as the minimum of the effective potential in the direction where all the $SU(2)$ doublets vanish. At tree-level, it is given by$$\left\vert x\right\vert ^{2}=\upsilon _{x}^{2}+\left[ Q_{1}/Q_{S}+\lambda
^{2}/g^{\prime 2}Q_{S}^{2}\right] \upsilon _{1}^{2}+\left[
Q_{2}/Q_{S}+\lambda ^{2}/g^{\prime 2}Q_{S}^{2}\right] \upsilon _{2}^{2}+%
\frac{\sqrt{2}}{g^{\prime 2}Q_{S}^{2}}\frac{A_{\lambda }}{\upsilon _{x}}%
\upsilon _{1}\upsilon _{2}.  \label{X0}$$The value of $x$ in (\[X0\]) should be relaxed by including the one-loop corrections given below in Eq. (\[V1\]). In the case where the right hand side of (\[X0\]) is positive, the wrong vacuum ($0,0,x$) does exist, otherwise it should be ($0,0,0$). In both cases, ($\upsilon _{1},\upsilon
_{2},\upsilon _{x}e^{i\theta _{0}}$) must be the absolute minimum for the effective potential. This condition imposes additional constraints on the parameters of the model.

The one-loop effective potential at zero temperature is given in the $%
\overline{DR}$ scheme by [@NaLa]$$V^{1-l}\left( h_{1},h_{2},S\right) =V\left( h_{1},h_{2},S\right) +\sum_{i}%
\frac{n_{i}m_{i}^{4}}{64\pi ^{2}}\left( \log \frac{m_{i}^{2}}{\Lambda ^{2}}-%
\frac{3}{2}\right) ,  \label{V1}$$where $m_{i}\left( h_{1},h_{2},S\right) $ are the field-dependant masses, which are given in Appendix A, $\Lambda $ is the renormalization scale, which is taken to be $\Lambda =\upsilon =246~GeV,$ and $n_{i}$ are the fields multiplicities: $n_{W}=6$, $n_{Z}=n_{Z^{\prime }}=3$, $%
n_{h_{1}^{0}}=n_{h_{2}^{0}}=n_{S}=n_{A^{0}}=n_{G}=1$, $n_{t}=-12$, $n_{%
\tilde{t}_{L}}=n_{\tilde{t}_{R}}=6$, $n_{\tilde{\chi}}=-2$, $n_{\tilde{C}}=-4
$, where $A^{0}$, $G$, $\tilde{t}_{L,R},$ $\tilde{\chi}$ and $\tilde{C}$ denote the $CP$-odd Higgs, Goldstone boson, left- and right-handed 