Renormalization-group equations of neutrino masses and flavor mixing parameters in matter

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 2 root 2G(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 (m)over-tilde(i)(for i = 1, 2, 3). Given the standard parametrization of V, the RGEs for {(theta)over-tilde(12), (theta)over-tilde(13), (theta)over-tilde(2)(3), (delta)over-tilde} 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.