We consider a two-Higgs doublet scenario containing three $SU(2)_L$ singlet heavy neutrinos with Majorana masses. The second scalar doublet as well as the neutrinos are odd under a $Z_2$ symmetry. This scenario not only generates Majorana masses for the light neutrinos radiatively but also makes the lighter of the neutral $Z_2$-odd scalars an eligible dark matter candidate, in addition to triggering leptogenesis at the scale of the heavy neutrino masses. Taking two representative values of this mass scale, we identify the allowed regions of the parameter space of the model, which are consistent with all dark matter constraints. At the same time, the running of quartic couplings in the scalar potential to high scales is studied, thus subjecting the regions consistent with dark matter constraints to further requirements of vacuum stability, perturbativity and unitarity. It is found that part of the parameter space is consistent with all of these requirements all the way up to the Planck scale, and also yields the correct signal strength in the diphoton channel for the scalar observed at the Large Hadron Collider.
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