\right) H
 -
 {\rm i} A
 \right\} \nonumber \\
& &+
 \frac{\sqrt{2} m_{\tau}}{v \cos^{2}\beta}
 \left(
 \kappa^{L}_{3i}
 \overline{\tau} {\rm P}_{L} \nu_{i} 
 +
 \kappa^{R}_{i3}
 \overline{\ell_{i}} {\rm P}_{L} \nu_{\tau}
 \right)  H^{-}
 +
 {\rm H.c.},
\label{eq:tauLFV}\end{aligned}$$ where ${\rm P}_{L}$ is the projection operator to the left-handed field, and $\ell_1$ and $\ell_2$ respectively represent $e$ and $\mu$. In general, the LFV parameters $\kappa^{L,R}_{ij}$ can be expressed in terms of $\epsilon^{L,R}_{ij}$ and $\tan\beta$.[^3] We here take these $\kappa^{L,R}_{ij}$ as effective couplings, and investigate their phenomenological consequences. We note that Eq. (\[eq:tauLFV\]) is exact in the limit of $m_{\ell_{i}}^{} \to 0$. The terms of $\kappa_{i3}^L$ and $\kappa_{3i}^R$ ($i=1, 2$) are proportional to $m_{\ell_i}^{}$, so that they decouple in this approximation.

We briefly discuss relationship between $\kappa^{L,R}_{ij}$ and new physics models beyond the cut-off scale of the [*effective*]{} THDM. When a new physics model is specified at the high energy scale, $\kappa^{L,R}_{ij}$ can be predicted as a function of the model parameters. For example, in the MSSM, slepton mixing can be a source of LFV. Notice that the induced LFV Higgs interactions do not necessarily decouple in the limit where the SUSY particles are sufficiently heavy, because their couplings only depend on the ratio of the SUSY parameters. 