Correlations of errors in measurements ofCPviolation at neutrino factories

Using $\Delta {\chi }^2$ which is defined by the difference of the number of events with the $\mathrm{CP}$ phase $\delta$ and the hypothetical one with $\delta =0,$ we discuss correlations of errors of the $\mathrm{CP}$ 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 $\delta =0\mathrm{}$ at $3\sigma \mathrm{C}.\mathrm{L}.\mathrm{}$ The optimum muon energy and the baseline depend on the magnitude of ${\theta }_{13},$ the background fraction, and the uncertainty of the normalization of the matter effect, but in general lie in the range $20\mathrm{GeV}\lesssim E_{\mu }\lesssim 50\mathrm{GeV},$ $1000\mathrm{km}\lesssim L\lesssim 3000\mathrm{km}.$ If we assume that the uncertainty of the matter effect is as large as 20% then the optimum values may be modified to $E_{\mu }\lesssim 10\mathrm{GeV},$ $L\lesssim 1000\mathrm{km}$ due to the strong correlation of $\delta$ and the matter effect. We show analytically that sensitivity to $\mathrm{CP}$ violation is lost for $E_{\mu }\ll 10\mathrm{GeV}$ or for $E_{\mu }\gg 50\mathrm{GeV}.$ We also discuss the possibility of measuring $\mathrm{CP}$ violation at the upgraded JHF experiment by taking all the error correlations into account, and show that it is possible to demonstrate $\delta \ne 0$ at $3\sigma \mathrm{C}.\mathrm{L}.\mathrm{}$ for ${\theta }_{13}\gtrsim 3°.$