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_∗≡ηΔ₃₁+(1−η)Δ₃₂ is introduced, where Δ_ji≡m_j²−m_i² for ji=21,31,32 are the ordinary neutrino mass-squared differences and 0≤η≤1 is a real and positive parameter. Expanding neutrino oscillation probabilities in terms of α≡Δ₂₁/Δ_∗, we demonstrate that the analytical formulas can be remarkably simplified for η=cos² θ₁₂, with θ₁₂ 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 O(α²). 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.