CP and T violation in neutrino oscillations and invariance of Jarlskogʹs determinant to matter effects

Terrestrial matter effects in neutrino propagation are T-invariant, so that any observed T violation when neutrinos pass through the Earth, such as an asymmetry between the transition probabilities P(νμ→νe) and P(νe→νμ), would be a direct indication of T violation at the fundamental level. Matter effects do however modify the magnitudes of T-violating asymmetries, and it has long been known that resonant enhancement can lead to large effects for a range of plausible values of the relevant parameters. We note that the determinant of the commutator of the lepton mass matrices is invariant under matter effects and use this fact to derive a new expression for the T-violating asymmetries of neutrinos propagating through matter. We give some examples which could have physical relevance.