Symmetries of Baryons and Mesons

Abstract

The system of strongly interacting particles is discussed, with electromagnetism, weak interactions, and gravitation considered as perturbations. The electric current ${j}_{\ensuremath{\alpha}}$, the weak current ${J}_{\ensuremath{\alpha}}$, and the gravitational tensor ${\ensuremath{\theta}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ are all well-defined operators, with finite matrix elements obeying dispersion relations. To the extent that the dispersion relations for matrix elements of these operators between the vacuum and other states are highly convergent and dominated by contributions from intermediate one-meson states, we have relations like the Goldberger-Treiman formula and universality principles like that of Sakurai according to which the $\ensuremath{\rho}$ meson is coupled approximately to the isotopic spin. Homogeneous linear dispersion relations, even without subtractions, do not suffice to fix the scale of these matrix elements; in particular, for the nonconserved currents, the renormalization factors cannot be calculated, and the universality of strength of the weak interactions is undefined. More information than just the dispersion relations must be supplied, for example, by field-theoretic models; we consider, in fact, the equal-time commutation relations of the various parts of ${j}_{4}$ and ${J}_{4}$. These nonlinear relations define an algebraic system (or a group) that underlies the structure of baryons and mesons. It is suggested that the group is in fact $U(3)\ifmmode\times\else\texttimes\fi{}U(3)$, exemplified by the symmetrical Sakata model. The Hamiltonian density ${\ensuremath{\theta}}_{44}$ is not completely invariant under the group; the noninvariant part transforms according to a particular representation of the group; it is possible that this information also is given correctly by the symmetrical Sakata model. Various exact relations among form factors follow from the algebraic structure. In addition, it may be worthwhile to consider the approximate situation in which the strangeness-changing vector currents are conserved and the Hamiltonian is invariant under $U(3)$; we refer to this limiting case as "unitary symmetry." In the limit, the baryons and mesons form degenerate supermultiplets, which break up into isotopic multiplets when the symmetry-breaking term in the Hamiltonian is "turned on." The mesons are expected to form unitary singlets and octets; each octet breaks up into a triplet, a singlet, and a pair of strange doublets. The known pseudoscalar and vector mesons fit this pattern if there exists also an isotopic singlet pseudoscalar meson ${\ensuremath{\chi}}^{0}$. If we consider unitary symmetry in the abstract rather than in connection with a field theory, then we find, as an attractive alternative to the Sakata model, the scheme of Ne'eman and Gell-Mann, which we call the "eightfold way"; the baryons $N$, $\ensuremath{\Lambda}$, $\ensuremath{\Sigma}$, and $\ensuremath{\Xi}$ form an octet, like the vector and pseudoscalar meson octets, in the limit of unitary symmetry. Although the violations of unitary symmetry must be quite large, there is some hope of relating certain violations to others. As an example of the methods advocated, we present a rough calculation of the rate of ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}+\ensuremath{\nu}$ in terms of that of ${\ensuremath{\pi}}^{+}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}+\ensuremath{\nu}$.