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 2  − m 2 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 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 $$ mathcal{o}left({alpha}^2right) $$ . 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.