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 ${\theta }_{12}^{}\to {\theta }_{12}^{}-\pi /2$ 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 ${\Delta }_{\ast }^{}\equiv \eta {\Delta }_{31}^{}+(1-\eta ){\Delta }_{32}^{}$ for $0\le \eta \le 1$, where ${\Delta }_{ji}^{}\equiv m_j^2-m_i^2$ 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 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.

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