SU(3) Clebsch-Gordan coefficients with definite permutation symmetry

Abstract

Multiquark wave functions representing physical particles must not only have the correct transformation properties under the Poincaré, SU(3) flavor and SU(3) color groups, but must also be antisymmetric under the interchange of quark labels induced by these groups. Under the interchange of just space-time, flavor, or color labels, the multiquark wave functions may be of mixed permutation symmetry, as long as the effect of interchanging all types of labels is antisymmetric. It is shown how to construct multiquark wave functions as irreducible representations of SU(3) (or more generally any compact group) that have a definite permutation symmetry. The multiquark wave functions are realized as polynomials in a generalized Bargmann space, and it is shown that the Clebsch-Gordan coefficients arising in the n-fold tensor product decomposition can be obtained by differentiating these polynomials in a prescribed way. Tables are given for two to six multiquark states with definite SU(3) and permutation symmetry. The question of decoupling these multiquark states into quark-antiquark or three-quark states times ocean states is also discussed.