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Recently, a complete set of differential equations for the effective neutrinomasses and mixing parameters in matter have been derived to characterize theirevolution with respect to the ordinary matter term $a \equiv 2\sqrt{2}G^{}_{\rmF} N^{}_e E$, in analogy with the renormalization-group equations (RGEs) forrunning parameters. Via series expansion in terms of the small ratio$\alpha^{}_{\rm c} \equiv \Delta^{}_{21}/\Delta^{}_{\rm c}$, we obtainapproximate analytical solutions to the RGEs of the effective neutrinoparameters and make several interesting observations. First, at the leadingorder, $\widetilde{\theta}^{}_{12}$ and $\widetilde{\theta}^{}_{13}$ are givenby the simple formulas in the two-flavor mixing limit, while$\widetilde{\theta}^{}_{23} \approx \theta^{}_{23}$ and $\widetilde{\delta}\approx \delta$ are not changed by matter effects. Second, the ratio of thematter-corrected Jarlskog invariant $\widetilde{\cal J}$ to its counterpart invacuum ${\cal J}$ approximates to $\widetilde{\cal J}/{\cal J} \approx1/(\widehat{C}^{}_{12} \widehat{C}^{}_{13})$, where $\widehat{C}^{}_{12} \equiv\sqrt{1 - 2 A^{}_* \cos 2\theta^{}_{12} + A^2_*}$ with $A^{}_* \equiva/\Delta^{}_{21}$ and $\widehat{C}^{}_{13} \equiv \sqrt{1 - 2 A^{}_{\rm c} \cos2\theta^{}_{13} + A^2_{\rm c}}$ with $A^{}_{\rm c} \equiv a/\Delta^{}_{\rm c}$.Finally, after taking higher-order corrections into account, we find compactand simple expressions of all the effective parameters.

Analytical solutions to renormalization-group equations of effective neutrino masses and mixing parameters in matter