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Semisimple Lie algebras; Cartan decomposition; CP-violating phases; Kobayashi-Maskawa matrix; decomposition theorem; quark mass diagonalisation

It is proved that the groupG=SU(n) has a decompositionG=FCF whereF is a maximal abelian subgroup andC is an (n − 1)² parameter subset of matrices. The result is applied to the problem of absorbing the maximum possible number of phases in the mass-diagonalising matrix of the charged weak current into the quark fields; i.e., of determining the exact number of CP-violating phases for arbitrary number of generations. The inadequacies of the usual way of solving this problem are discussed. Then=3 case is worked out in detail as an example of the constructive procedure furnished by the proof of the decomposition theorem.

P P Divakaran¹ R Ramachandran¹

1. Tata Institute of Fundamental Research, Bombay - 400 005

Published

January 1980

Manuscript received

24 September 1979