Abstract
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.
Highlights
Neutrino oscillation experiments have firmly established that neutrinos are massive and lepton flavors are significantly mixed [1,2]
We derive the renormalization-group equations (RGEs) of the WB invariants in leptonic sector in the effective theory, which have rarely been investigated in the literature
We reach the conclusion that if the phases of the matrix elements of MD fulfill the conditions in Eq (3.18) in the basis where Ml and MR are diagonal, there will be no CP violation in the canonical seesaw model and all the CP asymmetries εiα vanish in the decays of heavy Majorana neutrinos
Summary
Neutrino oscillation experiments have firmly established that neutrinos are massive and lepton flavors are significantly mixed [1,2]. Since the discovery of leptonic CP violation is the primary goal of future neutrino oscillation experiments and it may be connected to cosmological matter-antimatter asymmetry [13,14,15,16], it is interesting to establish the sufficient and necessary conditions for CP conservation in the leptonic sector with massive Majorana neutrinos.
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