 CKM unitarity triangle

Column orthogonality of the CKM matrix gives three complex numbersDrawing these as successive vectors in the complex plane produces a closed triangle. Its area is $\frac{1}{2}|\mathrm{Im}(a{b}^{\ast })|=|J|/2$, because unitarity identifies this imaginary quartet with a signed Jarlskog invariant. A nonzero area therefore detects physical CP violation for nondegenerate quark masses.

For a worked example, choose the exact parametrization of the CKM matrix with $s_{12}=1/4$, $s_{23}=1/10$, $s_{13}=1/100$ and $\delta =\pi /3$. These are illustrative choices, not fitted quark mixing parameters. They give $J\simeq 2.085597\times 10^{-4}$. The unscaled triangle has area approximately $1.042799\times 10^{-4}$.

Divide the closure relation by $-b$, assuming $b\ne 0$, and define the complex numberThe normalized triangle has vertices $0,1,z$, with $z\simeq 0.195376+0.335788i$ for these choices. Dividing each side by $|b|$ divides area by $|b{|}^2$: its area is $|J|/(2|V_{cd}{V}_{cb}^{\ast }{|}^2)=|\bar{\eta }|/2$, approximately $0.167894$. The familiar normalized picture therefore does not itself have area $|J|/2$.

Under quantum field rephasings, $a,b,c$ all acquire the same phase, so the original triangle merely rotates and $z$ stays fixed. Ratios and angles can thus compare quark mixing measurements without dependence on field conventions. If the Jarlskog invariant vanishes, the triangle collapses; additional quark mass degeneracies require extra care in interpreting the converse.