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 $(\delta, \alpha^{}_1, \alpha^{}_2)$ with energy scales in the realistic limit $\theta^{}_{13} \to 0$. If $m^{}_3$ vanishes, we find that the one-loop renormalization-group equation (RGE) of $\delta$ does not diverge and its running has no quasi-fixed point. When $m^{}_3 \neq 0$ holds, we show that the continuity condition derived by Antusch {\it et al} is always valid, no matter whether the $\tau$-dominance approximation is taken or not. The RGE running of $\delta$ undergoes a quasi-fixed point determined by a nontrivial input of $\alpha^{}_2$ in the limit $m^{}_1 \to 0$. If three neutrino masses are nearly degenerate, it is also possible to arrive at a quasi-fixed point in the RGE evolution of $\delta$ 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.