ground state, however this mixing is proportional to $\sin \delta $. This will ensure the $CP$ conservation at tree-level in the scalar sector. Indeed, in addition to the phase appearance in the scalar sector due to the deviation of Arg($A_{\lambda }$) from $-\theta _{0}$, the $CP$ violation effect could be seen in this model at one-loop [@CP1], and also through the dependence on $\theta $ and not $\delta $ can be seen in the superpartner masses (\[stop\]), (\[cha\]) and (\[xi\]).

The parameters $m_{1}^{2}$, $m_{2}^{2}$, and $m_{S}^{2}$ can be eliminated by taking ($\upsilon _{1},\upsilon _{2},\upsilon _{x}e^{i\theta _{0}}$) as a minimum of the effective potential. The vacuum stability of (\[Vv\]) requires the condition:$$\lambda ^{4}-\lambda ^{2}\left[ \left( g_{1}^{2}+g_{2}^{2}\right)
/2+2g^{\prime 2}\left( Q_{S}^{2}-Q_{1}Q_{2}\right) \right] +\frac{3}{4}%
g^{\prime 2}\left( g_{1}^{2}+g_{2}^{2}\right) Q_{S}^{2}>0,  \label{Vsta}$$where the $U(1)^{\prime }$ gauge invariance condition $Q_{1}+Q_{2}+Q_{S}=0$is taken into account.

In models that include singlets, like the present one, one needs to be careful about whether the minimum ($\upsilon _{1},\upsilon _{2},\upsilon
_{x}e^{i\theta _{0}}$) is the absolute one. This can be checked by comparing the effective potential value at this minimum with its value at the wrong vacuum, which could be, in singlet models, ($0,0,x$) rather than the origin ($0,0,0$). The wrong vacuum can be defined 