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.