$s$-wave amplitude for the process of $\phi_{1} \phi_{2} \rightarrow \phi_{3}
\phi_{4}$ with $\phi_{a}$ ($a$=1-4) denoting Higgs bosons and longitudinal components of weak gauge bosons. We take the criterion $\xi$ to be 1 (and also 1/2 for comparison). The experimental constraints are provided by the LEP precision data[@LEP], the $b \rightarrow s \gamma$ results[@bsg-ex], and the direct search results for the Higgs bosons[@LEP]. The LEP precision data provide the strong constraint on the new physics structure via the gauge-boson two-point functions. The constraint on $\rho$ parameter indicates that the Higgs sector is approximately custodial $SU(2)$ symmetric. This requirement is satisfied when (i) $m_{H^{\pm}}^{} \simeq m_{A}^{}$, (ii) $m_{H^{\pm}}^{} \simeq m_{H}^{}$ with $\sin^{2} (\alpha - \beta)\simeq 1$, and (iii) $m_{H^{\pm}}^{} \simeq m_{h}^{}$ with $\cos^{2} (\alpha - \beta) \simeq 1$. It is known that in Model II, the $b \rightarrow s \gamma $ result gives the lower bound on the charged Higgs boson mass. We here take into account this bound by requiring $m_{H^{\pm}}^{} \gtrsim 350$ GeV.

Next, we consider the Yukawa interaction for charged leptons as $$\begin{aligned}
-\mathcal{L}_{\text{lepton}}=
\overline{\ell}_{R i} 
 \left\{
 Y_{\ell_{i}} \delta_{ij} \Phi_{1}  
 +
 \left(
 Y_{\ell_{i}} \epsilon^{L}_{ij} 
 +
 \epsilon^{R}_{ij} Y_{\ell_{j}}
 \right) \Phi_{2} 
 \right\}
 \cdot
 L_{j} + {\rm H.c.},
\label{eq:Llepton}\end{aligned}$$ where $\ell_{R i}$ 