Sufficient and Necessary Conditions for CP Conservation in the Case of Degenerate Majorana Neutrino Masses

TL;DR

This paper systematically investigates the conditions for CP conservation in leptonic Majorana neutrino models, focusing on mass degeneracies and their impact on CP-violating phases, and explores the connection between low- and high-energy CP violation through renormalization-group equations.

Contribution

It provides a comprehensive analysis of CP conservation conditions in Majorana neutrino models with mass degeneracies and derives RG equations linking CP violation across energy scales.

Findings

Conditions for CP conservation in degenerate neutrino masses identified

CP-violating phases analyzed in partial and complete degeneracy scenarios

Renormalization-group equations connect low- and high-energy CP violation

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.