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 $\theta^{}_{12} \to \theta^{}_{12} - \pi/2$ and the exchange of first two neutrino masses $m^{}_1 \leftrightarrow m^{}_2$, 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 $\eta$-gauge neutrino mass-squared difference $\Delta^{}_* \equiv \eta \Delta^{}_{31} + (1-\eta)\Delta^{}_{32}$ for $0 \leq \eta \leq 1$, where $\Delta^{}_{ji} \equiv m^2_j - m^2_i$ 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 renromalization-group equations. The validity of this approach becomes transparent when an analogy is made between the parameter $\eta$ herein and the renormalization scale $\mu$ in relativistic quantum field theories.