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 η-gauge neutrino mass-squared difference Δ∗ ≡ ηΔ31 + (1 − η)Δ32 is introduced, where Δji≡mj2 − mi2 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 = cos2θ12, 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.