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

<p>We apply the general idea of renormalization-group equations (RGEs) to understand how neutrino masses and flavor mixing parameters evolve when neutrinos propagate in a medium. Taking the matter parameter <em>a</em>≡22√<em>G</em>F<em>N</em><em>e</em><em>E</em> to be an arbitrary scale-like variable , we derive a complete set of differential equations for the effective neutrino mixing matrix <em>V</em> and the effective neutrino masses <em>m</em><em>i</em> (for <em>i</em>=1,2,3) without any {\it a priori} phase convention or assumption. In the standard parametrization of <em>V</em> the RGEs for the effective mixing angles and the CP-violating phase in matter are presented. 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 <em>μ</em>-<em>τ</em> asymmetries, the off-diagonal asymmetries and the sides of unitarity triangles of <em>V</em> are also obtained as a by-product.