Editing Cabibbo–Kobayashi–Maskawa matrix (section)

==The unitarity triangles==
The remaining constraints of unitarity of the CKM-matrix can be written in the form

:<math>\sum_k V_{ik}V^*_{jk} = 0 ~.</math>

For any fixed and different {{mvar|i}} and {{mvar|j}}, this is a constraint on three complex numbers, one for each {{mvar|k}}, which says that these numbers form the sides of a triangle in the [[complex plane]]. There are six choices of {{mvar|i}} and {{mvar|j}} (three independent), and hence six such triangles, each of which is called a ''unitary triangle''. Their shapes can be very different, but they all have the same area, which can be related to the [[CP violation|CP violating]] phase. The area vanishes for the specific parameters in the Standard Model for which there would be no [[CP violation]]. The orientation of the triangles depend on the phases of the quark fields.

A popular quantity amounting to twice the area of the unitarity triangle is the '''Jarlskog invariant''' (introduced by [[Cecilia Jarlskog]] in 1985), 
:<math> J = c_{12}c_{13}^2 c_{23}s_{12}s_{13}s_{23}\sin \delta \approx 3\cdot10^{-5} ~.</math> 
For Greek indices denoting up quarks and Latin ones down quarks, the 4-tensor <math>\;(\alpha,\beta;i,j)\equiv \operatorname{Im} (V_{\alpha i} V_{\beta j} V^*_{\alpha j} V_{\beta i}^{*}) \;</math> is doubly antisymmetric,
:<math>(\beta,\alpha;i,j) = -(\alpha,\beta;i,j)=(\alpha,\beta;j,i) ~.</math>
Up to antisymmetry, it only has {{nowrap| 9 {{=}} 3 × 3 }} non-vanishing components, which, remarkably, from the unitarity of {{mvar|V}}, can be shown to be ''all identical in magnitude'', that is,
:<math>
(\alpha,\beta;i,j)= J   ~  \begin{bmatrix} \;~~0 & \;~~1 & -1 \\ -1 & \;~~0 & \;~~1 \\ \;~~1 & -1 & \;~~0 \end{bmatrix}_{\alpha \beta} \otimes \begin{bmatrix} \;~~0 & \;~~1 & -1 \\ -1 & \;~~0 & \;~~1 \\ \;~~1 & -1 & \;~~0 \end{bmatrix}_{ij} \;,
</math>
so that 
:<math>J = (u,c;s,b) = (u,c;d,s) = (u,c;b,d) = (c,t;s,b) = (c,t;d,s) = (c,t;b,d)
 = (t,u;s,b) = (t,u;b,d) = (t,u;d,s) ~.</math>

Since the three sides of the triangles are open to direct experiment, as are the three angles, a class of tests of the Standard Model is to check that the triangle closes. This is the purpose of a modern series of experiments under way at the Japanese [[Belle experiment|BELLE]] and the American [[BaBar experiment|BaBar]] experiments, as well as at [[LHCb]] in CERN, Switzerland.