Overall Feature of CP dependence for Neutrino Oscillation Probability in Arbitrary Matter Profile

Abstract

We study the CP dependence of neutrino oscillation probability for all channels in arbitrary matter profile within three generations. We show that an oscillation probability for ν_e \to ν_μcan be written in the form P(ν_e \to ν_μ) =A_{eμ} cos δ+ B_{eμ} sin δ+ C_{eμ} without any approximation using the CP phase δ. This result holds not only in constant matter but also in arbitrary matter. Another probability for ν_μ\to ν_τcan be written in the form P(ν_μ\to ν_τ)= A_{μτ} cos δ+ B_{μτ} sin δ + C_{μτ} + D_{μτ} cos 2δ+ E_{μτ} sin 2δ. The term which is proportional to sin 2δdisappear, namely E_{μτ}=0, in symmetric matter. It means that the probability reduces to the same form as in constant matter. As for other channels, probabilities in arbitrary matter are at most the quadratic polynomials of sin δand cos δas in the above two channels. In symmetric matter, the oscillation probability for each channel reduces to the same form with respect to δas that in constant matter.