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

TL;DR

This paper introduces a symmetric formulation of neutrino oscillations in matter using an $ extit{ exteta}$-gauge and links it to renormalization-group equations, improving the accuracy and simplicity of oscillation probability calculations.

Contribution

It proposes a symmetric effective Hamiltonian for neutrino oscillations in matter and connects it to renormalization-group equations, identifying optimal $ extit{ exteta}$-gauge choices.

Findings

The effective Hamiltonian is invariant under specific transformations.

Only certain $ extit{ exteta}$ values are physically allowed.

The $ extit{ exteta} = ext{ extcos}^2 heta_{12}$ choice yields the most accurate probabilities.

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 $\Delta^{}_* \equiv \eta \Delta^{}_{31} + (1-\eta)\Delta^{}_{32}$ 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.