Leptonic Dirac CP violation predictions from residual discrete symmetries

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 GfGf, and that GfGf is broken to specific residual symmetries GeGe and GνGν 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=Z2Ge=Z2 and Gν=ZnGν=Zn, n>2n>2 or Zn×ZmZn×Zm, n,m≥2n,m≥2; ii) Ge=ZnGe=Zn, n>2n>2 or Zn×ZmZn×Zm, n,m≥2n,m≥2 and Gν=Z2Gν=Z2; iii) Ge=Z2Ge=Z2 and Gν=Z2Gν=Z2; iv) GeGe is fully broken and Gν=ZnGν=Zn, n>2n>2 or Zn×ZmZn×Zm, n,m≥2n,m≥2; and v) Ge=ZnGe=Zn, n>2n>2 or Zn×ZmZn×Zm, n,m≥2n,m≥2 and GνGν is fully broken. For given GeGe and GνGν, the sum rules for cos⁡δcos⁡δ thus derived are exact, within the approach employed, and are valid, in particular, for any GfGf containing GeGe and GνGν as subgroups. We identify the cases when the value of cos⁡δcos⁡δ 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 cos⁡δcos⁡δ can be unambiguously predicted once the flavour symmetry GfGf is fixed. We present predictions for cos⁡δcos⁡δ in these cases for the flavour symmetry groups Gf=S4Gf=S4, A4A4, T′T′ and A5A5, requiring that the measured values of the 3-neutrino mixing parameters sin2⁡θ12sin2⁡θ12, sin2⁡θ13sin2⁡θ13 and sin2⁡θ23sin2⁡θ23, taking into account their respective 3σ uncertainties, are successfully reproduced.