Naturally Light Sterile Neutrinos from Theory of R-parity

The fate of R-parity is one of the central issues in the minimal supersymmetric standard model (MSSM). Gauged $B-L$ symmetry provides a natural framework for addressing this question. Recently, it was pointed out that the minimal such theory does not need any additional Higgs if the $B-L$ breaking is achieved through the VEVs of right-handed sneutrinos, which ties the new physics scale to the scale of the MSSM. We show here that this immediately leads to an important prediction of two light sterile neutrinos, which can play a significant role in the BBN and neutrino oscillations. We also discuss some new relevant phenomenology for the LHC, in the context of the minimal supersymmetric left-right symmetric theory which provides a natural setting for the gauged $B-L$ symmetry.