Running CP-violating Phases of Majorana Neutrinos

Abstract

Three CP-violating phases of the $3\times 3$ lepton flavor mixing matrix $V$, denoted as $\delta$, $\rho$ and $\sigma$ in a useful phase convention, are entangled with one another in the one-loop renormalization-group equations. We show that $\delta =90^\circ$ at the electroweak scale ($\Lambda_{\rm EW} \sim 10^{2}$ GeV) can be radiatively generated from $\delta =0^\circ$ at the seesaw scale ($\Lambda_{\rm SS} \sim 10^{14}$ GeV) in the minimal supersymmetric standard model, provided three neutrino masses are nearly degenerate. As for the Majorana phases $\rho$ and $\sigma$, it is also possible to radiatively generate $\rho =90^\circ$ or $\sigma = 90^\circ$ at $\Lambda_{\rm EW}$ from $\rho =0^\circ$ or $\sigma = 0^\circ$ at $\Lambda_{\rm SS}$. This observation opens a new and interesting window for model building, in order to understand possible connection between the phenomena of CP violation at low- and high-energy scales.