Two-Loop $Z_4$ Dirac Neutrino Masses and Mixing, with Self-Interacting Dark Matter

Choosing how gauge $U(1)_\chi$ breaks in the context of $SO(10) \to SU(5) \times U(1)_\chi$, $Z_4$ lepton number may be obtained which maintains neutrinos as Dirac fermions. Choosing $\Delta(27)$ as the family symmetry of leptons, tree-level Dirac neutrino masses may be forbidden. Choosing a specific set of self-interacting dark-matter particles, Dirac neutrino masses and mixing may then be generated in two loops. This framework allows the realization of cobimaximal neutrino mixing, i.e. $\theta_{13} \neq 0$, $\theta_{23} = \pi/4$, $\delta_{CP} = \pm \pi/2$, as well as the desirable feature that the light scalar mediator of dark-matter interactions decays only to neutrinos, thereby not disrupting the cosmic microwave background (CMB).