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 $\tilde{\mathcal{J}}$ for leptonic CP violation in three-flavor neutrino oscillations in matter to its counterpart $\mathcal{J}$ 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$A^{}{\rm c} \equiv a/\Delta^{}{\rm c}$\mathrm{.}Here$\Delta^{}{ij} \equiv m^2_i - m^2_j$(for$ij = 21, 31, 32$)standfortheneutrinomass-squareddifferencesinvacuumand$\theta^{}{ij}$(for$ij = 12, 13, 23$) are the neutrino mixing angles in vacuum, while $\Delta^{}{\rm c} \equiv \Delta^{}{31}\cos^2\theta^{}{12} + \Delta^{}{32} \sin^2 \theta^{}{12}$andthematterparameter$a \equiv 2\sqrt{2}G^{}{\rm F} N^{}_e E$ 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 $\tilde{\mathcal{J}}$.