Model Calculations and the Equal Mass Spacing of the Decuplet

Abstract

The response of the decuplet masses to octet perturbations of the meson and baryon masses and couplings is studied in an $\frac{N}{D}$ model of baryon-meson scattering. It is found that the equal mass spacing of the decuplet is satisfied in the model, even for large values of the symmetry breaking. The numerical inaccuracy of several forms of perturbation theory indicates that the physical baryon masses represent a large symmetry breaking in the calculation of the decuplet; for physical baryon and meson masses and $\mathrm{SU}(3)$-symmetric couplings, the first-order mass shift is about one-half the value given by the exact solution of the model. This suggests that the octet output of the masses is specially enhanced in the exact solution of the model, just as in the first-order Dashen-Frautschi theory. The rate at which the first-order formulas become accurate as the degenerate mass of the external particles is increased is examined. The octet sum rules of $S$-matrix elements break down quickly as the octet perturbations are increased.