Pseudoscalar-Meson Mass Differences in a Self-Consistent Model

A bootstrap model of the $P$ (pseudoscalar) and $V$ (vector) meson octets is considered, in which the $P$ and $V$ are bound states or resonances in two-particle states of the type $\mathrm{PV}$ and $\mathrm{PP}$, respectively. Unitary symmetry is assumed. Several approximations are made in the many-channel, partial-wave dispersion relations of the model, in order that simple self-consistency relations among the various $P$ and $V$ meson mass ratios may be obtained. If it is required that the $P$ octet not be degenerate or nearly degenerate, near self-consistency can be obtained only if the pion mass is small compared to the $K$ and $\eta$ masses. It is argued that the $S{U}_3$ scheme of $P$, $V$, and baryon octets, and a $j^P={\frac{3}{2}}^{+}$ baryon decuplet, with mass splittings similar to those observed experimentally, is unusually well suited to a nondegenerate solution of a bootstrap model, because of the strong mutual coupling that exists among the lightest members of the various multiplets.