mechanism introduces an extension of the standard Higgs potential in the commutative case, in fact the Higgs scalar field $\phi$ and the $\tilde{X}_A^\mu$, $\tilde{X}_B^\mu$ fields are present together in the potential. In this way the gauge field takes part in the definition of the vacuum. Another important property of the action, considering the $\tilde{X}_A^\mu$, $\tilde{X}_B^\mu$ as independent, is the invariance under the translations: $$\phi(x)\to\phi(x+a),\ X^\mu_A(x)\to X_A^\mu(x+a), \  X^\mu_B(x)\to X^\mu_B(x+a) , \ X^\mu_0(x)\to X^\mu_0(x+a)$$ which in other $\phi^4$-renormalizable theory is broken. Beside, the action is invariant under $U(1) \times U(1)$ transformations: $$\phi \to u_A\star\phi\star\overline{u_B}, \ \tilde{X}\to u_A\star \tilde{X}^\mu_A\star \overline{u_A}, \ \tilde{X}^\mu_B\to u_B\star\tilde{X}^\mu_B\star \overline{u_B} \label{gauge}$$ In field theory the ground state can be defined through the minimum of the action, the relevant part of the for the minimization is: $$\begin{aligned}
&&S(\mathcal{D}_A)=  \nonumber \\
&&+\frac{1}{(1+\Omega^2)^2} \int d^4z \Bigg\{\left(\frac{(1-\Omega^2)^2}{2}-\frac{(1-\Omega^2)^4}{6(1+\Omega^2)^2}\right)(F^A_{\mu\nu}\star F_A^{\mu\nu}+F^B_{\mu\nu}\star F_B^{\mu\nu})\nonumber \\
&&+\left(\bar{\phi}\star\phi+ \frac{4\Omega^2}{1+\Omega^2}\tilde{X}^A_\mu\star\tilde{X}_A^\mu-\frac{\chi_{-1}}{\chi_0}\right)^2\nonumber \\
&&+\left(\phi\star\bar{\phi}+ \frac{4\Omega^2}{1+\Omega^2}\tilde{X}^B_\mu\star\tilde{X}_B^\mu-\frac{\chi_{-1}}{\chi_0}\right)^2\nonumber \\
&& -2\left(\frac{4\Omega^2}{1+\Omega^2}\tilde{X}^\mu_0\star\tilde{X}^0_\mu-\frac{\chi_{-1}}{\chi_0} \right)^2+2(1+\Omega^2)D_\mu\phi\star\overline{D^\mu\phi}\Bigg\}(z)+\mathcal{O}(\chi_1) \label{faction1}\end{aligned}$$ Where we have omitted the constant part and we have rescaled the coefficient in front of the integral. Considering the fields $X_A^\mu$, $X_B^\mu$ as fields variables instead $A^\mu$, $B^\mu$ we can state that each terms of the action is semi-positive defined, so in order to find the minimum it is sufficient to 