Phases of flavor neutrino masses and CP-violation

For flavor neutrino masses MijPDG (i,j=e,μ,τ) compatible with the phase convention defined by Particle Data Group (PDG), if neutrino mixings are controlled by small corrections to those with sinθ13=0 denoted by sinθ13δMeτPDG and sinθ13δMττPDG, CP-violating Dirac phase δCP is calculated to be δCP≈arg[(MμτPDG⁎/tanθ23+MμμPDG⁎)δMeτPDG+MeePDGδMeτPDG⁎−tanθ23MeμPDGδMττPDG⁎] (mod π), where θij (i,j=1,2,3) denotes an i-j neutrino mixing angle. If possible neutrino mass hierarchies are taken into account, the main source of δCP turns out to be δMeτPDG except for the inverted mass hierarchy with m˜1≈−m˜2, where m˜i=mie−iφi (i=1,2) stands for a neutrino mass mi accompanied by a Majorana phase φi for φ1,2,3 giving two CP-violating Majorana phases. We can further derive that δCP≈arg(MeμPDG)−arg(MμμPDG) with arg(MeμPDG)≈arg(MeτPDG) for the normal mass hierarchy and δCP≈arg(MeePDG)−arg(MeτPDG)+π for the inverted mass hierarchy with m˜1≈m˜2. For specific flavor neutrino masses Mij whose phases arise from Meμ,eτ,ττ, these phases can be connected with arg(MijPDG) (i,j=e,μ,τ). As a result, numerical analysis suggests that Dirac CP-violation becomes maximal as |arg(Meμ)| approaches to π/2 for the inverted mass hierarchy with m˜1≈m˜2 and for the degenerate mass pattern satisfying the inverted mass ordering and that Majorana CP-violation becomes maximal as |arg(Mττ)| 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 δCP and two CP-violating Majorana phases.