The three flavor linear sigma model is studied as a “toy model” for understanding the role of possible light scalar mesons in the $\pi \pi ,$ $\pi K$ and $\pi \eta$ scattering channels. The approach involves computing the tree level partial wave amplitude for each channel and unitarizing by a simple K-matrix prescription which does not introduce any new parameters. If the renormalizable version of the model is used there is only one free parameter. While this highly constrained version has the right general structure to explain $\pi \pi$ scattering, it is “not quite” right. A reasonable fit can be made if the renormalizability (for the effective Lagrangian) is relaxed while chiral symmetry is maintained. The occurrence of a Ramsauer-Townsend mechanism for the $f_0\mathrm{}\left(980\right)\mathrm{}$ region naturally emerges. The effect of unitarization is very important and leads to “physical” masses for the scalar nonet all less than about 1 GeV. The $a_0\mathrm{}\left(1450\right)\mathrm{}$ and $K_0^{*}\mathrm{}\left(1430\right)\mathrm{}$ appear to be “outsiders” in this picture and to require additional fields. Comparison is made with a scattering treatment using a more general nonlinear sigma model approach. In addition some speculative remarks and a highly simplified larger toy model are devoted to the question of the quark substructure of the light scalar mesons.

• Received 20 December 2000

DOI:https://doi.org/10.1103/PhysRevD.64.014031