$\tau$- and $\mu$-physics in a general two Higgs doublet model with $\mu-\tau$ flavor violation

Motivated by the recent CMS excess in a flavor violating Higgs decay $h \rightarrow \mu \tau$ as well as the anomaly of muon anomalous magnetic moment (muon g-2), we consider a scenario where both the excess in $h \rightarrow \mu \tau$ and the anomaly of muon g-2 are explained by the $\mu-\tau$ flavor violation in a general two Higgs doublet model. We study various processes involving $\mu$ and $\tau$, and then discuss the typical predictions and constraints in this scenario. Especially, we find that the prediction of $\tau \rightarrow \mu \gamma$ can be within the reach of the Belle II experiment. We also show that the lepton non-universality between $\tau \rightarrow \mu \nu \bar{\nu}$ and $\tau \rightarrow e \nu \bar{\nu}$ can be sizable, and hence the analysis of the current Belle data and the future experimental improvement would have an impact on this model. Besides, processes such as $\tau \rightarrow \mu l^+ l^-~(l=e,~\mu)$, $\tau \rightarrow \mu \eta$, $\mu \rightarrow e \gamma$, $\mu \rightarrow 3e$, and muon EDM can be accessible, depending on the unknown Yukawa couplings. On the other hand, the processes like $\tau \rightarrow e \gamma$ and $\tau \rightarrow e l^+ l^-~ (l=e,~\mu)$ could not be sizable to observe because of the current strong constraints on the $e-\mu$ and $e-\tau$ flavor violations. Then we also conclude that contrary to $h \rightarrow \mu \tau$ decay mode, the lepton flavor violating Higgs boson decay modes $h\rightarrow e \mu$ and $h\rightarrow e\tau$ are strongly suppressed, and hence it will be difficult to observe these modes at the LHC experiment.