Partially Conserved Axial-Vector Current, Charge Commutators, Off-Mass-Shell Correction, and the BrokenSU(3)Symmetry

Abstract

We point out that the extension of the PCAC (partially conserved axial-vector current) relation ${\ensuremath{\partial}}_{\ensuremath{\mu}}{{A}_{\ensuremath{\mu}}}^{\ensuremath{\pi}}={C}_{\ensuremath{\pi}}{\ensuremath{\varphi}}^{\ensuremath{\pi}} \mathrm{to} {\ensuremath{\partial}}_{\ensuremath{\mu}}{{A}_{\ensuremath{\mu}}}^{K}={C}_{K}{\ensuremath{\varphi}}^{K}$ and the use of charge commutators typified by ${A}_{K}=[{V}_{K},{A}_{\ensuremath{\pi}}]$ are useful in the study of broken $\mathrm{SU}(3)$ symmetry. The use of the ${\ensuremath{\partial}}_{\ensuremath{\mu}}{{A}_{\ensuremath{\mu}}}^{K}={C}_{K}{\ensuremath{\varphi}}^{K}$ condition usually confronts us with a considerable off-mass-shell extrapolation ${m}_{K}\ensuremath{\rightarrow}0$. However, by using the above charge commutators and the approximation we propose, the off-mass-shell extrapolation ${m}_{K}\ensuremath{\rightarrow}0$ may be replaced by a more comfortable one, ${m}_{\ensuremath{\pi}}\ensuremath{\rightarrow}0$, effectively to first order in the symmetry-breaking interaction. This approach is applied to the study of the $\mathrm{SU}(3)$ symmetry breaking. Encouraging results have been obtained in the case of $V\ensuremath{\rightarrow}P+P$ (i.e., ${K}^{*}\ensuremath{\rightarrow}K+\ensuremath{\pi}$ and $\ensuremath{\rho}\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}$) decays and in the direct determination of the $f\ensuremath{-}{f}^{\ensuremath{'}}$ mixing angle from their decay widths. We also make some estimate of the off-shell extrapolation ${m}_{K}\ensuremath{\rightarrow}0$ compared with the case ${m}_{\ensuremath{\pi}}\ensuremath{\rightarrow}0$. Another useful application of the above charge commutators is for the weak leptonic decays. We can derive a set of sum rules for the axial-vector coupling constants of the leptonic decays of hyperons which seem to give new insight into the Cabibbo theory of leptonic interactions.