Looking into Analytical Approximations for Three-flavor Neutrino Oscillation Probabilities in Matter

TL;DR

This paper derives simplified analytical formulas for three-flavor neutrino oscillation probabilities in matter, improving accuracy and applicability across various energies and baselines, with a novel parameterization involving the $ heta_{12}$ mixing angle.

Contribution

The paper introduces a new set of compact formulas for neutrino oscillations in matter, utilizing an $ extit{ extbf{η}}$-gauge mass-squared difference and simplifying the expressions for specific parameter choices.

Findings

Formulas are valid for arbitrary neutrino energies and baselines.

Analytical expressions are simplified for $ ext{η} = ext{cos}^2 heta_{12}$.

High accuracy maintained across different experimental conditions.

Abstract

Motivated by tremendous progress in neutrino oscillation experiments, we derive a new set of simple and compact formulas for three-flavor neutrino oscillation probabilities in matter of a constant density. A useful definition of the $\eta$-gauge neutrino mass-squared difference $\Delta^{}_* \equiv \eta \Delta^{}_{31} + (1-\eta) \Delta^{}_{32}$ is introduced, where $\Delta^{}_{ji} \equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 \leq \eta \leq 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $\alpha \equiv \Delta^{}_{21}/\Delta^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $\eta = \cos^2 \theta^{}_{12}$, with $\theta_{12}^{}$ being the solar mixing angle. As a by-product, the mapping from neutrino oscillation parameters in vacuum to their counterparts in matter is obtained at the order of ${\cal O}(\alpha^2)$. Finally, we show that our approximate formulas are not only valid for an arbitrary neutrino energy and any baseline length, but also still maintaining a high level of accuracy.