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

## TL;DR

This paper derives approximate analytical solutions to the renormalization-group equations for effective neutrino masses and mixing parameters in matter, revealing simple formulas and key behaviors of these parameters under matter effects.

## Contribution

The paper provides the first approximate analytical solutions to the RGEs of neutrino parameters in matter using series expansion, clarifying their behavior and relations.

## Key findings

- Leading order formulas for f0f1 and f0f3 are simple and two-flavor like.
- Matter effects do not significantly alter f0f2f1 and f0f2f3 at leading order.
- The ratio f0f4f1/f0f4f3 of Jarlskog invariants approximates to 1/(f0f4f4f1 f0f4f3f1) under matter effects.

## 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 $\alpha^{}_{\rm c} \equiv \Delta^{}_{21}/\Delta^{}_{\rm c}$, we obtain approximate analytical solutions to the RGEs of the effective neutrino parameters and make several interesting observations. First, at the leading order, $\widetilde{\theta}^{}_{12}$ and $\widetilde{\theta}^{}_{13}$ are given by 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 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 $\widehat{C}^{}_{12} \equiv \sqrt{1 - 2 A^{}_* \cos 2\theta^{}_{12} + A^2_*}$ with $A^{}_* \equiv a/\Delta^{}_{21}$ and $\widehat{C}^{}_{13} \equiv \sqrt{1 - 2 A^{}_{\rm c} \cos 2\theta^{}_{13} + A^2_{\rm c}}$ with $A^{}_{\rm c} \equiv a/\Delta^{}_{\rm c}$. Finally, after taking higher-order corrections into account, we find compact and simple expressions of all the effective parameters.