($i$=1-3) are right-handed charged leptons, and $L_{i}$ ($i$=1-3) denote the lepton doublets and $Y_{\ell_{i}} $ are the Yukawa couplings for $\ell_{i}$. This interaction is reduced to be of Model II[@HHG] in the limit $\epsilon^{L,R}_{ij} \rightarrow 0$ with the discrete symmetry under $e_{R}^{i} \rightarrow + e_{R}^{i}$, $L_{i} \rightarrow + L_i$, $\Phi_{1} \rightarrow + \Phi_{1}$, and $\Phi_{2} \rightarrow - \Phi_{2}$. Nonzero values of $\epsilon_{ij}^{L,R}$ ($i\neq j$) yield the LFV Yukawa couplings after the diagonalization of the mass matrix. We note that in supersymmetric standard models, the Yukawa interaction for leptons is of Model II at the tree level, and $\epsilon_{ij}^{L,R}$ can be induced at the loop level due to slepton mixing[@MSSMRN; @BK; @DER]. For the quark sector, Model II Yukawa interactions are assumed to suppress FCNC, imposing the invariance under the transformation of $u_{R}^{i} \rightarrow - u_{R}^{i}$, $d_{R}^{i} \rightarrow + d_{R}^{i}$, $q_{L}^{i} \rightarrow + q_{L}^{i}$, $\Phi_{1} \rightarrow + \Phi_{1}$, and $\Phi_{2} \rightarrow -\Phi_{2}$.

The tau lepton associated LFV interactions in Eq. (\[eq:Llepton\]) can be reduced in the mass eigenbasis of each field to $$\begin{aligned}
-\mathcal{L}_{\tau{\rm LFV}}
&=&
\frac{m_{\tau}}{v \cos^{2}\beta}
 \left(
 \kappa^{L}_{3i} \overline{\tau} {\rm P}_{L} \ell_{i}
 +
 \kappa^{R}_{i3} \overline{\ell}_i {\rm P}_{L} \tau
 \right)
 \left\{
 \cos\left(\alpha - \beta \right) h
 +
 \sin\left(\alpha - \beta 