values of $\tan\beta$ ($\tan\beta \lesssim 30$), where rare tau decay results cannot give substantial upper limit on $|\kappa_{32}|^{2}$, ${\rm Br}(h \rightarrow \tau^{\pm}\mu^{\mp})_{\text{max}}$ can be sufficiently large. We find that the possible maximal values of the branching ratio can be greater than $\mathcal{O}(10^{-3})$ in a wide rage of the theoretically allowed region.

For relatively lower $\tan\beta$ values, the experimental upper limits on $|\kappa_{32}|^2$ from rare tau decays are weaker, and $\text{Br}(h\rightarrow\tau^{\pm} \mu^{\mp})$ can be sufficiently large ($\gtrsim \mathcal{O}(10^{-3})$ for $m_h \sim 120$ GeV). It is expected that a sufficient number of such light $h$ can be produced at future colliders such as CERN LHC, currently planned International Linear Collider (ILC) and CERN CLIC. It has been pointed out that the decay process $h \to \tau^\pm\mu^\mp$ can easily be detected at ILC with the luminosity of 1 ab$^{-1}$, when $m_h \sim 120$ GeV and ${\rm Br}(h\rightarrow\tau^\pm\mu^\mp) \gtrsim \mathcal{O}(10^{-3})$ via the Higgsstrahlung process by using the recoil momentum of $Z$ boson[@Osaka]. Therefore, the LFV search via the decay $h \to \tau^\pm\mu^\mp$ at ILC is complementary to that via rare tau decays at (super) B factories, and the both cover a wide region of the parameter space of the lepton flavor violating THDM.

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