Running CP-violating Phases of Majorana Neutrinos

Abstract

Three CP-violating phases of the $3\times 3$ lepton flavor mixing matrix $V$, denoted as $δ$, $ρ$ and $σ$ in a useful phase convention, are entangled with one another in the one-loop renormalization-group equations. We show that $δ=90^\circ$ at the electroweak scale ($Λ_{\rm EW} \sim 10^{2}$ GeV) can be radiatively generated from $δ=0^\circ$ at the seesaw scale ($Λ_{\rm SS} \sim 10^{14}$ GeV) in the minimal supersymmetric standard model, provided three neutrino masses are nearly degenerate. As for the Majorana phases $ρ$ and $σ$, it is also possible to radiatively generate $ρ=90^\circ$ or $σ= 90^\circ$ at $Λ_{\rm EW}$ from $ρ=0^\circ$ or $σ= 0^\circ$ at $Λ_{\rm SS}$. 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.