Baryon Mass Splittings in anSU(6)×O(3)Quark Model

Abstract

The positive- and negative-parity baryon mass shifts are investigated under the assumption that these particles belong, respectively, to the representations (56,1) and (70,3) of the group $\mathrm{SU}\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$, which are dynamically realizable from a 3-quark model with totally symmetric ($S$) functions, as found earlier by one of the authors. Two different types of $\mathrm{SU}\left(2\right)\mathrm{}$-invariant central forces $V^{\mathrm{}\left(1\right)\mathrm{}}$ and $V^{\mathrm{}\left(2\right)\mathrm{}}$, each of which is shown to be in conformity with the usual mass relations for the 56 states, are employed. One of these forces ($V^{\mathrm{}\left(2\right)\mathrm{}}$) is, however, found to violate the Gell-Mann-Okubo formula for certain negative-parity octets. It is also found that appreciable mixtures of both $V^{\mathrm{}\left(1\right)\mathrm{}}$ and $V^{\mathrm{}\left(2\right)\mathrm{}}$ are necessary even for a qualitative representation of the experimental masses. The effect of an $\mathrm{SU}\left(2\right)\mathrm{}$-invariant spin-orbit force of the type ${\tau }_1-{\tau }_2$ of modest strength (∼25 MeV) is found to be very helpful in producing a reasonably good fit to the actual masses of the negative-parity baryons. Such a force has, however, no first-order effect on the 56 masses, on the assumption of orbital $S$ functions, which can be constructed only with $S$-wave $Q-Q$ pairs. The significance of this result is briefly discussed in connection with the question of quark statistics.