Determining the Dirac CP Violation Phase in the Neutrino Mixing Matrix from Sum Rules

Abstract

Using the fact that the neutrino mixing matrix $U = U^\dagger_{e}U_ν$, where $U_{e}$ and $U_ν$ result from the diagonalisation of the charged lepton and neutrino mass matrices, we analyse the sum rules which the Dirac phase $δ$ present in $U$ satisfies when $U_ν$ has a form dictated by flavour symmetries and $U_e$ has a "minimal" form (in terms of angles and phases it contains) that can provide the requisite corrections to $U_ν$, so that reactor, atmospheric and solar neutrino mixing angles $θ_{13}$, $θ_{23}$ and $θ_{12}$ have values compatible with the current data. The following symmetry forms are considered: i) tri-bimaximal (TBM), ii) bimaximal (BM) (or corresponding to the conservation of the lepton charge $L' = L_e - L_μ- L_τ$ (LC)), iii) golden ratio type A (GRA), iv) golden ratio type B (GRB), and v) hexagonal (HG). We investigate the predictions for $δ$ in the cases of TBM, BM (LC), GRA, GRB and HG forms using the exact and the leading order sum rules for $\cosδ$ proposed in the literature, taking into account also the uncertainties in the measured values of $\sin^2θ_{12}$, $\sin^2θ_{23}$ and $\sin^2θ_{13}$. This allows us, in particular, to assess the accuracy of the predictions for $\cosδ$ based on the leading order sum rules and its dependence on the values of the indicated neutrino mixing parameters when the latter are varied in their respective 3$σ$ experimentally allowed ranges.