Self-Consistent Sets of Mesons in a Bootstrap Model

Consideration is given to the possibility that the self-consistency requirements of some dispersion-theoretic bootstrap model may specify uniquely the set of strongly-interacting particles found in nature. The discussion is based on a particular, approximate model of the $P$ (pseudoscalar) and $V$ (vector) mesons. In this model the $V$ exchange forces in the $P+P$ states produce the $V$ mesons, and the $V$ exchange forces in the $P+V$ states produce the $P$ mesons. Attention is limited to systems in which the particles arising in a particular way are degenerate or nearly degenerate, and are represented by a small number of irreducible representations of a simple Lie group of first, second, or third rank. Three plausible self-consistency requirements are postulated. The smallest meson set that satisfies all three postulates corresponds to the group S${\mathrm{U}}_3$. The predicted particles in this scheme are a $P$-meson octet, a $V$ octet, a $V$ singlet, and a singlet particle of spin and parity $2^{-}$. One of the self-consistency postulates is concerned with deviations from degeneracy, and leads to the Gell-Mann-Okubo sum rule for the S${\mathrm{U}}_3$ scheme. The double-septet scheme of the group $G_2$ does not satisfy any of the postulates.