The matter effect to T-violation at a neutrino factory

We analyzed T-violation in neutrino oscillation by using perturbation methods with respect to $\Delta m_{21}^2L/2E$ and $\delta a(x)L/2E$, where $\delta a(x)$ represents the matter density fluctuation from its average value. We found that the matter contribution to T-violation arises from interferences between $\Delta m_{21}^2L/2E$ and $\delta a(x) L/2E$. In the 2nd order, the symmetric and asymmetric matter density fluctuations give effects to the $\sin \delta$ (intrinsic) and the $\cos \delta$ (fake) parts of T-violation. We give their analytic forms and analyze the matter contribution to the $\sin \delta$ and $\cos \delta$ terms. We found that, for L=3000km, both the symmetric and asymmetric matter density fluctuations give negligible contributions to T-violation, and that thus the constant (average) matter density gives a good approximation. On the other hand, we argue that, for L=7000km or longer length, T-violation turns out to become very small due to cancellation between the 1st and the 2nd order terms. This shows that the constant (average) matter approximation is not valid.