Renormalization of the neutrino mass matrix

The renormalization group equations for the general 2 by 2, complex, neutrino mass matrix are shown to have exact, analytic solutions. Simple formulas are given for the physical mixing angle, complex phase and mass ratio in terms of their initial values and the energy scales. We also establish a (complex) renormalization invariant relating these parameters. The qualitative features of the physical parameters' renormalization are clearly illustrated in vector field plots. In both the SM and MSSM, maximal mixing is a saddle point.