Radiative Generation of Leptonic CP Violation

Abstract

Three CP-violating phases of the $3\times 3$ lepton flavor mixing matrix $V$ are entangled with one another in the renormalization-group evolution from the seesaw scale ($\Lambda_{\rm SS} \sim 10^{14}$ GeV) to the electroweak scale ($\Lambda_{\rm EW} \sim 10^2$ GeV). Concerning the Dirac phase $\delta$, we show that $\delta =90^\circ$ at $\Lambda_{\rm EW}$ can be radiatively generated from $\delta =0^\circ$ at $\Lambda_{\rm SS}$ in the minimal supersymmetric standard model, if 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 $\Lambda_{\rm SS}$. The one-loop renormalization-group equations for the Jarlskog invariant and two off-diagonal asymmetries of $V$ are derived, and their running behaviors from $\Lambda_{\rm SS}$ to $\Lambda_{\rm EW}$ are numerically illustrated.