Field theory and the nonrelativistic quark model: A parametrization of the baryon magnetic moments and masses

We show that the results of calculations of physical quantities (e.g., the magnetic moments and the masses) in a relativistic field theory can be parametrized in a way typical of the nonrelativistic quark model (NRQM). For a relativistic field theory that, as QCD, satisfies the two properties that (a) the electromagnetic current is carried only by the quarks and (b) flavor breaking is due only to the mass difference between the quarks, the most general expression of the magnetic moments of the baryon octet contains ten different types of terms and therefore ten parameters. If one neglects terms that break flavor at an order higher than the first, the ten terms reduce to seven and these are determined from the seven known magnetic moments. We find that (1) among the seven parameters determined by fitting exactly the data, the two parameters characteristic of the simplest NRQM description are indeed the largest ones (their values, in spite of the presence of five more parameters, are almost unchanged with respect to the NRQM), and (2) the ${\Sigma }^0$→Λγ rate (to first order in flavor breaking) is related to the magnetic moments of the octet baryons; the prediction is consistent with the present data on the rate. For the masses, in the flavor-breaking approximation to first order, we obtain a five-parameter formula containing the Gell-Mann–Okubo relation of the octet and the two equal-spacing relations of the decuplet. Comparing this general mass formula with that obtained with the NRQM (with the potential between quarks used by De Rújula, Georgi, and Glashow) it turns out that (1) one of the above five parameters does not intervene in the NRQM (it happens to be by far the smallest among the five) and (2) noting that in the NRQM the ratio between two of our parameters is related to the mass difference Δm between the λ and scrP quarks, the ratio Δm/$m_{\lambda }$ determined in this way is 0.31; that obtained from the magnetic moments is 0.35. These results, and others on the M1 and E2 Δ→Pγ transition, show that the NRQM provides an approximation to the exact solution correct to ≃15%.