Sufficient and necessary conditions for C P 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., ${m}_{1}={m}_{2}\ensuremath{\ne}{m}_{3}$) 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., ${M}_{1}={M}_{2}\ensuremath{\ne}{M}_{3}$) 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.