Correlation between leptonic CP violation and μ-τ symmetry breaking

Considering the mu-tau symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic CP violation by determining three eigenvectors associated with M=(M nu M nu)-M-dagger for a complex flavor neutrino mass matrix M-nu in the flavor basis. Since the Dirac CP violation is absent in the mu-tau symmetric limit, leptonic CP violation is sensitive to the mu-tau symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase (delta) arises from the mu-tau symmetry breaking part of M-e mu,M-e tau and an additional phase (rho) is associated with the mu-tau symmetric part of M-e mu,M-e tau, where M-ij stands for an ij matrix element (i,j=e,mu,tau). The phase rho is redundant and can be removed but leaves its effect in the Dirac CP violation characterized by sin(delta+rho). The perturbative results suggest the exact formula of mixing parameters including that of delta and rho, 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 theta(13)=0 or cos(delta+rho)=0, the latter of which indicates maximal CP violation, where theta(13) is the nu(e)-nu(tau) mixing angle.