Schizophrenic neutrinos and ν-less double beta decay

We point out a new possibility for neutrinos where all neutrino flavors can be part Dirac and part Majorana. Our primary motivation comes from an attempt to use supersymmetric seesaw models to tie inflation, baryon asymmetry of the Universe and dark matter to the neutrino sector. The idea however could stand on its own, with or without supersymmetry. We present a realization of this possibility within an S3 family symmetry for neutrino masses, where we obtain tri-bi-maximal mixing for neutrinos to the leading order. The model predicts that for the case of inverted hierarchy, the lower limit on the neutrino mass measured in neutrinoless double beta decay experiments is about a factor of two larger than the usual Majorana case.