LAUSR

Abstract In this paper, we show that the ratio of the effective Jarlskog invariant J ˜ for leptonic CP violation in three-flavor neutrino oscillations in matter to its counterpart J in vacuum J ˜ / J ≈ 1 / ( C ˆ 12 C ˆ 13 ) holds as an excellent approximation, where C ˆ 12 ≡ 1 − 2 A ˆ ⁎ cos ⁡ 2 θ 12 + A ˆ ⁎ 2 with A ˆ ⁎ ≡ a cos 2 ⁡ θ 13 / Δ 21 and C ˆ 13 ≡ 1 − 2 A c cos ⁡ 2 θ 13 + A c 2 with A c ≡ a / Δ c . Here Δ i j ≡ m i 2 − m j 2 (for i j = 21 , 31 , 32 ) stand for the neutrino mass-squared differences in vacuum and θ i j (for i j = 12 , 13 , 23 ) are the neutrino mixing angles in vacuum, while Δ c ≡ Δ 31 cos 2 ⁡ θ 12 + Δ 32 sin 2 ⁡ θ 12 and the matter parameter a ≡ 2 2 G F N e E are defined. This result has been explicitly derived by improving the previous analytical solutions to the renormalization-group equations of effective neutrino masses and mixing parameters in matter. Furthermore, as a practical application, such a simple analytical formula has been implemented to understand the existence and location of the extrema of J ˜ .