in the $\tan\beta$-$m_{\Phi}^{}$ plane ($m_\Phi^{}\equiv m_H^{}=m_A^{}=m_{H^\pm}^{}$) for $m_h^{}=120$ GeV and $\sin(\alpha-\beta)=-1$ with (a) $M=m_\Phi^{}$, (b) $M=m_\Phi^{}/\sqrt{2}$, and (c) $M=0$. The dark (light) shaded area indicates the excluded region by the theoretical requirements of vacuum stability and perturbative unitarity with $\xi=1$ ($\xi=1/2$). []{data-label="Fig:Br-Htm"}](BrHzero.eps "fig:"){width="4.52cm"}]{} (2,0.5)[(c)]{} (4,0.7)[$\tan\beta$]{}

In Figs. \[Fig:Br-Htm\]-(a), \[Fig:Br-Htm\]-(b) and \[Fig:Br-Htm\]-(c), contour plots of ${\rm Br}(H \to \tau^\pm\mu^\mp)_{\rm max}$, the upper limit of ${\rm Br}(H \to \tau^\pm\mu^\mp)$ under the rare tau decay results, are shown in the $\tan\beta$-$m_\Phi^{}$ plane for $M=m_\Phi^{}$, $m_\Phi^{}/\sqrt{2}$ and $0$, respectively. As expected, the contours are insensitive to the values of $M$, and approximately the same in Figs. \[Fig:Br-Htm\]-(a), \[Fig:Br-Htm\]-(b) and \[Fig:Br-Htm\]-(c). It is shown that ${\rm Br}(H \to \tau^\pm\mu^\mp)_{\rm max}$ can be larger than $10^{-3}$ except for large $\tan\beta$ values with relatively small $m_\Phi^{}$. Therefore, it turns out to be no substantial upper limit on the ${\rm Br}(H \to \tau^\pm\mu^\mp)$ in the relatively low $\tan\beta$ region ($\tan\beta \lesssim 20$) from the LFV rare tau decay results. When $M$ is smaller than $m_\Phi$, where the heavier Higgs boson partially receive their masses from the vacuum expectation value, the allowed parameter region is strongly constrained by the requirements of vacuum stability and perturbative unitarity. In particular, for 