An explanation of one loop induced $h\to \mu \tau$ decay

We discuss a possibility to explain the excess of $h\to \mu \tau$ at one-loop level. We introduce three generation of vector-like lepton doublet $L^{\mathrm{\prime }}$ and two singlet scalars $S_{1,2}$ which are odd under $Z_2$, while all the standard model fields are even under this discrete symmetry. We show that $S_1$ can be a good dark matter candidate. We show that we can explain the dark matter relic abundance, large part of the discrepancy of muon $g-2$ between experiments and the standard model predictions, as well as the $h\to \mu \tau$ excess of $\sim 1$ %, while evading constraints from experiments of dark matter direct detection and charged lepton flavor violating processes. We also consider prospects of production of $S_2$ at LHC with energy $\sqrt{s}=14$ TeV.