Bosonic Operator Methods for the Quark Model

The non-relativistic quark model has been a useful tool in the study of hadrons. Baryons and mesons are described by quantum mechanical wavefunctions for nonrelativistic constituent quarks. The lowest lying baryons, the 81/2 and 103/2, are three quark states with wavefunctions which are completely antisymmetric in color, and completely symmetric in position and spin-flavor. The properties of baryons can be computed in a systematic expansion in powers of 1/Nc, where Nc = 3 is the number of colors in QCD. There is an exact SU(6) symmetry for baryons in the Nc → ∞ limit of QCD, which is broken by 1/Nc corrections [1]. The systematic application of the 1/Nc expansion requires computing properties of SU(6) baryon representations [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The non-relativistic quark model also has SU(6) symmetry, and is a convenient way to keep track of SU(6) group theory. Applications of the quark model to compute baryon matrix elements, and to compute SU(6) Clebsch-Gordan coefficients, require evaluting quark operator matrix elements between baryon states. It is useful to have an efficient means of computing these quark model matrix elements. The traditional approach has been to write down color ⊗ spin ⊗ flavor wavefunctions, and use them to compute matrix elements [12, 13]. The wavefunction is completely antisymmetric in color, with the quark color indices contracted using ǫabc. As a result, the three quarks must enter as one of each of the three colors, and the wavefunction is written by ordering the quarks by their color index. As an example, the proton (Jz = 1/2) wavefunction is