Explaining the 3.5 keV X-ray Line in a ${L_{\mu}-L_{\tau}}$ Extension of the Inert Doublet Model

We explain the existence of neutrino masses and their flavor structure, dark matter relic abundance and the observed 3.5 keV X-ray line within the framework of a gauged $U(1)_{L_{\mu} - L_{\tau}}$ extension of the "scotogenic" model. In the $U(1)_{L_{\mu} - L_{\tau}}$ symmetric limit, two of the the RH neutrinos are degenerate in mass, while the third is heavier. The $U(1)_{L_{\mu} - L_{\tau}}$ symmetry is broken spontaneously. Firstly, this breaks the $\mu-\tau$ symmetry in the light neutrino sector. Secondly, this results in mild splitting of the two degenerate RH neutrinos, with their mass difference given in terms of the $U(1)_{L_{\mu} - L_{\tau}}$ breaking parameter. Finally, we get a massive $Z_{\mu\tau}$ gauge boson. Due to the added $Z_2$ symmetry under which the RH neutrinos and the inert doublet are odd, the canonical Type-I seesaw is forbidden and the tiny neutrino masses are generated radiatively at one loop. The same $Z_2$ symmetry also ensures that the lightest RH neutrino is stable and the other two can only decay into the lightest one. This makes the two nearly-degenerate lighter neutrinos a two-component dark matter, which in our model are produced by the freeze-in mechanism via the decay of the $Z_{\mu\tau}$ gauge boson in the early universe. We show that the next-to-lightest RH neutrino has a very long lifetime and decays into the lightest one at the present epoch explaining the observed 3.5 keV line.