Flavor projection operators and applications toCPviolation with any number of families

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Abstract
Basis-independent flavor projection operators are constructed from the quark mass matrices in the standard electroweak model. These operators keep track of the identity of the quarks in any arbitrary basis for the mass matrices. The method is applied to a study of the basis-invariant conditions for CP symmetry with any number of families, where it is shown that the condition can immediately be written down in an invariant fashion, with the help of the projection operators. For the case of 4 families the CP conditions are worked out in detail and their symmetry properties are exhibited, with the use of the appropriate reduction formulas. The question of mass degeneracies, which is essential in discussing CP symmetry, is also studied. Finally, for the case of 3 families, an invariant measure of CP violation is given.

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