General Property of Neutrino Mass Matrix and CP Violation

It is found that the atmospheric neutrino mixing angle of θ_{atm} is determined to be \tanθ_{atm}=Im(B)/Im(C) for B=M_{ν_eν_μ} and C=M_{ν_eν_τ}, where M_{ij} is the ij element of M^\dagger_νM_νwith M_νas a complex symmetric neutrino mass matrix in the (ν_e, ν_μ, ν_τ)-basis. Another mixing angle, θ_{13}, defined as U_{e3} = \sinθ_{13}e^{-iδ} is subject to the condition: \tan 2θ_{13} \propto |\sinθ_{atm}B+\cosθ_{atm}C| and the CP-violating Dirac phase of δis identical to the phase of \sinθ_{atm}B^\ast+\cosθ_{atm}C^\ast. The smallest value of |\sin θ_{13}| is achieved at \tanθ_{atm}=-Re(C)/Re(B) that yields the maximal CP-violation and that implies C=-κB^\ast for the maximal atmospheric neutrino mixing of \tanθ_{atm}=κ=\pm 1. The generic smallness of |\sin θ_{13}| can be ascribed to the tiny violation of the electron number conservation.

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