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 ${\hat{C}}_{12}^{}\equiv \sqrt{1-2{\hat{A}}_{\ast }^{}\cos 2{\theta }_{12}^{}+{\hat{A}}_{\ast }^2}$ with ${\hat{A}}_{\ast }^{}\equiv a{\cos }^2{\theta }_{13}^{}/{\Delta }_{21}^{}$ and ${\hat{C}}_{13}^{}\equiv \sqrt{1-2A_{\mathrm{c}}^{}\cos 2{\theta }_{13}^{}+A_{\mathrm{c}}^2}$ with $A_{\mathrm{c}}^{}\equiv a/{\Delta }_{\mathrm{c}}^{}$. Here ${\Delta }_{ij}^{}\equiv m_i^2-m_j^2$ (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 ${\Delta }_{\mathrm{c}}^{}\equiv {\Delta }_{31}^{}{\cos }^2{\theta }_{12}^{}+{\Delta }_{32}^{}{\sin }^2{\theta }_{12}^{}$ 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 $\tilde{\mathcal{J}}$.

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