Neutrino masses and mixed dark matter from doublet and singlet scalars

We consider the extension of the Standard Model with an inert scalar doublet, three right-handed neutrinos, and singlet scalar fields, $\varphi$ and $S$. In this model, neutrino masses are zero in the limit of the unbroken $Z_4$ discrete symmetry. We show that when the singlet scalar field $\varphi$ gets a VEV, the $Z_4$ symmetry is broken to $Z_2$, and neutrino masses are generated at one-loops due to the mixings between the neutral components of the inert scalar doublet and the singlet scalar field $S$. There is a dark matter candidate from the lightest neutral scalar field, which is a mixture of the inert scalar doublet and the singlet scalar field $S$, in general. The $Z_4$ breaking mass terms are constrained by electroweak precision data and direct detection (DD) bounds for dark matter, favoring small mixings or almost degenerate masses for the DM scalars. As a result, we discuss the implications of the results for small neutrino masses and DD-safe dark matter.