Correlations of errors in measurements of CP violation at neutrino factories

Using $\Delta\chi^2$ which is defined by the difference of the number of events with the CP phase $\delta$ and the hypothetical one with $\delta=0$, we discuss correlations of errors of the CP phase and other oscillation parameters as well as the matter effect in measurements at neutrino factories. By varying the oscillation parameters and the normalization of the matter effect, we evaluated the data size required to reject a hypothesis with $\delta=0$ at 3$\sigma$CL. The optimum muon energy and the baseline depends on the magnitude of $\theta_{13}$, the background fraction, the uncertainty of the normalization of the matter effect, but in general lie in the ranges 20GeV $\lesssim E_\mu\lesssim$ 50GeV, 1000km$\lesssim L\lesssim$3000km. If we assume that the uncertainty of the matter effect is as large as 20% then the optimum values may be modified to $E_\mu\lesssim$ 10GeV, $L\lesssim$1000km due to the strong correlation of $\delta$ and the matter effect. We show analytically that sensitivity to CP violation is lost for $E_\mu\ll$ 10GeV or for $E_\mu\gg$ 50GeV. We also discuss the possibility of measuring CP violation at the upgraded JHF experiment by taking all the error correlations into account, and show that it is possible to demonstrate $\delta\ne0$ at 3$\sigma$CL for $\theta_{13}\gtrsim 3^\circ$.