Higgs-$\mu$-$\tau$ Coupling at High and Low Energy Colliders

There is no tree-level flavor changing neutral current (FCNC) in the standard model (SM) which contains only one Higgs doublet. If more Higgs doublets are introduced for various reasons, the tree level FCNC would be inevitable except extra symmetry was imposed. Therefore FCNC processes are the excellent probes for the physics beyond the SM (BSM). In this paper, we studied the lepton flavor violated (LFV) decay processes $h\rightarrow\mu\tau$ and $\tau\rightarrow\mu\gamma$ induced by Higgs-$\mu$-$\tau$ vertex. For $\tau\rightarrow\mu\gamma$, its branching ratio is also related to the $ht\bar{t}$, $h\tau^+\tau^-$ and $hW^+W^-$ vertices. We categorized the BSM into two scenarios for the Higgs coupling strengths near or away from SM. For the latter scenario, we took the spontaneously broken two Higgs doublet model (Lee model) as an example. We considered the constraints by recent data from LHC and B factories, and found that the measurements gave weak constraints. At LHC Run II, $h\rightarrow\mu\tau$ will be confirmed or set stricter limit on its branching ratio. Accordingly, $\textrm{Br}(\tau\rightarrow\mu\gamma)\lesssim\mathcal{O}(10^{-10}-10^{-8})$ for general chosen parameters. For the positive case, $\tau\rightarrow\mu\gamma$ can be discovered with $\mathcal{O}(10^{10})$ $\tau$ pair samples at SuperB factory, Super $\tau$-charm factory and new Z-factory. The future measurements for $\textrm{Br}(h\rightarrow\mu\tau)$ and $\textrm{Br}(\tau\rightarrow\mu\gamma)$ will be used to distinguish these two scenarios or set strict constraints on the correlations among different Higgs couplings, please see Table II in the text for details.