A decomposition theorem for SU(n) and its application to CP-violation through quark mass diagonalisation

Abstract

It is proved that the group G=SU(n) has a decomposition G=FCF where F is a maximal abelian subgroup and C 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. The n=3 case is worked out in detail as an example of the constructive procedure furnished by the proof of the decomposition theorem.