$h,Z\to \ell_i \bar\ell_j$, $\Delta a_{\mu}$, $\tau\to (3\mu,\mu \gamma)$ in generic two-Higgs-doublet models

Inspired by a significance of $2.4\sigma$ in $h\to \mu \tau$ decay observed by CMS at $\sqrt{s} = 8$ TeV, we investigate the Higgs lepton flavor violating effects in the generic two-Higgs-doublet model (GTHDM), where the lepton flavor changing neutral currents are induced at tree level and arisen from Yukawa sector. We revisit the constraints for GTHDM by considering theoretical requirements, precision measurements of $\delta \rho$ and oblique parameters $S$, $T$, $U$ and Higgs measurements. The bounds from Higgs data now play the major role. With the values of parameters that simultaneously satisfy the Higgs bounds and the excess of Higgs coupling to $\mu$-$\tau$ at CMS, we find that the tree-level $\tau\to 3\mu$ and loop-induced $\tau\to \mu \gamma$ could be consistent with current experimental upper limits; the discrepancy in muon $g-2$ between experiment and standard model prediction could be solved; and an interesting relation between muon $g-2$ and branching ratio (BR) for $\mu\to e \gamma$ is found. The GTHM results that the ratio $BR(h\to e\tau)/BR(h\to \mu \tau)$ should be smaller than $10^{-4}$ in order of magnitude. Additionally, we also study the rare decay $Z\to \mu \tau$ and get $BR(Z\to \mu\tau)< 10^{-6}$.