Self-Consistent Particle Sets in a Baryon Bootstrap Model

An $\mathrm{SU}\left(n\right)\mathrm{}$-symmetric bootstrap model is considered in which the baryons are meson-baryon bound states. A postulate used previously in similar models of mesons is applied: that a solution to the dynamical equations should exist in which particles of the same type are degenerate. It is shown that this postulate is difficult to satisfy unless the baryons correspond to a single irreducible representation of the $\mathrm{SU}\left(n\right)\mathrm{}$ group. The $\pi -N-N^{*}$ system is discussed in detail. The condition that the baryons correspond to a single irreducible representation does not favor a particular value of $n$ for the underlying $\mathrm{SU}\left(n\right)\mathrm{}$ symmetry, but does rule out the fundamental representation for the baryons, and favors the group $\mathrm{SU}\left(\mathrm{mn}\right)\mathrm{}$ over $\mathrm{SU}\left(m\right)\mathrm{}\otimes \mathrm{SU}\left(n\right)\mathrm{}$. Several relations between $\mathrm{SU}\left(4\right)\mathrm{}$ and $\mathrm{SU}\left(6\right)\mathrm{}$ symmetry schemes are pointed out.