on $|\kappa_{31}|^2$. Therefore, $|\kappa_{31}^{\rm max}|^2$ is determined from one of $\tau \to e \eta$, $\tau \to e \gamma$ and $\tau^- \to e^-K^+K^-$ depending on parameter regions.

Lepton flavor violating Higgs boson decays {#Sec:LFV-Hdecay}
==========================================

As shown in the previous section, the LFV Yukawa couplings can be tested only in the large $\tan\beta$ region by searching for rare tau decays. In order to cover the region unconstrained by rare tau decay results, we here consider LFV via the decay of the neutral Higgs bosons; i.e., $\phi^{0} \rightarrow \tau^\pm \ell_i^\mp$ ($\phi^{0}=h,H$ and $A$). Branching ratios for these decays are calculated[@Rossi; @Assamagan; @kot; @Osaka; @Herrero] to be $$\begin{aligned}
{\rm Br}(h\rightarrow \tau^{-} \ell^{+}_i) =&
 \frac{1}{16\pi} 
 \frac{m_{\tau}^{2} \cos^{2} \left(\alpha - \beta \right)}
 {v^{2} \cos^{4}\beta}
 \left| \kappa_{3i} \right|^{2}
 \frac{m_{h}^{} \left(1- \frac{m_{\tau}^{2}}{m_{h}^{2}}\right)^{2}}
 {\Gamma(h\rightarrow\text{all})}, 
\label{eq:Brhtm}\\
{\rm Br}(H\rightarrow \tau^{-} \ell^{+}_i) =&
 \frac{1}{16\pi} 
 \frac{m_{\tau}^{2} \sin^{2} \left(\alpha - \beta \right)}
 {v^{2} \cos^{4}\beta}
 \left| \kappa_{3i} \right|^{2}
 \frac{m_{H}^{} \left(1- \frac{m_{\tau}^{2}}{m_{H}^{2}}\right)^{2}}
 {\Gamma(H\rightarrow\text{all})},
\label{eq:BrhtmH}\\
{\rm Br}(A\rightarrow \tau^{-} \ell^{+}_i) =&
 \frac{1}{16\pi} 
 \frac{m_{\tau}^{2}}
 {v^{2} \cos^{4}\beta}
 \left| \kappa_{3i} \right|^{2}
 \frac{m_{A}^{} \left(1- \frac{m_{\tau}^{2}}{m_{A}^{2}}\right)^{2}}
 {\Gamma(A\rightarrow\text{all})},
\label{eq:BrhtmA}\end{aligned}$$ where $\Gamma (\phi^{0} \rightarrow \text{all}) $ is the total width for corresponding neutral Higgs boson $\phi^{0}$. We here neglect terms of $\mathcal{O}(m_{\ell_i}^{2}/m_{\phi^{0}}^{2})$. Branching ratios for $\phi^0 \to \tau^+\ell_i^-$ coincide with those for $\phi^0 \to 