ChiralSU(3)×SU(3)and the Decayη→3π. II

In the preceding paper (paper I) we examined the decay $\eta \to 3\pi$ in a very general way based on the current algebra of $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ and partial conservation of axial-vector current, but making use of nonlinear chiral Lagrangian techniques. Our results restated the failure of this approach to $\eta$ decay which was already known from the work of Sutherland, but in a new, quantitative form which we referred to as the "scale difficulty." In the present paper we continue to apply the same approach, the same techniques, and the same symmetry-breaking model to the same problem. In order to alleviate the scale difficulty, we investigate an extension of the theory in which $\eta$ decay is mainly due to a new term which is also contained in the symmetry-breaking $(3,\overline{3})+(\overline{3},3)$ representation. We find, firstly, that the new contribution to $\eta$ decay does not alter the good value for the slope. Secondly, using the soft-pion result for $m_{{\pi }^{+}}^{}{}_{}{}^2-m_{{\pi }^0}^{}{}_{}{}^2$ we can again uniquely determine the phenomenological Lagrangian, and we find that the scale difficulty disappears. Thirdly, we present a simple condition on the symmetry-breaking Lagrangian, plus a speculative test of that condition, that agrees with the new solution. A modification of the tadpole theory of Coleman and Glashow, in which the octet of tadpole mesons belongs to a nonlinear realization of $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$, is explored in the light of the new theory of $\eta$ decay. This new tadpole theory, unlike the old, has no difficulty in explaining octet dominance in the weak nonleptonic decays. It also explains why the non non- $(3,\overline{3})+(\overline{3},3)$ part of the pseudoscalar-meson Lagrangian is small compared to the $(3,\overline{3})+(\overline{3},3)$ part, as well as why it is predominantly octet. A new numerical fit to the tadpole theory is presented, which shows that the baryon Lagrangian must also be predominantly $(3,\overline{3})+(\overline{3},3)$.