Matter Enhancement of T Violation in Neutrino Oscillation

We study the matter enhancement of T violation in neutrino oscillation with three generations. The magnitude of T violation is proportional to Jarlskog factor J. Recently, the elegant relation, (\Delta_m)_{12}(\Delta_m)_{23}(\Delta_m)_{31}J_{m} = \Delta_{12}\Delta_{23}\Delta_{31}J, was derived, where \Delta_{ij}=\Delta m^2_{ij}/(2E) and subscript m implies the quantities in matter. Using this relation, we reconsider how J_m changes as a function of the matter potential a under the approximation |\Delta m^2_{12}| \ll |\Delta m^2_{13}|. We show that the number of maxima for J_{m} depends on the magnitude of \sin^2 2 \theta_{13} and there are two maxima considering the constraint on \sin^2 2\theta_{13} from the CHOOZ experiment. One maximum of $J_{m} at a = O(\Delta_{12}) is given by J/\sin2\theta_{12}, which leads to the large enhancement of J_m in the case of the SMA MSW solution. The other maximum at a = O(\Delta_{13}) is |\Delta_{12}/\Delta_{13}|J/\sin2\theta_{13}, and the enhancement is possible, if \sin 2 \theta_{13} is small enough. These maximal values are consistent with the results obtained by other methods.