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

Abstract

In this article, we point out that the effective Hamiltonian for neutrino oscillations in matter is invariant under the transformation of the mixing angle $θ^{}_{12} \to θ^{}_{12} - π/2$ 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 $η$-gauge neutrino mass-squared difference $Δ^{}_* \equiv ηΔ^{}_{31} + (1-η)Δ^{}_{32}$ for $0 \leq η\leq 1$, where $Δ^{}_{ji} \equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$, and show that only $η= 1/2$, $η= \cos^2θ^{}_{12}$ or $η= \sin^2 θ^{}_{12}$ is allowed. Furthermore, we prove that $η= \cos^2 θ^{}_{12}$ 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 $η$ herein and the renormalization scale $μ$ in relativistic quantum field theories.

12 pages, no figures, minor changes, to be published in the Special Issue "Emerging Leaders" in J. Phys. G