Abstract
We show how a minimal (littlest) seesaw model involving two right-handed neutrinos and a very constrained Dirac mass matrix, with one texture zero and two independent Dirac masses, may arise from $S_4\times U(1)$ symmetry in a semi-direct supersymmetric model. The resulting CSD3 form of neutrino mass matrix only depends on two real mass parameters plus one undetermined phase. We show how the phase may be fixed to be one of the cube roots of unity by extending the $S_4\times U(1)$ symmetry to include a product of $Z_3$ factors together with a CP symmetry, which is spontaneously broken leaving a single residual $Z_3$ in the charged lepton sector and a residual $Z_2$ in the neutrino sector, with suppressed higher order corrections. With the phase chosen from the cube roots of unity to be $-2\pi/3$, the model predicts a normal neutrino mass hierarchy with $m_1=0$, reactor angle $\theta_{13}=8.7^\circ$, solar angle $\theta_{12}=34^\circ$, atmospheric angle $\theta_{23}=44^\circ$, and CP violating oscillation phase $\delta_{\rm CP}=-93^\circ$, depending on the fit of the model to the neutrino masses.
Highlights
Scheme with normal hierarchy seems to be a two right-handed model with a Dirac mass matrix involving one texture zero and a particular pattern of couplings, together with a diagonal right-handed neutrino mass matrix [18], mD = a 3b, ab MR =
In this paper, guided by the principles of minimality and symmetry, we have been led to a highly predictive theory of neutrino mass and lepton mixing in which all CP phases are fixed and the neutrino masses and the entire lepton mixing matrix are determined by only two real input mass parameters
Our main achievement is to show that the new version of CSD3 can be obtained from symmetry arguments based on S4, working in the basis where the diagonal T generator can enforce the diagonality of the charged lepton mass matrix due to a residual Z3 symmetry, while the preserved S4 subgroup SU in the neutrino sector with a residual Z2 symmetry is instrumental in enforcing TM1 mixing
Summary
Before getting into too many technicalities of symmetry and model building, it is useful to give a sketch of the type of model we will present in this paper. It is clear that both types of CSD3 give good predictions for lepton mixing angles, assuming that η = ±2π/3 In both examples in table 1 the CP phase is predicted to be δCP ≈ −π/2.4 For the original CSD3, η = 2π/3 is identified as the leptogenesis phase and the baryon asymmetry of the universe leads to a determination of the lighter atmospheric neutrino mass Matm = 4 × 1010 GeV [21]. In appendix B, the mass matrix in eq (2.4) is generalised to a new type of CSDn, and analytic formulas for neutrino masses and lepton mixing parameters are presented for any real value of n ( we are only interested in n = 3 here). The results may be compared to the numerical results in [29] and the analytic formulas in [30] for the original version of CSDn based on a generalisation of eq (1.2)
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