On the Properties of the Effective Jarlskog Invariant for Three-flavor Neutrino Oscillations in Matter

In this paper, we show that the ratio of the effective Jarlskog invariant $\widetilde{\cal J}$ for leptonic CP violation in three-flavor neutrino oscillations in matter to its counterpart ${\cal J}$ in vacuum $\widetilde{\cal J}/{\cal J} \approx 1/(\hat{C}^{}_{12} \hat{C}^{}_{13})$ holds as an excellent approximation, where $\hat{C}^{}_{12} \equiv \sqrt{1 - 2 \hat{A}^{}_* \cos 2\theta^{}_{12} + \hat{A}^2_*}$ with $\hat{A}^{}_* \equiv a\cos^2 \theta^{}_{13}/\Delta^{}_{21}$ and $\hat{C}^{}_{13} \equiv \sqrt{1 - 2 A^{}_{\rm c} \cos 2\theta^{}_{13} + A^2_{\rm c}}$ with $A^{}_{\rm c} \equiv a/\Delta^{}_{\rm c}$. Here $\Delta^{}_{ij} \equiv m^2_i - m^2_j$ (for $ij = 21, 31, 32$) stand for the neutrino mass-squared differences in vacuum and $\theta^{}_{ij}$ (for $ij = 12, 13, 23$) are the neutrino mixing angles in vacuum, while $\Delta^{}_{\rm c} \equiv \Delta^{}_{31}\cos^2\theta^{}_{12} + \Delta^{}_{32} \sin^2 \theta^{}_{12}$ and the matter parameter $a \equiv 2\sqrt{2}G^{}_{\rm F} N^{}_e E$ are defined. This result has been explicitly derived by improving the previous analytical solutions to the renormalization-group equations of effective neutrino masses and mixing parameters in matter. Furthermore, as a practical application, such a simple analytical formula has been implemented to understand the existence and location of the extrema of $\widetilde{\cal J}$.