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., $m^{}_1 = m^{}_2 \neq m^{}_3$) 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., $M^{}_1 = M^{}_2 \neq M^{}_3$) 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.