On the quasi-fixed point in the running of CP-violating phases of Majorana neutrinos

Taking the standard parametrization of three-flavor neutrino mixing, we carefully examine the evolution of three CP-violating phases (δ,α1,α2) with energy scales in the realistic limit θ13→0. If m3 vanishes, we find that the one-loop renormalization-group equation (RGE) of δ does not diverge and its running has no quasi-fixed point. When m3≠0 holds, we show that the continuity condition derived by Antusch et al. is always valid, no matter whether the τ-dominance approximation is taken or not. The RGE running of δ undergoes a quasi-fixed point determined by a nontrivial input of α2 in the limit m1→0. If three neutrino masses are nearly degenerate, it is also possible to arrive at a quasi-fixed point in the RGE evolution of δ from the electroweak scale to the seesaw scale or vice versa. Furthermore, the continuity condition and the quasi-fixed point of CP-violating phases in another useful parametrization are briefly discussed.