Cabibbo-Kobayashi-Maskawa matrix: parameterizations and rephasing invariants

TL;DR

This paper explores the properties of the CKM matrix, introduces a recursive parameterization method for any number of generations, generalizes the Wolfenstein parameterization to four generations, and analyzes rephasing invariants.

Contribution

It presents a recursive construction method for the CKM matrix applicable to any number of generations and characterizes all rephasing invariant monomials.

Findings

Generalized Wolfenstein parameterization to 4 generations

Identified 30 fundamental phase invariant monomials

Proved all rephasing invariants can be expressed with up to 5 factors

Abstract

The paper is devoted to a discussion of general properties of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. First we propose a general method of a recursive construction of the CKM matrix for any number of generations. This allows to set up 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. Motivated by the rephasing invariance of the CKM observables we next consider 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 the Main Theorem we show that all rephasing invariant monomials can be expressed as a product of at most 5 factors: 4 of them are fundamental phase invariant monomials and the fifth factor consists of powers of squares of absolute values of the CKM matrix elements.