{6(1+\Omega^2)^2}\right)(F^A_{\mu\nu}\star F_A^{\mu\nu}+F^B_{\mu\nu}\star F_B^{\mu\nu})\nonumber \\
&&+\left(\bar{\phi}\star\phi+\frac{4\Omega^2}{1+\Omega^2}\tilde{X}^A_\mu\star\tilde{X}_A^\mu-\frac{\chi_{-1}}{\chi_0}\right)^2 \nonumber \\
&&+\left(\phi\star\bar{\phi}+ \frac{4\Omega^2}{1+\Omega^2}\tilde{X}^B_\mu\star\tilde{X}_B^\mu-\frac{\chi_{-1}}{\chi_0}\right)^2\nonumber \\
&&+ 2(1+\Omega^2)D_\mu\phi\star\overline{D^\mu\phi}\Bigg\}(z)+\mathcal{O}(\chi_1) \label{S}\end{aligned}$$ The omitted factor for the finite matrix model of size $N$ becomes constant so can be ignored. The minimum is obtained like before and formally is in this case the identity, of course, belongs to the matrix space. It is interesting to notice that the vacuum of the finite model, due to the Higgs field, is no longer invariant under the transformations , but is invariant under a subgroup of $U(N)\times U(N)$: $$u_A=\overline{u_B} \longrightarrow   u_A\star \textbf{I} \star\overline{u_B}=u_A\star \overline{u_A}=\textbf{I}$$ Having discretized the model will be performed a Monte Carlo simulation studying some statistical quantity such the energy density, specific heat, varying the parameters $\Omega, \frac{\chi_{-1}}{\chi_0}, \alpha$ and gathering some informations on the various contributions of the fields to the action. The simulations are quite cumbersome due to the complexity of the action and to the number of independent matrix to handle.

Discretization of the action
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The first step across the numerical analysis is to apply 