Sufficient and necessary conditions for CP conservation in the case of degenerate Majorana neutrino masses

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) not equal 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 not equal 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.