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

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 $η$-gauge neutrino mass-squared difference $Δ^{}_* \equiv ηΔ^{}_{31} + (1-η) Δ^{}_{32}$ is introduced, where $Δ^{}_{ji} \equiv m^2_j - m^2_i$ for $ji = 21, 31, 32$ are the ordinary neutrino mass-squared differences and $0 \leq η\leq 1$ is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of $α\equiv Δ^{}_{21}/Δ^{}_*$, we demonstrate that the analytical formulas can be remarkably simplified for $η= \cos^2 θ^{}_{12}$, with $θ_{12}^{}$ 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 ${\cal O}(α^2)$. 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.