Abstract
Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton) flavour symmetry, corresponding to a non-Abelian discrete symmetry group Gf, and that Gf is broken to specific residual symmetries Ge and Gν of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase δ of the neutrino mixing matrix U. The residual symmetries considered are: i) Ge=Z2 and Gν=Zn, n>2 or Zn×Zm, n,m≥2; ii) Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν=Z2; iii) Ge=Z2 and Gν=Z2; iv) Ge is fully broken and Gν=Zn, n>2 or Zn×Zm, n,m≥2; and v) Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν is fully broken. For given Ge and Gν, the sum rules for cosδ thus derived are exact, within the approach employed, and are valid, in particular, for any Gf containing Ge and Gν as subgroups. We identify the cases when the value of cosδ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of cosδ can be unambiguously predicted once the flavour symmetry Gf is fixed. We present predictions for cosδ in these cases for the flavour symmetry groups Gf=S4, A4, T′ and A5, requiring that the measured values of the 3-neutrino mixing parameters sin2θ12, sin2θ13 and sin2θ23, taking into account their respective 3σ uncertainties, are successfully reproduced.
Highlights
Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a flavour symmetry, corresponding to a non-Abelian discrete symmetry group Gf, and that Gf is broken to specific residual symmetries Ge and Gν of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase δ of the neutrino mixing matrix U
For the symmetry group A4 we find that none of the combinations of the residual symmetries Ge = Z2 and Gν = Z2 provide physical values of cos δ and phenomenologically viable results for the neutrino mixing angles simultaneously
The residual symmetries can constrain the forms of the 3×3 unitary matrices Ue and Uν, which diagonalise the charged lepton and neutrino mass matrices, and the product of which represents the PMNS neutrino mixing matrix U, U = Ue† Uν
Summary
The discrete symmetry approach to understanding the observed pattern of 3-neutrino mixing (see, e.g., [1]), which is widely explored at present (see, e.g., [2,3,4,5]), leads to specific correlations between the values of at least some of the mixing angles of the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) neutrino mixing matrix U and, either to specific fixed trivial or maximal values of the CP violation (CPV) phases present in U (see, e.g., [6,7,8,9,10] and references quoted therein), or to a correlation between the values of the neutrino mixing angles and of the Dirac. These two forms appear in a large class of theoretical models of flavour and theoretical studies, in which the generation of charged lepton masses is an integral part (see, e.g., [17, 32,33,34,35,36,37]) In this setting with Uν having one of the five symmetry forms, TBM, BM, GRA, GRB and HG, and Ue given by eq (11), the Dirac phase δ of the PMNS matrix was shown in [11] to satisfy the following sum rule: cos δ = tan θ23 sin 2θ12 sin θ13 cos 2θ1ν2 + sin θ12 − cos θ1ν2. Appendices A, B, C, D and E contain technical details related to the study
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