The role of residual symmetries in dark matter stability and the
  neutrino nature
  We consider the class of models where Dirac neutrino masses at one loop and
the dark matter stability can be obtained using only the global $U(1)_{B-L}$
symmetry already present in Standard Model. We discuss how the residual
$\mathcal{Z}_n$ subgroup, left unbroken after the breaking of $U(1)_{B-L}$,
dictates the neutrino nature, namely if they are Dirac or Majorana particles,
as well as determines the stability of the dark matter candidate in such
models. In particular, we show that without the correct breaking of
$U(1)_{B-L}$ to an appropriate residual $\mathcal{Z}_n$ symmetry, the Dirac
nature of neutrinos and/or dark matter stability might be lost. For
completeness we also provide some examples where the dark matter stability is
accidental or lost completely. Finally, we discuss one example model where the
Dirac neutrinos with naturally small one loop masses as well as dark matter
stability, are both protected by the same residual $\mathcal{Z}_6$ subgroup,
without need for adding any new explicit or accidental symmetries beyond the
Standard Model symmetries.
