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, (Δm)12(Δm)23(Δm)31Jm=Δ12Δ23Δ31J, was derived, where Δij=Δm2ij/(2E) and subscript m implies the quantities in matter. Using this relation, we reconsider how Jm changes as a function of the matter potential a under the approximation |Δm212|⪡|Δm213|. We show that the number of maxima for Jm depends on the magnitude of sin22θ13 and there are two maxima considering the constraint on sin22θ13 from the CHOOZ experiment. One maximum of Jm at a=O(Δ12) is given by J/sin2θ12, which leads to the large enhancement of Jm in the case of the SMA MSW solution. The other maximum at a=O(Δ13) is |Δ12/Δ13|J/sin2θ13, and the enhancement is possible, if sin2θ13 is small enough. These maximal values are consistent with the results obtained by other methods.