Dirac neutrinos, dark matter stability and flavour predictions from Lepton Quarticity

We propose to relate dark matter stability to the possible Dirac nature of neutrinos. The idea is illustrated in a simple scheme where small Dirac neutrino masses arise from a type-I seesaw mechanism as a result of a $Z_4$ discrete lepton number symmetry. The latter implies the existence of a viable WIMP dark matter candidate, whose stability arises from the same symmetry which ensures the Diracness of neutrinos. The symmetry groups ${\mathrm{\Delta }}_{27}$ and $A_4$ are then used to extract a rich variety of flavour predictions.