Matrix representation of symmetries in flavor space, invariant functions of mass matrices, and applications

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Abstract
A formalism suitable for studying the symmetry properties of the standard electroweak model in the flavor space is given. In this formalism the (quark) mass matrices play the leading role. It is shown that any amplitude in this model may be written in a matrix form in flavor space, whereby the amplitude contains two kinds of matrices: γ matrices which stand for the space-time properties and mass matrices which represent the flavor-symmetry properties. In this matrix representation the analytic expressions obtained for the amplitudes are concise and the flavor-symmetry properties are more transparent than they are usually, where the summations over internal flavors tend to obscure these symmetries. Furthermore, the concept of invariant functions of mass matrices is discussed and it is shown how the measurable quantities may be expressed as invariant functions. Examples given are cross sections, form factors, and measurables of the quark mixing matrix. Finally, the relationship of the present work and the commutator formalism for CP violation in the standard model with three families is discussed.