Jarlskog invariant ($J$)

The Jarlskog invariant is a measure of CP violation that survives quantum field rephasings. Fix its sign by the quartetFor the exact three-generation CKM matrix, direct multiplication givesThe CKM unitarity triangle's illustrative parameters give $J=89991\sqrt{55}/3200000000\simeq 2.085597\times 10^{-4}$. Setting $s_{13}=0$ or $\delta =0$ makes $J=0$: the existence of three quark generations is necessary for this mechanism, but not sufficient for CP violation.

To see the invariance directly, the phases multiplying $V_{ud}{V}_{cs}{V}_{us}^{\ast }{V}_{cd}^{\ast }$ sum toUnder CP, the CKM matrix is complex conjugated and $J$ changes sign. In an exactly unitary three-generation CKM matrix, quartets formed from any distinct pair of rows and columns have imaginary parts equal to $J$ up to sign. This follows by multiplying a row orthogonality relation by a suitable pair of entries and taking its imaginary part: its diagonal product is real, forcing the two remaining imaginary parts to cancel. The same Jarlskog invariant therefore determines the areas of all six unscaled CKM unitarity triangles.