Phases of Flavor Neutrino Masses and CP Violation

Abstract

For flavor neutrino masses M^{PDG}_{ij} (i,j=e,mu,tau) compatible with the phase convention defined by Particle Data Group (PDG), if neutrino mixings are controlled by small corrections to those with sin(theta_{13})=0 denoted by sin(theta_{13})deltaM^{PDG}_{e tau} and sin(theta_{13})deltaM^{PDG}_{tau tau}, CP-violating Dirac phase delta{CP} is calculated by using these corrections. If possible neutrino mass hierarchies are taken into account, the main source of delta{CP} turns out to be deltaM_{e tau}^{PDG} except for the inverted mass hierarchy with {m}_1 approx -{m}_2, where {m}_i=m_ie^{-i varphi_i} (i=1,2) stands for a neutrino mass m_i accompanied by a Majorana phase varphi_i for varphi_{1,2,3} giving two CP-violating Majorana phases. We can further derive that delta_{CP} approx arg(M_{e mu}^{PDG})-arg(M_{mu mu}^{PDG}) with arg (M_{e mu}^{PDG}) approx arg(M_{e tau}^{PDG}) for the normal mass hierarchy and delta_{CP} approx arg(M_{ee}^{PDG})-arg(M_{e tau}^{PDG})+pi for the inverted mass hierarchy with {m}_1 approx {m}_2. For specific flavor neutrino masses M_{ij} whose phases arise from M_{e mu,e tau,tau tau}, these phases can be connected with arg(M_{ij}^{PDG}) (i,j=e,mu,tau). As a result, numerical analysis suggests that Dirac CP-violation becomes maximal as |arg(M_{e mu})| approaches to pi/2 for the inverted mass hierarchy with {m}_1 approx {m}_2 and for the degenerate mass pattern satisfying the inverted mass ordering and that Majorana CP-violation becomes maximal as |arg(M_{tau tau})| approaches to its maximal value around 0.5 for the normal mass hierarchy. Alternative CP-violation induced by three CP-violating Dirac phases is compared with the conventional one induced by delta{CP} and two CP-violating Majorana phases.