\\
& +\left[ \lambda ^{2}-\left( g_{1}^{2}+g_{2}^{2}\right) /4+g^{\prime
2}Q_{1}Q_{2}\right] h_{1}^{2}h_{2}^{2}/4+\left[ \lambda ^{2}+g^{\prime
2}Q_{1}Q_{S}\right] h_{1}^{2}S^{2}/4  \notag \\
& +\left[ \lambda ^{2}+g^{\prime 2}Q_{2}Q_{S}\right]
h_{2}^{2}S^{2}/4+m_{1}^{2}h_{1}^{2}/2+m_{2}^{2}h_{2}^{2}/2+m_{S}^{2}S^{2}/2+%
\{A_{\lambda }Sh_{1}h_{2}e^{i\delta }/2\sqrt{2}+h.c\}.  \label{Vv}\end{aligned}$$The neutral scalar sector in this model contains three $CP$-even scalars and one $CP$-odd scalar. The tadpole minimization conditions at the ground state along the $CP$-odd scalar forces the relative $CP$ phase $\theta $, to be canceled by the phase of the parameter $A_{\lambda }$ in (\[tree\]), which must be taken as $A_{\lambda }=\pm \left\vert A_{\lambda }\right\vert
e^{-i\theta _{0}}$. This makes the ground state at tree-level independent of this relative phase, and the phase $\delta $ in (\[Vv\]) is just $%
\delta =\theta -\theta _{0}$. It is clear that the dependence of (\[Vv\]) on the phases disappears at the ground state. However, when considering the one-loop corrections in the tadpole minimization condition along the $CP$-odd scalar, the tree-level phase cancelation is no longer valid at one-loop, and the argument of $A_{\lambda }$ differs slightly than $-\theta _{0}$ (or $-\theta _{0}+\pi $).

The structure of the tree-level potential (\[tree\]) implies the presence of a mixing between the submatrices ($CP$-odd scalars, Goldstone bosons, and $CP$-even scalars) at tree-level for any values of the scalar fields except at the 