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

Abstract

Recently, a complete set of differential equations for the effective neutrino masses and mixing parameters in matter have been derived to characterize their evolution with respect to the ordinary matter term $a \equiv 2\sqrt{2}G^{}_{\rm F} N^{}_e E$, in analogy with the renormalization-group equations (RGEs) for running parameters. Via series expansion in terms of the small ratio $α^{}_{\rm c} \equiv Δ^{}_{21}/Δ^{}_{\rm c}$, we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, $\widetildeθ^{}_{12}$ and $\widetildeθ^{}_{13}$ are given by the simple formulas in the two-flavor mixing limit, while $\widetildeθ^{}_{23} \approx θ^{}_{23}$ and $\widetildeδ \approx δ$ are not changed by matter effects. Second, the ratio of the matter-corrected Jarlskog invariant $\widetilde{\cal J}$ to its counterpart in vacuum ${\cal J}$ approximates to $\widetilde{\cal J}/{\cal J} \approx 1/(\widehat{C}^{}_{12} \widehat{C}^{}_{13})$, where $\widehat{C}^{}_{12} \equiv \sqrt{1 - 2 A^{}_* \cos 2θ^{}_{12} + A^2_*}$ with $A^{}_* \equiv a/Δ^{}_{21}$ and $\widehat{C}^{}_{13} \equiv \sqrt{1 - 2 A^{}_{\rm c} \cos 2θ^{}_{13} + A^2_{\rm c}}$ with $A^{}_{\rm c} \equiv a/Δ^{}_{\rm c}$. Finally, after taking higher-order corrections into account, we find compact and simple expressions of all the effective parameters.

20 pages, 8 figures. More discussions added, to be published in JHEP