Unitarity condition method for global fits of the Cabibbo-Kobayashi-Maskawa matrix

Abstract

We report on an exact method for global fits of the CKM matrix by using the necessary and sufficient conditions the data have to satisfy in order to find a unitary matrix compatible with them, and this method can be applied to both quark and lepton CKM matrices. The key condition writes $-1\le\cos\delta\le 1$ where $\delta$ is the phase that encodes the CP violation, and it is obtained when one describes the CKM matrix in terms of four moduli of its entries. We use it to reconstruct many approximate unitary matrices from the PDG data, and by using the double stochasticity property of the unitary matrices we find a strong dependence of $\sin 2\beta$ on $|U_{ub}|$, that implies also strong correlations between $|U_{ij}|$ themselves, proving the constraining power of this approach.