Continuous and Discrete Symmetries of Renormalization Group Equations for Neutrino Oscillations in Matter

Three-flavor neutrino oscillations in matter can be described by three effective neutrino masses $\widetilde{m}^{}_i$ (for $i = 1, 2, 3$) and the effective mixing matrix $V^{}_{\alpha i}$ (for $\alpha = e, \mu, \tau$ and $i = 1, 2, 3$). When the matter parameter $a \equiv 2\sqrt{2} G^{}_{\rm F} N^{}_e E$ is taken as an independent variable, a complete set of first-order ordinary differential equations for $\widetilde{m}^2_i$ and $|V^{}_{\alpha i}|^2$ have been derived in the previous works. In the present paper, we point out that such a system of differential equations possesses both the continuous symmetries characterized by one-parameter Lie groups and the discrete symmetry associated with the permutations of three neutrino mass eigenstates. The implications of these symmetries for solving the differential equations and looking for differential invariants are discussed.