can be related to the 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, For Greek indices denoting up quarks and Latin ones down quarks, the 4-tensor is doubly antisymmetric, Up to antisymmetry, it only has non-vanishing components, which, remarkably, from the unitarity of , can be shown to be all identical in magnitude, that is, so that 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 and the American BaBar experiments, as well as at LHCb in CERN, Switzerland. Parameterizations Four independent parameters are required to fully define the CKM matrix. Many parameterizations have been proposed, and three of the most common ones are shown below. KM parameters The original parameterization of Kobayashi and Maskawa used three angles (, , ) and a CP-violating phase angle (). is the Cabibbo angle. Cosines and sines of the angles are denoted and , for respectively. "Standard" parameters A "standard" parameterization of the CKM matrix uses three Euler angles (, , ) and one CP-violating phase (). is the Cabibbo angle. Couplings between quark generations and vanish if . Cosines and sines of the angles are denoted and , respectively. The 2008 values for the standard parameters were: = , = , = and = radians = . Wolfenstein parameters A third parameterization of the CKM matrix was introduced by Lincoln Wolfenstein with the four parameters , , , and , which would all 'vanish' (would be zero) if there were no coupling. The four Wolfenstein parameters have the property that all are of order 1 and are related to the 'standard' parameterization: {| |- | | |- | | |- | | |} Although the Wolfenstein parameterization of the CKM matrix can be as exact as desired when carried to high order, it is mainly used for generating convenient approximations to the standard parameterization. The approximation to order , good to better than 0.3% accuracy, is: Rates of CP violation