  We investigate some consequences if neutrinoless double beta decays of nuclei
are dominated by short range interactions. To illustrate our results, we assume
that such decays proceed mainly through short range interactions involving
two-W-bosons exchanges and confine ourselves to only include new scalars
without new gauge interactions for SM fermions. For the neutrino mass problem
we propose to solve it by adopting that the active light neutrinos have
predominantly Dirac masses and the small Majorana masses induced by the new
scalars render them pseudo(quasi)-Dirac particles. This particular aspect of
neutrinos may be detectable in the next generation of neutrino oscillation
experiments and/or neutrino telescopes. If so this opens a new connection
between neutrinoless double beta decays and neutrino physics. We also noted the
new physics signals such as high charged scalar states that can be explored in
hadron colliders. In particular, we find that a high energy e^- e^- collider
will be very useful in testing the origin of lepton number violation which
complements neutrinoless double decays studies.
