\tau^-\ell_i^+$ given in Eqs. (\[eq:Brhtm\]), (\[eq:BrhtmH\]) and (\[eq:BrhtmA\]). In the following, we concentrate on the decays into a $\tau$-$\mu$ pair. We take the values of the SM parameters as $\alpha_{\text{em}}=0.007297$, $G_{F}=1.166 \times 10^{-5}$ $\text{GeV}^{-2}$, $m_{Z}^{}=91.19$ GeV, $m_{\tau} = 1.777$ GeV, $m_{\mu}=0.1057$ GeV, $m_{b}=4.1$ GeV, $m_{t}=174.3$ GeV, $m_{c}=1.15$ GeV, $m_{s}=0.120$ GeV.

A search for the LFV decays $h\rightarrow \tau^\pm\ell_i^\mp$ can give important information for extended Higgs sectors and thus for the structure of new physics, even when only $h$ is found and any other direct signals for the extended Higgs sector are not obtained by experiments. We here evaluate the possible maximal value of the branching ratio ${\rm Br}(h \rightarrow \tau^{-} \mu^{+})_{\text{max}}$ under the results of the rare tau decay search, by inserting $|\kappa_{32}^{\text{max}}|^{2}$ of Eq. (\[eq:kappaSq-max\]) into the $|\kappa_{32}|^{2}$ in Eq. (\[eq:Brhtm\]).

(7.8,6) (0,1)[![Contour plots of ${\rm Br}(h \rightarrow \tau^{\pm} \mu^{\mp})_{\text{max}}$, the possible maximal values for the branching ratio under the tau rare decay results, are shown (a) in the $\tan\beta$-$m_{A}$ plane and (b) in the $\tan\beta$-$\sin(\alpha-\beta)$ plane. The parameters are taken as the same as Figs. \[Fig:allowed-kappa\]-(a) and \[Fig:allowed-kappa\]-(b), respectively. The dark (light) shaded area indicates the excluded region by the theoretical requirements of vacuum stability and perturbative unitarity 