Sufficient and Necessary Conditions for CP Conservation in the Case of Degenerate Majorana Neutrino Masses

Abstract

In this paper, we carry out a systematic study of the sufficient and necessary conditions for CP conservation in the leptonic sector with massive Majorana neutrinos. In particular, the emphasis is placed on the number of CP-violating phases in the presence of a partial mass degeneracy (e.g., $m_1^{}=m_2^{}\ne m_3^{}$) or a complete mass degeneracy $m_1^{}=m_2^{}=m_3^{}$, where $m_i^{}$ (for $i=1,2,3$) stand for the masses of three ordinary neutrinos. In the canonical seesaw model with three right-handed neutrino singlets, CP-violating phases in the special case of a partial (e.g., $M_1^{}=M_2^{}\ne M_3^{}$) or complete (i.e., $M_1^{}=M_2^{}=M_3^{}$) mass degeneracy of three heavy Majorana neutrinos are also examined. In addition, we derive the renormalization-group equations of the weak-basis invariants in the effective theory with a general mass spectrum of Majorana neutrinos, to which the solutions establish the direct connection between CP violation at low- and high-energy scales.

Comments: 28 pages, 2 tables, more discussions and references added, accepted for publication in Phys. Rev. D