Dynamical Model of SU(3) for Baryons Based on a Gravitationally Bound Linear Rotator

A model of baryons is presented according to which each baryon consists of 3 real massive constituent particles (quarks or unitons) bound together by superstrong gravitational forces. Such strong gravitational forces arise because the free mass of each constituent particle is of the order of 1/2(ħc/G)1/2, whereG is the universal gravitational constant. Applying Lagrange’s analysis of the gravitational 3-body problem to this model, we obtain the straight-line configuration (linear rotator or gravitational string) as the only stable solution (the other solution is the equilateral triangle) that is consistent with the Pauli principle and with the observed data for the proton and neutron. This gravitational model accounts for the 3-quark saturation of baryons, since no stable gravitational configuration consisting of more than 3 particles exists. The very strong gravitational binding of the three quarks-of the order of 1019 GeV-accounts for the fact that no free quarks have been knocked out of baryons in even the most energetic scattering experiments performed thus far. With two of the bound quarks in our model in one rotational state (at the end points) and the third at the center of our linear configuration it is unnecessary to introduce para-Fermi statistics to satisfy the exclusion principle so that colored quarks are not required. If we introduce three distinct quarks p, n, λ, where n and λ have the same charge, but differ in strangeness, we obtain a total of 27 linear configurations by considering all possible triplet combinations. Only 18 of these survive as distinct however, since the interchange of the two end quarks in any one of the linear configurations gives no new configuration. We show that this group of 18 distinct configurations breaks up quite naturally into an octet and a decimet, so that a correct count of baryon states, with their correct quantum numbers, is a natural consequence of our model. Applying a straightforward first-order Bohr-type calculation to our rigid-rotator model for a nucleon, we obtain 1.02 · 10−13 cm for the nucleon radius and + 0.8 MeV for the neutron-proton mass difference, which is about e of the correct value. In the same way, assuming that both the p and n quarks haveg-factors of 1, we obtain +2.5 and −2.0 nuclear magnetons for the magnetic moments of the proton and neutron, respectively. If, however, theg-factor of the n-quark is 2 and that of the p-quark is 1, for which we present strong theoretical evidence, the two magnetic moments are +3.0 and −2.0 nuclear magnetons. We show that the matrix representation of theSU3-group for baryons can be factorized into a product of threeSU2 rotations. This leads to a natural dynamical interpretation of the baryon quantum numbers, if we assume that each of the three matrix factors represents the rotational degrees of freedom of one of the quarks in our dynamical model. In particular, we find that «strangeness» is related to the relativistic precession of the end quark in the strong gravitational field of the center quark. This conclusion is strengthened by two facts: 1) the strangeness quantum number enters into the Gell-Mann-Okubo mass formula in exactly the same way as the precession of an atom in a diatomic molecule enters into the energy level formula for the diatomic molecule, 2) the relativistic precession of the quark in our model leads us again to the quantization conditionGm2 ≈ħc.