Scotogenic Cobimaximal Dirac Neutrino Mixing from $Δ(27)$ and $U(1)_χ$

In the context of $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_χ$, where $U(1)_χ$ comes from $SO(10) \to SU(5) \times U(1)_χ$, supplemented by the non-Abelian discrete $Δ(27)$ symmetry for three lepton families, Dirac neutrino masses and their mixing are radiatively generated through dark matter. The gauge $U(1)_χ$ symmetry is broken spontaneously. The discrete $Δ(27)$ symmetry is broken softly and spontaneously. Together, they result in two residual symmetries, a global $U(1)_L$ lepton number and a dark symmetry, which may be $Z_2$, $Z_3$, or $U(1)_D$ depending on what scalar breaks $U(1)_χ$. Cobimaximal neutrino mixing, i.e. $θ_{13} \neq 0$, $θ_{23} = π/4$, and $δ_{CP} = \pm π/2$, may also be obtained.

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