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

Abstract

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 $\tilde{\mathcal{J}}/\mathcal{J}\approx 1/({\hat{C}}_{12}^{}{\hat{C}}_{13}^{})$ holds as an excellent approximation, where ${\hat{C}}_{12}^{}\equiv \sqrt{1-2{\hat{A}}_{\ast }^{}\cos 2\hat{I}{¸}_{12}^{}+{\hat{A}}_{\ast }^2}$ with ${\hat{A}}_{\ast }^{}\equiv a{\cos }^2\hat{I}{¸}_{13}^{}/\hat{I}{”}_{21}^{}$ and ${\hat{C}}_{13}^{}\equiv \sqrt{1-2A_{\mathrm{c}}^{}\cos 2\hat{I}{¸}_{13}^{}+A_{\mathrm{c}}^2}$ with $A_{\mathrm{c}}^{}\equiv a/\hat{I}{”}_{\mathrm{c}}^{}$. Here $\hat{I}{”}_{ij}^{}\equiv m_i^2-m_j^2$ (for $ij=21,31,32$) stand for the neutrino mass-squared differences in vacuum and $\hat{I}{¸}_{ij}^{}$ (for $ij=12,13,23$) are the neutrino mixing angles in vacuum, while $\hat{I}{”}_{\mathrm{c}}^{}\equiv \hat{I}{”}_{31}^{}{\cos }^2\hat{I}{¸}_{12}^{}+\hat{I}{”}_{32}^{}{\sin }^2\hat{I}{¸}_{12}^{}$ and the matter parameter $a\equiv 2\sqrt{2}{G}_{\mathrm{F}}^{}{N}_e^{}E$ 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 $\tilde{\mathcal{J}}$.

16 pages, 3 figures. More discussions and references added, to be published in NPB