Symmetric formulation of neutrino oscillations in matter and its intrinsic connection to renormalization-group equations

In this article, we point out that the effective Hamiltonian for neutrino oscillations in matter is invariant under the transformation of the mixing angle theta(12).-> theta(12) - pi/2 and the exchange of first two neutrino masses m(1) <-> m(2), if the standard parametrization of lepton flavor mixing matrix is adopted. To maintain this symmetry in perturbative calculations, we present a symmetric formulation of the effective Hamiltonian by introducing an.-gauge neutrino mass-squared difference Delta(*) equivalent to eta Delta(31) +(1 -eta) Delta(32) for 0 <= eta <= 1, where Delta(ji) equivalent to m(j)(2) - m(i)(2) for ji = 21, 31, 32, and show that only eta = 1 2, eta = cos(2) theta(12) or eta = sin(2)theta(12) is allowed. Furthermore, we prove that eta = cos(2) theta(12) is the best choice to derive more accurate and compact neutrino oscillation probabilities, by implementing the approach of renromalizationgroup equations. The validity of this approach becomes transparent when an analogy is made between the parameter. herein and the renormalization scale mu in relativistic quantum field theories.