$h\to\mu\tau$ and muon g-2 in the alignment limit of two-Higgs-doublet model

We examine the $h\to \mu\tau$ and muon g-2 in the exact alignment limit of two-Higgs-doublet model. In this case, the couplings of the SM-like Higgs to the SM particles are the same as the Higgs couplings in the SM at the tree level, and the tree-level lepton-flavor-violating coupling $h\mu\tau$ is absent. We assume the lepton-flavor-violating $\mu\tau$ excess observed by CMS to be respectively from the other neutral Higgses, $H$ and $A$, which almost degenerates with the SM-like Higgs at the 125 GeV. After imposing the relevant theoretical constraints and experimental constraints from the precision electroweak data, $B$-meson decays, $\tau$ decays and Higgs searches, we find that the muon g-2 anomaly and $\mu\tau$ excess favor the small lepton Yukawa coupling and top Yukawa coupling of the non-SM-like Higgs around 125 GeV, and the lepton-flavor-violating coupling is sensitive to another heavy neutral Higgs mass. In addition, if the $\mu\tau$ excess is from $H$ around 125 GeV, the experimental data of the heavy Higgs decaying into $\mu\tau$ favor $m_A>230$ GeV for a relatively large $H\bar{t}t$ coupling.