Abstract
In the context of $$SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_\chi $$SU(3)C×SU(2)L×U(1)Y×U(1)χ, where $$U(1)_\chi $$U(1)χ comes from $$SO(10) \rightarrow SU(5) \times U(1)_\chi $$SO(10)→SU(5)×U(1)χ, supplemented by the non-Abelian discrete $$\Delta (27)$$Δ(27) symmetry for three lepton families, Dirac neutrino masses and their mixing are radiatively generated through dark matter. The gauge $$U(1)_\chi $$U(1)χ symmetry is broken spontaneously. The discrete $$\Delta (27)$$Δ(27) symmetry is broken softly and spontaneously. Together, they result in two residual symmetries, a global $$U(1)_L$$U(1)L lepton number and a dark symmetry, which may be $$Z_2$$Z2, $$Z_3$$Z3, or $$U(1)_D$$U(1)D depending on what scalar breaks $$U(1)_\chi $$U(1)χ. Cobimaximal neutrino mixing, i.e. $$\theta _{13} \ne 0$$θ13≠0, $$\theta _{23} = \pi /4$$θ23=π/4, and $$\delta _{CP} = \pm \pi /2$$δCP=±π/2, may also be obtained.
Bound-state dark matter and Dirac neutrino masses