CPviolation versus the matter effect in long-baseline neutrino oscillation experiments

We investigate, within the framework of three generations of neutrinos, the effects of $\mathrm{CP}$ violation in long-baseline neutrino oscillation experiments. We aim at illuminating the global feature of the interplay between a genuine effect due to the $\mathrm{CP}$-violating phase and a fake one due to the Earth matter effect. To this goal, we develop a formalism based on the adiabatic approximation and perturbative treatment of the matter effect which allows us to obtain approximate analytic expressions of the oscillation probabilities. We present an order-of-magnitude estimation and a detailed numerical computation of the absolute and the relative magnitudes of the $\mathrm{CP}$ violations under the mass hierarchy suggested by the atmospheric neutrino anomaly and the cosmological dark matter. We find that the genuine $\mathrm{CP}$-violating effect is at most $\sim$1%, and the matter effect dominates over the intrinsic $\mathrm{CP}$ violation only in a region of parameters where the oscillation probability of ${\nu }_{\mu }\to {\nu }_e$ is large.