Neutrino Mixing and Geometric CP Violation with Delta(27) Symmetry

Predictive spontaneous CP violation is possible if it is obtained geometrically through a non-Abelian discrete symmetry. I propose such a model of neutrino mass and mixing based on Delta(27).

Since the experimental determination of nonzero θ 13 in neutrino oscillations, the next big question in neutrino physics is CP violation. Theoretically, this should be understood together with the mixing angles themselves. Whereas non-Abelian discrete symmetries (the first [1,2,3,4] of which was A 4 ) are useful in obtaining tribimaximal mixing [5] which requires θ 13 = 0 and no CP violation, the data now require either a modification or a new approach. In the former, CP violation may be incorporated by allowing nonzero θ 13 and complex Yukawa couplings. A simple example is a variation [6] of the original A 4 model [4] for tribimaximal mixing. In the latter, the discrete symmetry may be extended to include generalized CP transformations [7], which in the case [8] of S 4 could lead to maximal CP violation as well as maximal θ 23 . Another possible approach in this category is spontaneous geometric CP violation [9] using ∆(27), which has recently been applied [10] successfully to the quark sector. This paper deals with the lepton sector [11,12,13] and how it may be related [14] to dark matter.
The non-Abelian discrete symmetry ∆(27) has 27 elements, with nine one-dimensional irreducible representations 1 i (i = 1, ..., 9) and two three-dimensional ones 3 and 3 * . Its 11 × 11 character table as well as the 27 defining 3 × 3 matrices of its 3 representation are given in Ref. [11]. The group multiplcation rules are The important property to notice is that 3 × 3 × 3 has three invariants: 123 + 231 + 312 − In this paper, the assignments of the lepton and Higgs fields are different from previous studies [11,12,13], with the new requirement that CP be spontaneously broken geometri- where ω = exp(2πi/3) = −1/2 + i √ 3/2. This M l is identical in form to that of the original A 4 model of Ref. [1]. The new feature here is that CP conservation is imposed on the Lagrangian (so that all the Yukawa couplings are real) but it is spontaneously broken by the vacuum, i.e. [9,10] ( Hence where m e = √ 3f e v, etc.
For the neutrino mass matrix, three Higgs doublets are added so that the dimension-five operator Λ −1 (ννφ 0 )ζ 0 for the 3 × 3 Majorana neutrino mass matrix has six invariants, i.e.
where Λ −1 v ζ 0 i have been absorbed into the definitions of the f parameters.
Using Eq. (5), the neutrino mass matrix in the tribimaximal basis is now given by where is reached for c = e = f = 0. To lowest order, c = 0 implies tan 2 θ 12 > 0.5 and θ 13 = 0; e = 0 implies tan 2 θ 12 can be greater or less than 1/2 and θ 13 = 0; f = 0 implies tan 2 θ 12 < 1/2 and θ 13 = 0. Given that data prefer the last choice, it will be assumed from now on that c and e are negligible and only nonzero f is considered. The immediate consequence [6] of this is that θ 12 and θ 13 are related, and that given θ 13 and θ 23 , | tan δ CP | is determined.
Since c = e = 0 has been assumed, M (1,2,3) ν is diagonalized by Since a, b, d, f are real, this implies With this structure, | sin θ 13 | = | sin θ|/ √ 3, which implies which agrees very well [6] with data. As for the phase φ, it is given by the condition tan 2 θ 23 = 1 − Since m 2 2 and m 2 3 are corrected by terms proportional to f 2 which are small, the following approximation for the neutrino masses is valid for the analysis below, i.e.
The invariant CP violating parameter J CP = Im(U µ3 U * e3 U e2 U * µ2 ) is simply given in this model by Using sin θ 13 ≃ 0.16 and | sin φ| > 1/ √ 2, the allowed range is thus obtained. As for the effective neutrino mass in neutrinoless double beta decay, its allowed range is approximately given by 0.03 < m ee < 0.07 eV.
Thus this model has two very specific predictions: (1) |J CP | is between 0.026 and 0.036, and (2) m ee is between 0.03 and 0.07 eV.
The dimension-five operator [16] for Majorana neutrino mass considered in the above may be implemented [14] in has been called "scotogenic", from the Greek "scotos" meaning darkness. Because of the Fig. 1 can be computed exactly [14], i.e.
A good dark-matter candidate is η R as first pointed out in Ref. [14], whereas its stabilty was already anticipated in Ref. [17]. It was subsequently proposed by itself as dark matter in Ref. [18] (to render the standard-model Higgs boson very heavy, which is now ruled out by data) and studied in detail in Ref. [19]. The η doublet has become known as the "inert" Higgs doublet, but it does have gauge and scalar interactions even if it is the sole addition to the standard model. In principle, the lightest N is also a possible dark-matter candidate [20], but its mass and couplings may be severely restricted by the experimental limit on µ → eγ decay, unless a symmetry exists to suppress it, which is possible in this case.
To accommodate the ∆(27) symmetry, the external φ 0 φ 0 lines are replaced by φ 0 i ζ 0 j , and the internal η 0 (N) lines are replaced by η 0 i , N i ∼ 3 on one side, and η 0 ∼ 1, N i ∼ 3 * on the other.
In conclusion, a special mechanism of CP violation has been implemented in a complete model of charged-lepton and neutrino masses and mixing, using the non-Abelian discrete symmetry ∆(27). The Lagrangian is required to conserve CP resulting in real Yukawa couplings, but the Higgs vacuum breaks CP spontaneously and geometrically. The resulting model has some very specific predictions, as given by Eqs. (12) to (22).