Seesaw lepton masses and muon $g-2$ from heavy vector-like leptons.

We propose a model for the vector-like lepton to explain the small muon mass by a see-saw mechanism, based on lepton-specific two Higgs doublet models with a local $U(1)'$ symmetry. There is no bare muon mass for a nonzero $U(1)'$ charge of the leptophilic Higgs doublet, so the physical muon mass is generated due to the mixing between the vector-like lepton and the muon after the leptophilic Higgs doublet and the dark Higgs get VEVs. In this scenario, the non-decoupling effects of the vector-like lepton give rise to leading contributions to the muon $g-2$, thanks to the light $Z'$ and the light dark Higgs boson. We discuss various constraints on the model from lepton flavor violation, electroweak precision and Higgs data, as well as collider searches.