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We borrow the general idea of renormalization-group equations (RGEs) tounderstand how neutrino masses and flavor mixing parameters evolve whenneutrinos propagate in a medium, highlighting a meaningful possibility that thegenuine flavor quantities in vacuum can be extrapolated from theirmatter-corrected counterparts to be measured in some realistic neutrinooscillation 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 neutrinomixing 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 forthe first time. We demonstrate some useful differential invariants which retainthe same form from vacuum to matter, including the well-known Naumov and Toshevrelations. The RGEs of the partial $\mu$-$\tau$ asymmetries, the off-diagonalasymmetries and the sides of unitarity triangles of $V$ are also obtained as aby-product.

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