ChiralSU(3)×SU(3)and the Decayη→3π. I

Abstract

The difficulties with the current-algebra treatment of the decay of the $\eta$ meson into three $\pi$ mesons are analyzed within the framework of a nonlinear Lagrangian calculation based on the chiral group $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$. The most general $\eta$-decay Lagrangian with the symmetry properties of the effective second-order electromagnetic interaction, and containing no more than two derivaties, is constructed and is shown to involve only one arbitrary parameter, the coupling constant $g$. The energy dependence, or slope, of the $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ matrix element depends only on the choice of representation for the $\mathrm{SU}\left(3\right)\mathrm{}$ and $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{}$ (collectively called medium-strong) symmetry-breaking terms. The simplest choice for these terms in a theory containing explicit partial conservation of axial-vector current — namely, a common $(3,\overline{3})+(\overline{3},3)$ representation for both — gives good agreement with experiment, Furthermore, exactly three parameters exist for adjusting the coupling constant and the electromagnetic mass differences $m_{{\pi }^{+}}^{}{}_{}{}^2-m_{{\pi }^0}^{}{}_{}{}^2$ and $m_{K^{+}}^{}{}_{}{}^2-m_{K^0}^{}{}_{}{}^2$ to their measured values. However, the coefficients of terms in the resulting phenomenological Lagrangian are unnaturally large; they give rise to the small mass differences only because of extensive cancellations (a condition which we refer to as a "scale difficulty"). The same techniques used for $\eta$ decay are also applied to the nonleptonic weak decays of the $K$ mesons. The usual current-algebra results are obtained as a check that our methods are exactly equivalent to the conventional ones.