Two-loop radiative seesaw, muon $g-2$, and $\tau$-lepton-flavor violation with DM constraints.

The quartic scalar coupling $\lambda_5$ term, which violates the lepton-number by two units in the Ma-model, is phenomenologically small when the model is applied to the lepton-flavor violation (LFV) processes. In order to dynamically generate the $\lambda_5$ parameter through quantum loop effects and retain the dark matter (DM) candidate, we extend the Ma-model by adding a $Z_2$-odd vector-like lepton doublet and a $Z_2$-even Majorana singlet. With the new couplings to the Higgs and gauge bosons, the observed DM relic density can be explained when the upper limits from the DM-nucleon scattering cross sections are satisfied. In addition to the neutrino data and LFV constraints, it is found that the DM relic density can significantly exclude the free parameter space. Nevertheless, the resulting muon $g-2$ mediated by the inert charged-Higgs can fit the $4.2\sigma$ deviation between the experimental measurement and the SM result, and the branching ratio for $\tau\to \mu \gamma$ can be as large as the current upper limit when the rare $\mu\to (e \gamma, 3 e)$ decays are suppressed. In addition, it is found that the resulting $BR(\tau \to \mu \rho)$ can reach the sensitivity of Belle II with an integrated luminosity of 50 $ab^{-1}$.