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

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 $\{\widetildeθ^{}_{12}, \widetildeθ^{}_{13}, \widetildeθ^{}_{23}, \widetildeδ\}$ 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 $μ$-$τ$ asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of $V$ are also obtained as a by-product.