Relativistic Quark Model Based on the Veneziano Representation. II. General Trajectories

The model previously proposed is extended to include multiquark trajectories. Once any trajectories with more than a single quark and antiquark are included, it is necessary to include trajectories where the number of quarks plus the number of antiquarks, which we call the total quark number, is arbitrarily large. The necessary factorization properties of the multiparticle Veneziano amplitudes will hold provided the intercept of the leading trajectory is a polynomial function of the total quark number, and the degeneracy of the levels on all but the leading trajectory will increase with the order of the polynomial. It is possible to construct two different models depending on whether one allows nonplanar duality diagrams. The model with nonplanar diagrams resembles more closely the nonrelativistic harmonic-oscillator quark model, and the nonplanar duality diagrams must be associated with the nonplanar Veneziano amplitudes discussed in a previous paper. One can introduce $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry-breaking by making the intercept depend on the number of strange and nonstrange quarks separately, and one then obtains a modified Gell-Mann-Okubo mass formula.