General Treatment of the Breaking of Chiral Symmetry and Scale Invariance in theSU(3) σModel

We study the model of nine scalar and nine pseudoscalar fields interacting by means of the most general nonderivative chiral ${\pi }^{+}$ invariant and any particular symmetry-breaking term. Two basic "generating" equations which express the complete content of chiral symmetry are derived. The masses and coupling constants of arbitrary order for this model are simply found by differentiating the "generating" equations an arbitrary number of times and using an equation which expresses the stability of the "ground" state. In this way, previous results on this model can be easily recaptured and a systematic framework for the investigation of different symmetry-breaking terms is provided. Numerical estimates are made for scalar-meson widths. $\pi \pi$ and $\pi K$ scattering lengths. and ${\eta }^{\prime }\to \eta 2\pi$ decays. The consequences of imposing scale invariance on the invariant part of the interaction are also investigated by writing down a scale invariance "generating equation." Finally, we discuss the relation between our approach and the method of using the divergences of currents and trace of the energy-momentum tensor.