Measure of the size of CP violation in extended models

In this letter we introduce a possible measure of the size of CP violation in the Standard Model and its extensions, based on quantities invariant under the change of weak quark basis. We also introduce a measure of the " average size " of CP violation in a model, which can be used to compare the size of CP violation in models involving extra sequential or vector-like quarks, or left-right symmetry. The definition of " Maximal CP Violation " and the related problem of finding an adequate measure of the size of CP violation in the Standard Model (SM) has been object of interest [1, 2, 3, 4, 5] since experiment [6] revealed that the phase appearing in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [7] had to be large to explain the observed CP violation in the K 0-¯ K 0 system. As shown by Wolfenstein [1] any definition of maximal CP violation based on maximizing the phase appearing in the CKM matrix does not make sense because it depends on the parametrization of the CKM matrix itself. A phase of π/2 in one parametrization does not correspond to a phase of π/2 in another parametrization. Gronau and Schechter [2] used the construction of the CKM matrix as a product of three 2 × 2 unitary matrices (i. e. the Murnaghan construction) to find a certain combination of phases (the " invariant phase ") that remained invariant under most rephasings of the quark fields. However, this invariant phase depends on the adoption of the Murnaghan construction as well as on the order in which the 2 × 2 matrices are multiplied. The formulation in terms of rephasing invariants of the CKM matrix was carried out independently by Jarlskog [3, 4] and Dunietz, Greenberg and Wu [5] leading to two different definitions of measures. In Refs. [3, 4] the condition of maximality is that some of the quantities a CP = 2 Im (αβ *)/(|α| 2