Symmetry, dark matter and LHC phenomenology of the minimal νSM

A sterile neutrino with a mass of a few keV can play the role of a warm dark matter(DM). This can be realized in seesaw models with 3 left- and 3 right-handed neutrinos. It is possible to identify the keV neutrino to be one of the right-handed neutrinos leaving the other two to be much more heavier, the $\nu$SM model. We show that with this realization of keV neutrino DM, the model has an approximate Friedberg-Lee symmetry providing a natural explanation for the lightness of the right-handed neutrino. We also find that in this model the mixing parameters couple light and heavy neutrinos are strongly correlated, and can be large enough to have testable effects at the LHC for the two heavy right-handed neutrinos to be in the hundred-GeV range.