Enhancement of CP Violating terms for Neutrino Oscillation in Earth Matter

We investigate the $\nu_e \to \nu_{\mu}$ oscillation in the framework of three generations when neutrinos pass through the earth. The oscillation probability is represented by the form, $P(\nu_e \to \nu_\mu)=A\cos \delta+B\sin \delta+C$ in arbitrary matter profile by using the leptonic CP phase $\delta$. We compare our approximate formula in the previous paper with the formula which includes second order terms of $\alpha=\Delta m_{21}^2/\Delta m_{31}^2$ and $s_{13}=\sin \theta_{13}$. Non-perturbative effects of $\alpha$ and $s_{13}$ can be taken into account in our formula and the precision of the formula is rather improved around the MSW resonance region. Furthermore, we compare the earth matter effect of $A$ and $B$ with that of $C$ studied by other authors. We show that the magnitude of $A$ and $B$ can reach a few ten % of $C$ around the main three peaks of $C$ in the region $E>1$ GeV by numerical calculation. We give the qualitative understanding of this result by using our approximate formula. The mantle-core effect, which is different from the usual MSW effect, appears not only in $C$ but also in $A$ and $B$, although the effect is weakened.