Toy Model for Two Chiral Nonets

Motivated by the possibility that nonets of scalar mesons might be described as mixtures of two quark and four quark components, we further study a toy model in which corresponding chiral nonets (containing also the pseudoscalar partners) interact with each other. Although the two quark and four quark chiral fields transform identically under SU(3)_L \times SU(3)_R transformations they transform differently under the U(1)_A transformation which essentially counts total (quark + antiquark) content of the mesons. To implement this we formulate an effective Lagrangian which mocks up the U(1)_A behavior of the underlying QCD. We derive generating equations which yield Ward identity type relations based only on the assumed symmetry structure. This is applied to the mass spectrum of the low lying pseudoscalars and scalars. as well as their excitations . Assuming isotopic spin invariance, it is possible to disentangle the amount of two quark vs. four quark content in the pseudoscalar \pi, K ,\eta type states and in the scalar \kappa type states. It is found that a small four quark content in the lightest pseudoscalars is consistent with a large four quark content in the lightest of the scalar \kappa mesons. The present toy model also allows one to easily estimate the strength of a four quark vacuum condensate. There seems to be a rich and interesting structure