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We investigate neutrino oscillations in constant matter within the context ofthe standard three neutrino scenario. We derive an exact and simple formula forthe oscillation probability applicable to all channels. In the standardparametrization, the probability for $\nu_e$ $\to$ $\nu_{\mu}$ transition canbe written in the form $P(\nu_e \to\nu_{\mu})=A_{e\mu}\cos\delta+B_{e\mu}\sin\delta+C_{e\mu}$ without anyapproximation using CP phase $\delta$. For $\nu_{\mu}$ $\to$ $\nu_{\tau}$transition, the linear term of $\cos 2\delta$ is added and the probability canbe written in the form $P(\nu_{\mu} \to\nu_{\tau})=A_{\mu\tau}\cos\delta+B_{\mu\tau}\sin\delta+C_{\mu\tau}+D_{\mu\tau}\cos 2\delta$. We give the CP dependences ofthe probability for other channels. We show that the probability for eachchannel in matter has the same form with respect to $\delta$ as in vacuum. Itmeans 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 arereproduced from the effective mixing angles and the effective CP phasecalculated by Zaglauer and Schwarzer. Through the calculation, a new identityis obtained by dividing the Naumov-Harrison-Scott identity by the Toshevidentity.Comment: 12 pages, RevTeX4 style, changed title, minor correction

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Exact formulas and simple<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>CP</mml:mi></mml:math>dependence of neutrino oscillation probabilities in matter with constant density

Exact formulas and simpleCPdependence of neutrino oscillation probabilities in matter with constant density