CP violation and matter effect in long-baseline neutrino oscillation in the four-neutrino model

We investigate the CP violation effect and the matter effect in the long-baseline neutrino oscillations in the four-neutrino model with mass scheme of the two pairs of two close masses separated by a gap of the order of 1 eV by using the constraints on the mixing matrix derived from the solar neutrino deficit, atmospheric neutrino anomaly, LSND experiments, and the other terrestrial neutrino oscillation experiments. We also use the results of the combined analyses by Gonzalez-Garcia, Maltoni, and Pe~na-Garay of the solutions to the solar and atmospheric neutrino problems with the recent SNO solar neutrino data. For the solution of close-to-active solar neutrino oscillations plus close-to-sterile atmospheric neutrino oscillations, the pure CP violation part of the oscillation probability difference between the CP-conjugate channels could attain as large as $0.10-0.25$ in the neutrino energy range of $E=6-15$ GeV at the baseline $L=730$ km for ${\nu }_{\mu }\to {\nu }_{\tau }$ oscillation, and the matter effect is at the $8-15$ level of the pure CP violation effect, while for the solution of near-pure-sterile solar neutrino oscillations plus near-pure-active atmospheric neutrino oscillations, the pure CP violation effect in $\mathrm{\Delta }P({\nu }_{\mu }\to {\nu }_{\tau })$ is very small $(\sim 0.01)$ and is comparable to the matter effect. For ${\nu }_{\mu }\to {\nu }_e$ oscillation, the pure CP violation effect is independent of the active-sterile admixture and is at most 0.05 in $E=1.5-3$ GeV at $L = 290$kmandthemattereffectisatthe$15-30%$ level.