Phenomenological implications of sterile neutrinos in the $\mu \nu$SSM and dark matter

We analyze the role of sterile neutrinos in the framework of the $\mu \nu$SSM, where the presence of right-handed neutrinos provides a simultaneous solution to $\mu$- and $\nu$-problems in supersymmetry. We adopt a minimalistic approach, reproducing light neutrino masses and mixing angles at tree level using just two right-handed neutrinos as part of the seesaw mechanism. A third right-handed neutrino does not contribute significantly to the mass of the three active ones, behaving as a sterile neutrino with a mass in the range keV$-$MeV. Furthermore, a sterile neutrino of about $10$ keV can be a good candidate for dark matter with a lifetime larger than the age of the Universe. In particular, the three-body decay to active neutrinos gives the dominant contribution to its lifetime. The one-loop decay to gamma and active neutrino is subdominant, but relevant for observations such as astrophysical X-rays. We find regions of the parameter space of the $\mu \nu$SSM, with different values of the sterile neutrino mass, fulfilling not only these constraints but also collider constraints from the Higgs sector.