Quark unitarity triangles

Abstract

The angles and the lengths of the sides of all unitarity triangles of the Cabibbo–Kobayashi–Maskawa (CKM) matrix are determined with very high precision from the experimental data. Our analysis is independent of parameterization of the CKM matrix and it is based on the predictions of the unitarity for the angles and the areas of the unitarity triangles. We note that the lengths of the sides of the three unitarity triangles determined from the experimental data do not form a triangle. We resolve this incompatibility by performing a constrained fit, assuming the equality of the area of the unitarity triangles. We demonstrate that the measured data are compatible with the predictions of the unitarity of the CKM matrix. We show that the angles of the unitarity triangles obtained by the multiplication of the rows of the CKM matrix can be obtained from the angles obtained by the multiplication of the columns. The equality of those two types of angles is a simple, but a very powerful test of the general structure of the Standard Model.