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

We point out that the extension of the PCAC (partially conserved axial-vector current) relation ${\partial }_{\mu }{A_{\mu }}^{\pi }=C_{\pi }{\phi }^{\pi } \mathrm{to} {\partial }_{\mu }{A_{\mu }}^K=C_K{\phi }^K$ and the use of charge commutators typified by $A_K=[V_K,A_{\pi }]$ are useful in the study of broken $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry. The use of the ${\partial }_{\mu }{A_{\mu }}^K=C_K{\phi }^K$ condition usually confronts us with a considerable off-mass-shell extrapolation $m_K\to 0$. However, by using the above charge commutators and the approximation we propose, the off-mass-shell extrapolation $m_K\to 0$ may be replaced by a more comfortable one, $m_{\pi }\to 0$, effectively to first order in the symmetry-breaking interaction. This approach is applied to the study of the $\mathrm{SU}\left(3\right)\mathrm{}$ symmetry breaking. Encouraging results have been obtained in the case of $V\to P+P$ (i.e., $K^{*}\to K+\pi$ and $\rho \to \pi +\pi$) decays and in the direct determination of the $f-f^{\prime }$ mixing angle from their decay widths. We also make some estimate of the off-shell extrapolation $m_K\to 0$ compared with the case $m_{\pi }\to 0$. 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.