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) -> 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) not equal 0 holds, we show that the continuity condition derived by Antusch 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) -> 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. (c) 2006 Elsevier B.V. All rights reserved.