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

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 \nu_e \to \nu_\mu can be written in the form P(\nu_e \to \nu_\mu) =A_{e\mu} cos \delta + B_{e\mu} sin \delta + C_{e\mu} without any approximation using the CP phase \delta. This result holds not only in constant matter but also in arbitrary matter. Another probability for \nu_\mu \to \nu_\tau can be written in the form P(\nu_\mu \to \nu_\tau)= A_{\mu\tau} cos \delta + B_{\mu\tau} sin \delta

• C_{\mu\tau} + D_{\mu\tau} cos 2\delta + E_{\mu\tau} sin 2\delta. The term which is proportional to sin 2\delta disappear, namely E_{\mu\tau}=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 \delta and cos \delta as in the above two channels. In symmetric matter, the oscillation probability for each channel reduces to the same form with respect to \delta as that in constant matter.