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

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 ${\Delta }_{\ast }^{}\equiv \eta {\Delta }_{31}^{}+(1-\eta ){\Delta }_{32}^{}$ is introduced, where ${\Delta }_{ji}^{}\equiv m_j^2-m_i^2$ for $ji=21,31,32$ are the ordinary neutrino mass-squared differences and $0\le \eta \le 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $\alpha \equiv {\Delta }_{21}^{}/{\Delta }_{\ast }^{}$, 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 $\mathcal{O}({\alpha }^2)$. 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.

Comments: 34 pages, 5 figures; v2: reference added, to appear in JHEP