Correlation between leptonic CP violation and μ-τ symmetry breaking

Considering the μ-τ symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic CP violation by determining three eigenvectors associated with M=M_ν^†M_ν for a complex flavor neutrino mass matrix M_ν in the flavor basis. Since the Dirac CP violation is absent in the μ-τ symmetric limit, leptonic CP violation is sensitive to the μ-τ symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase (δ) arises from the μ-τ symmetry breaking part of M_eμ,eτ and an additional phase (ρ) is associated with the μ-τ symmetric part of M_eμ,eτ, where M_ij stands for an ij matrix element (i,j=e,μ,τ). The phase ρ is redundant and can be removed but leaves its effect in the Dirac CP violation characterized by sin⁡(δ+ρ). The perturbative results suggest the exact formula of mixing parameters including that of δ and ρ, which turns out to be free from the effects of the redundant phases. As a result, it is generally shown that the maximal atmospheric neutrino mixing necessarily accompanies either sin⁡θ₁₃=0 or cos⁡(δ+ρ)=0, the latter of which indicates maximal CP violation, where θ₁₃ is the ν_e-ν_τ mixing angle.