Renormalization-Group Equations of Neutrino Masses and Flavor Mixing Parameters in Matter

TL;DR

This paper develops renormalization-group equations to describe how neutrino masses and mixing parameters evolve in matter, enabling the extrapolation of vacuum properties from matter-affected measurements in neutrino oscillation experiments.

Contribution

It derives the first complete set of differential equations for neutrino mixing parameters in matter, including matter-modified mixing angles and CP phase, based on a novel RGEs approach.

Findings

Formulated RGEs for neutrino mixing angles and CP phase in matter.

Identified invariants like Naumov and Toshev relations in matter.

Derived RGEs for unitarity triangle parameters.

Abstract

We borrow the general idea of renormalization-group equations (RGEs) to understand how neutrino masses and flavor mixing parameters evolve when neutrinos propagate in a medium, highlighting a meaningful possibility that the genuine flavor quantities in vacuum can be extrapolated from their matter-corrected counterparts to be measured in some realistic neutrino oscillation experiments. Taking the matter parameter $a \equiv 2\sqrt{2} \ G^{}_{\rm F} N^{}_e E$ to be an arbitrary scale-like variable with $N^{}_e$ being the net electron number density and $E$ being the neutrino beam energy, we derive a complete set of differential equations for the effective neutrino mixing matrix $V$ and the effective neutrino masses $\widetilde{m}^{}_i$ (for $i = 1, 2, 3$). Given the standard parametrization of $V$, the RGEs for $\{\widetilde{\theta}^{}_{12}, \widetilde{\theta}^{}_{13}, \widetilde{\theta}^{}_{23}, \widetilde{\delta}\}$ in matter are formulated for the first time. We demonstrate some useful differential invariants which retain the same form from vacuum to matter, including the well-known Naumov and Toshev relations. The RGEs of the partial $\mu$-$\tau$ asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of $V$ are also obtained as a by-product.