Starting with the modern developments of current algebra and the hypothesis of partially conserved axial-vector current, it has gradually become apparent that the strong interactions are almost invariant under the group $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$. In the limit that symmetry breaking is neglected, $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ does not appear as a symmetry of the particle states as $\mathrm{SU}\left(3\right)\mathrm{}$ does, but rather as a symmetry realized by eight Goldstone bosons, i.e., the pseudoscalar octet. Most papers on $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ symmetry have been concerned with soft-meson theorems and their connection with effective Lagrangians. This paper is devoted to other aspects of the symmetry. Part of the paper is frankly pedagogical. The physics behind a symmetry realized by way of Goldstone bosons is brought out through a study of the $\sigma$ model. Then the general principles are stated abstractly and applied to the hadrons. One of the new results presented here is that there are two distinct ways in which $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$ can be realized. In both cases there is an octet of massless pseudoscalar mesons. The two possibilities differ in the residual symmetry of the hadron spectrum: In one case, it is only $\mathrm{SU}\left(3\right)\mathrm{}$; in the other, it is $\mathrm{SU}\left(3\right)\mathrm{}$ times a discrete symmetry, which leads to parity doublets. It is conjectured that some of the observed parity doubling in nucleon resonances is a consequence of this new discrete symmetry. Symmetry breaking is discussed in detail and is found to be very complex. In particular, it is shown that, at least for the pseudoscalar-meson masses, octet enhancement can never occur for first-order perturbations around an $\mathrm{SU}\left(3\right)\mathrm{}\otimes \mathrm{SU}\left(3\right)\mathrm{}$-symmetrical limit. Since octet enhancement is an empirical fact, one is forced to conclude that lowest-order perturbation theory is not a good approximation. In connection with octet enhancement, we show how one can use a principle of pole dominance in the angular momentum plane to replace scalar "tadpole" mesons with Regge trajectories.

• Received 9 January 1969

DOI:https://doi.org/10.1103/PhysRev.183.1245