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-tilde1≈−m-tilde2, where m-tildei=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-tilde1≈m-tilde2. 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-tilde1≈m-tilde2 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.