Symmetric Quark Model of Baryon Resonances

The symmetric quark model of baryon resonances is applied to the (56,$0^{+}$), (70,$1^{-}$), and (20,$1^{+}$) supermultiplets. Using a systematic $\mathrm{SU}\left(6\right)\mathrm{}$ analysis, octet dominance, and dominance of two-body contributions to the mass operator, the Gürsey-Radicati mass formula for the (56,$0^{+}$) is derived without use of perturbation theory. An equal-spacing relation is derived for the $\mathrm{SU}\left(6\right)\mathrm{}$-symmetric mass contribution for the (56,$0^{+}$), (70,$1^{-}$), and (20,$1^{+}$) in ascending order, with the (20,$1^{+}$) lying above 2 BeV. A detailed analysis of the (70,$1^{-}$), which yields a quantitative fit for the spin-orbit-split negative-parity resonances, is made, with the result that the magnitude of the octet spin-orbit term is about six times greater than the singlet one. The masses and mixing amplitudes for all the resonances in the (70,$1^{-}$) are calculated, and it is pointed out that because of the large mixings for $J^P={\frac{1}{2}}^{-} \mathrm{and} {\frac{3}{2}}^{-}$, the Gell-Mann-Okubo mass formula cannot be expected to hold for these resonances, so that there is no sense in trying to group them into octets and decuplets.