Lepton-Specific Two Higgs Doublet Model as a Solution of Muon $g-2$ Anomaly

We discuss the Type-X (lepton-specific) two Higgs doublet model as a solution of the anomaly of the muon $g-2$. We consider various experimental constraints on the parameter space such as direct searches for extra Higgs bosons at the LEP II and the LHC Run-I, electroweak precision observables, the decay of $B_s \to \mu^+\mu^-$, and the leptonic decay of the tau lepton. We find that the measurement of the tau decay provides the most important constraint, which excludes the parameter region that can explain the muon $g-2$ anomaly at the 1$\sigma$ level. We then discuss the phenomenology of extra Higgs bosons and the standard model-like Higgs boson ($h$) to probe the scenario favored by the $g-2$ data at the collider experiments. We find that the $4\tau$, $3\tau$ and $4\tau + W/Z$ signatures are expected as the main signal of the extra Higgs bosons at the LHC. In addition, we clarify that the value of the $h\tau\tau$ coupling is predicted to be the standard model value times about $-1.6$ to $-1.0$, and the branching fraction of the $h\to \gamma\gamma$ mode deviates from the standard model prediction by $-30\%$ to $-15\%$. Furthermore, we find that the exotic decay mode, $h$ decaying into the $Z$ boson and a light CP-odd scalar boson, is allowed, and its branching fraction can be a few percent. These deviations in the property of $h$ will be tested by the precision measurements at future collider experiments.