Radiative generation of leptonic $CP$ violation

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 (${\Lambda }_{\mathrm{SS}}\sim {10}^{14}$ GeV) to the electroweak scale (${\Lambda }_{\mathrm{EW}}\sim {10}^2$ GeV). Concerning the Dirac phase $\delta$, we show that $\delta =90°$ at ${\Lambda }_{\mathrm{EW}}$ can be radiatively generated from $\delta =0°$ at ${\Lambda }_{\mathrm{SS}}$ 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 $\rho =90°$ or $\sigma =90°$ at ${\Lambda }_{\mathrm{EW}}$ from $\rho =0°$ or $\sigma =0°$ at ${\Lambda }_{\mathrm{SS}}$. The one-loop renormalization-group equations for the Jarlskog invariant and two off-diagonal asymmetries of $V$ are derived, and their running behaviors from ${\Lambda }_{\mathrm{SS}}$ to ${\Lambda }_{\mathrm{EW}}$ are numerically illustrated.

Figure 1

The RGE running behaviors of three $CP$-violating phases and three rephasing-invariant quantities of $V$ from ${\Lambda }_{\mathrm{SS}}$ to ${\Lambda }_{\mathrm{EW}}$ in the MSSM, where the input values of relevant parameters can be found from Table .

Figure 2

The radiative generation of $\delta =90°$ at ${\Lambda }_{\mathrm{EW}}$ from ${\theta }_{13}=1°$ and different values of $\rho$ and $\sigma$ at ${\Lambda }_{\mathrm{SS}}$. The input values of other parameters can be found from Table .

Figure 3

The RGE running behaviors of three $CP$-violating phases and three rephasing-invariant quantities of $V$ from ${\Lambda }_{\mathrm{SS}}$ to ${\Lambda }_{\mathrm{EW}}$ in the MSSM, where the input values of relevant parameters can be found from Table .

Figure 4

The RGE running behaviors of three $CP$-violating phases and three rephasing-invariant quantities of $V$ from ${\Lambda }_{\mathrm{SS}}$ to ${\Lambda }_{\mathrm{EW}}$ in the MSSM, where the input values of relevant parameters can be found from Table .

Figure 5

The RGE running behaviors of three $CP$-violating phases and three rephasing-invariant quantities of $V$ from ${\Lambda }_{\mathrm{SS}}$ to ${\Lambda }_{\mathrm{EW}}$ in the MSSM, where the input values of relevant parameters can be found from Table .