On the properties of the effective Jarlskog invariant for three-flavor neutrino oscillations in matter

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/(Cˆ12Cˆ13) holds as an excellent approximation, where Cˆ12≡1−2Aˆ⁎cos⁡2θ12+Aˆ⁎2 with Aˆ⁎≡acos2⁡θ13/Δ21 and Cˆ13≡1−2Accos⁡2θ13+Ac2 with Ac≡a/Δc. Here Δij≡mi2−mj2 (for ij=21,31,32) stand for the neutrino mass-squared differences in vacuum and θij (for ij=12,13,23) are the neutrino mixing angles in vacuum, while Δc≡Δ31cos2⁡θ12+Δ32sin2⁡θ12 and the matter parameter a≡22GFNeE 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˜.