Exact Formulas and Simple CP dependence of Neutrino Oscillation Probabilities in Matter with Constant Density

Abstract

We investigate neutrino oscillations in constant matter within the context of the standard three neutrino scenario. We derive an exact and simple formula for the oscillation probability applicable to all channels. In the standard parametrization, the probability for $ν_e$ $\to$ $ν_μ$ transition can be written in the form $P(ν_e \to ν_μ)=A_{eμ}\cosδ+B_{eμ}\sinδ+C_{eμ}$ without any approximation using CP phase $δ$. For $ν_μ$ $\to$ $ν_τ$ transition, the linear term of $\cos 2δ$ is added and the probability can be written in the form $P(ν_μ \to ν_τ)=A_{μτ}\cosδ+B_{μτ} \sinδ+C_{μτ}+D_{μτ}\cos 2δ$. We give the CP dependences of the probability for other channels. We show that the probability for each channel in matter has the same form with respect to $δ$ as in vacuum. It means that matter effects just modify the coefficients $A$, $B$, $C$ and $D$. We also give the exact expression of the coefficients for each channel. Furthermore, we show that our results with respect to CP dependences are reproduced from the effective mixing angles and the effective CP phase calculated by Zaglauer and Schwarzer. Through the calculation, a new identity is obtained by dividing the Naumov-Harrison-Scott identity by the Toshev identity.