Cabibbo–Kobayashi–Maskawa matrix: rephasing invariants and parameterizations

Abstract

In this work we study two topics: the first one considers the general phase invariant monomials built out of the CKM matrix elements and their conjugates. We show, that there exist 30 fundamental phase invariant monomials and 18 of them are a product of 4 CKM matrix elements and 12 are a product of 6 CKM matrix elements. In our main result we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors with positive powers. Next, we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to construct a parameterization with desired properties. As an application we generalize the Wolfenstein parameterization to the case of 4 generations and obtain restrictions on the CKM suppression of the fourth generation.