Jarlskog invariant
= Jarlskog invariant
{c}
{title2=$J$}
{wiki}
The <Jarlskog invariant> is a measure of <CP violation> that survives <quantum field rephasings>. Fix its sign by the quartet
$$
J=\operatorname{Im}(V_{ud}V_{cs}V_{us}^*V_{cd}^*).
$$
For the exact three-generation <CKM matrix>, direct multiplication gives
$$
J=c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta.
$$
The <CKM unitarity triangle>'s illustrative parameters give $J=89991\sqrt{55}/3200000000\simeq2.085597\times10^{-4}$. 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 $V_{ud}V_{cs}V_{us}^*V_{cd}^*$ sum to
$$
(-\alpha_u+\beta_d)+(-\alpha_c+\beta_s)+(\alpha_u-\beta_s)+(\alpha_c-\beta_d)=0.
$$
Under <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>.