Is SU(3) × SU(3) Realized with Goldstone Bosons?

The linear $\sigma$ model is used to study how full chiral symmetry is realized as the (3,3*) + (3*,3) interaction is turned off. The Lagrangian which fits the observed spin-zero mass spectrum chooses either a normal realization with degenerate SU(3) × SU(3) multiplets or a Goldstone realization with an octet of massless pseudoscalar mesons, depending on whether the mass term of the Lagrangian is greater or less than a certain critical value. The value of the mass term depends sensitively on the masses of the $I=0\mathrm{}$ scalar resonances and present knowledge of these masses is compatible with either type of realization. However, the values of the nucleon $\sigma$ commutators calculated from meson-nucleon scattering data can resolve this issue. For the normal solution the $\pi -N\sigma$ commutator is predicted to be about 75 MeV, whereas for the Goldstone solution the prediction is 30 MeV. The symmetry-breaking parameters are $a=-0.91\mathrm{}$ and $b=-0.16\mathrm{}$ independent of how full symmetry is achieved.