expressed as $$\begin{aligned}
-\mathcal{L}_{\text{Higgs}} 
&=&
m_{1}^{2} \left| \Phi_{1} \right|^{2}
+
m_{2}^{2} \left| \Phi_{2} \right|^{2}
-
\left( m_{3}^{2} \Phi_{1}^{\dagger} \Phi_{2} + {\rm H.c.} \right)
+
\frac{\lambda_{1}}{2} \left|\Phi_{1}\right|^{4}
+
\frac{\lambda_{2}}{2} \left|\Phi_{2}\right|^{4} 
+
\lambda_{3} \left|\Phi_{1}\right|^{2} \left|\Phi_{2}\right|^{2}
+
\lambda_{4} \left|\Phi_{1}^{\dagger} \Phi_{2} \right|^{2}\nonumber \\
&&
+
\left\{
 \frac{\lambda_{5}}{2}
 \left( \Phi_{1}^{\dagger} \Phi_{2} \right)^{2} + {\rm H.c.}
\right\} + \left(
 \lambda_{6} \left|\Phi_{1}\right|^{2}
 \Phi_{1}^{\dagger}
 \Phi_{2} +{\rm H.c.} \right) 
 + \left(
 \lambda_{7} \left|\Phi_{2}\right|^{2}
 \Phi_{1}^{\dagger}
 \Phi_{2} +{\rm H.c.} \right), 
\label{eq:LHiggs}\end{aligned}$$ where $\Phi_{1}$ and $\Phi_{2}$ are the scalar iso-doublets with hypercharge $1/2$. In Eq. (\[eq:LHiggs\]), $m_{3}^{2}$, $\lambda_{5}$, $\lambda_{6}$ and $\lambda_{7}$ are complex in general. We here assume that all the parameters $m_{1-3}^{2}$ and $\lambda_{1-7}$ are real. The terms of $m_{3}^{2}$, $\lambda_{6}$ and $\lambda_{7}$ break the discrete symmetry explicitly. As we consider the model in which the discrete symmetry is explicitly broken only in the leptonic Yukawa interaction, we set the hard-breaking coupling constants to be zero in the Higgs potential; i.e., $\lambda_{6} = \lambda_{7} =0$[^2], and retain only the soft-breaking mass parameter $m_3^2$.

There are eight degrees of freedom in the two Higgs doublet fields. Three of them are absorbed by the weak gauge bosons via the Higgs mechanism, and remaining five are physical states. After the diagonalization of the mass 