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., $m_1=m_2\ne m_3$) 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.