We study the restrictions imposed on chiral symmetry breaking terms by a set of postulates, whose main content is furnished by certain ideas of Michel and Radicati. We then test certain particular cases by utilizing an effective Lagrangian and calculating meson-meson scattering lengths. A convenient technique for calculating traces of matrices which appear in our model is presented in an appendix.
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The strong breaking of hadronic internal symmetry is described in a natural fashion, as shown by Weinberg [l], by giving the tensorial structure of the symmetry breaking terms of the Lagrangian under the chiral group. In the SU3+ x SU3-framework, the (3,3) + (5, 3) model of Gell-Mann, Oakes and Renner [2] is the simplest possible choice, the weak mass of the pion reflecting the proximity of SU2+ x SU2-invariance. Meanwhile, there are at present no clear tests of this model, and, moreover, the controversy on the sigma term in the pionnucleon scattering leaves some room for near SU3 invariance rather than SU2+ x SU2-invariance.
In this context, it seems interesting to investigate other possibilities, but if one wants to have a theory with a certain predictive power, one can choose either one term belonging to a representation other than (3, S) + (7, 3), (in this case (8, 8) is the most commonly proposed [3,4]), or a mixture of several representations, adding a criterion in order to reduce the number of arbitrary parameters.
In this paper we continue the study, started in a preceding work [5], of the chiral symmetry breaking terms from the point of view of Michel and Radicati [6]. Indeed, they have shown that the directions of the leptonic and electromagnetic interactions in SU3+ x SU3-x (1, C, P, CP) can be characterized by the notion of idempotent or nilpotent of an algebra defined on the (1, 8) + (8, 1) representation, or in a related manner by the notion of critical orbits they generate. They