On the Quasi-fixed Point in the Running of CP-violating Phases of Majorana Neutrinos

Abstract

Taking the standard parametrization of three-flavor neutrino mixing, we carefully examine the evolution of three CP-violating phases $(\hat{I}´,\hat{I}{\pm }_1^{},\hat{I}{\pm }_2^{})$ with energy scales in the realistic limit $\hat{I}{¸}_{13}^{}\to 0$. If $m_3^{}$ vanishes, we find that the one-loop renormalization-group equation (RGE) of $\hat{I}´$ does not diverge and its running has no quasi-fixed point. When $m_3^{}\ne 0$ holds, we show that the continuity condition derived by Antusch {\it et al} is always valid, no matter whether the $\ddot{I}„$-dominance approximation is taken or not. The RGE running of $\hat{I}´$ undergoes a quasi-fixed point determined by a nontrivial input of $\hat{I}{\pm }_2^{}$ 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 $\hat{I}´$ 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.

RevTex 13 pages. Slight modifications made for the text, Table I and references. Results and conclusions unchanged. Accepted for publication in Phys. Lett. B