= CKM unitarity triangle
{c}
{title2=$V_{ud}V_{ub}^*+V_{cd}V_{cb}^*+V_{td}V_{tb}^*=0$}
Column <orthogonality> of the <CKM matrix> gives three <complex numbers>
$$
a=V_{ud}V_{ub}^*,\qquad b=V_{cd}V_{cb}^*,\qquad c=V_{td}V_{tb}^*,\qquad a+b+c=0.
$$
Drawing these as successive vectors in the <complex plane> produces a closed <triangle>. Its area is $\tfrac12|\operatorname{Im}(ab^*)|=|J|/2$, 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 $J\simeq2.085597\times10^{-4}$. The unscaled <triangle> has area approximately $1.042799\times10^{-4}$.
Divide the closure relation by $-b$, assuming $b\ne0$, and define the <complex number>
$$
z=\bar\rho+i\bar\eta=-\frac{a}{b}.
$$
The normalized <triangle> has vertices $0,1,z$, with $z\simeq0.195376+0.335788i$ for these choices. Dividing each side by $|b|$ divides area by $|b|^2$: its area is $|J|/(2|V_{cd}V_{cb}^*|^2)=|\bar\eta|/2$, approximately $0.167894$. The familiar normalized picture therefore does not itself have area $|J|/2$.
\Image[/ckm-exact-unitarity-triangle.png]
{title=An exactly unitary illustrative CKM matrix closes a triangle}
{description=The <CKM unitarity triangle> is shown after normalization to a unit base; the two area formulas distinguish the original <triangle> from its rescaled version. The chosen parameters are illustrative.}
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.
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