Soft A4 → Z3 Symmetry Breaking and Cobimaximal Neutrino Mixing

I propose a model of radiative charged-lepton and neutrino masses with A4 symmetry. The soft breaking of A4 to Z3 lepton triality is accomplished by dimension-three terms. The breaking of Z3 by dimension-two terms allows cobimaximal neutrino mixing (θ13 6= 0, θ23 = π/4, δCP = ±π/2) to be realized with only very small finite calculable deviations from the residual Z3 lepton triality. This construction solves a long-standing technical problem inherent in renormalizable A4 models since their inception. ar X iv :1 60 1. 00 13 8v 1 [ he pph ] 2 J an 2 01 6 For the past several years, some new things have been learned regarding the theory of neutrino flavor mixing. (1) Whereas the choice of symmetry, for example A4 [1, 2, 3], and its representations are obviously important, the breaking of this symmetry into specific residual symmetries, for example A4 → Z3 lepton triality [4, 5], is actually more important. (2) A mixing pattern may be obtained [6] independent of the masses of the charged leptons and neutrinos. (3) The clashing of residual symmetries between the charged-lepton, for example A4 → Z3, and neutrino, for example A4 → Z2, sectors is technically very difficult to maintain [7]. (4) The essential incorporation of CP transformations [8, 9] may be the new approach [10, 11, 12, 13, 14, 15] which will lead to an improved understanding of neutrino flavor mixing. In this paper, a model of radiative charged-lepton and neutrino masses is proposed with the following properties. (1) The masses are generated in one loop through dark matter [16], i.e. particles distinguished from ordinary matter by an exactly conserved dark symmetry. This is the so-called scotogenic mechanism. (2) The symmetry A4 × Z2 is imposed on all dimension-four terms of the renormalizable Lagrangian with particle content given in Table 1. (3) Dimension-three terms break A4 × Z2, but all such terms respect the residual Z3 lepton triality. (4) Dimension-two terms break Z3, which is nevertheless retained in dimension-three (and dimension-four) terms with only finite calculable deviations. This solves the problem of clashing residual symmetries. (5) The proposed specific model results in cobimaximal [15] neutrino mixing (θ13 6= 0, θ23 = π/4, δCP = ±π/2), which is consistent with the present data [17, 18]. It is also theoretically sound, because the residual Z3 is protected, unlike previous proposals. Cobimaximal mixing becomes thus a genuine prediction, robustly supported in the context of a complete renormalizable theory of neutrino mass and mixing. The dark U(1)D and Z2 symmetries are assumed to be unbroken. The other Z2 symmetry is used to forbid the dimension-four Yukawa couplings l̄LlRφ 0 so that charged leptons only