Scotogenic $Z_2$ or $U(1)_D$ Model of Neutrino Mass with $\Delta(27)$ Symmetry

The scotogenic model of radiative neutrino mass with $Z_2$ or $U(1)_D$ dark matter is shown to accommodate $\Delta(27)$ symmetry naturally. The resulting neutrino mass matrix is identical to either of two forms, one proposed in 2006, the other in 2008. These two structures are studied in the context of present neutrino data, with predictions of $CP$ violation and neutrinoless double beta decay.

To understand the pattern of neutrino mixing, non-Abelian discrete symmetries have been used frequently in the past several years, starting with A 4 [1,2,3,4,5,6].Another symmetry ∆(27) was also studied [7,8,9] some years ago.Using the fact that it admits geometric CP violation [10], it has been proposed recently for understanding the CP phases in the mixing of quarks [11,12] and of leptons [13,14].
In a parallel development, there is a large body of literature on the radiative generation of neutrino mass through dark matter.The simplest original (scotogenic) one-loop model [15] adds one extra scalar doublet (η + , η 0 ) and three neutral fermion singlets N 1,2,3 together with an exactly conserved Z 2 symmetry under which the new particles are odd and the standardmodel (SM) particles are even.The resulting one-loop diagram for Majorana neutrino mass is shown in Fig. 1.A variation of this mechanism was recently proposed [16], using two extra scalar doublets (η + 1,2 , η 0 1,2 ) transforming as ±1 under an U (1) D gauge symmetry together with three Dirac fermion singlets N 1,2,3 transforming as +1.The resulting one-loop diagram for Majorana neutrino mass is shown in Fig. 2.
Combining these two ideas, it is shown in this paper that ∆(27) is naturally adapted to realize the two neutrino mass matrices proposed earlier [7,9] without additional particle content in the loop, in either the Z 2 or U (1) D case.We then study their implications in the In both cases, the neutrino mass matrix is of the form where a, b, c are proportional to the three arbitrary vacuum expectation values of ζ.
Consider first the case where the charged-lepton mass matrix is diagonal.In Ref. [7], two solutions were found with θ 13 = 0; one with f 1, the other with f −0.5.The former turns out to be unacceptable because θ 13 is always very small.The latter has a solution as shown below.Let f = −0.5 + , a = b(1 + η) and c = b(1 − δ), then in the tribimaximal basis, the neutrino mass matrix becomes where , η, δ are all assumed to be small compared to one.We define δ + 2η = ζ and assume all parameters tio be real, then We consider also the case with δ purely imaginary, in which case sin 2 2θ 23 = 1 is guaranteed in the limit of a symmetry based on a generalized CP transformation [17].Using a complete numerical analysis, we plot in Fig. 3 the predictions of this model for m ee as a function of sin 2 2θ 12 for sin 2 2θ 13 = 0.095 ± 0.010.The higher (lower) band corresponds to δ real (purely imaginary), with the allowed region in between for any arbitrary phase.We plot in Fig. 4 the invariant J CP as a function of sin 2 2θ 13 for δ purely imaginary and sin 2 θ 12 = 0.857 ± 0.024.
Consider next the case where M l is given by Eq. ( 3).In the tribimaximal basis, Let where , η , δ are all assumed to be small compared with one.This turns out to have the same approximate solution as Eq. ( 5) with the following substitutions: The predicted m ee is also approximately the same.Thus the physical manifestations of this second model are indistinguiable from those of the first to a good approximation.

Figure 1 :
Figure 1: One-loop generation of neutrino mass with Z 2 symmetry.