Partially Conserved Axial-Vector Current and the NonleptonicK-Meson Decays

Current commutation relations and a partially conserved axial-vector current hypothesis (PCAC) are applied to the nonleptonic $K$ decays. On the assumption that the effective weak interactions should have the transformation property suggested by Gell-Mann in the quark model of the chiral symmetry, and that the axial-vector isospin charges to which the pions are related through PCAC should be generators of $U\left(2\right)\mathrm{}\times U\left(2\right)\mathrm{}$, the $K_{2\pi }$ and the $K_{3\pi }$ decay amplitudes, in which the four-momenta of all the final pions are continued to zero, are proved to obey the $\Delta I=\frac{1}{2}$ rule. In the approximation of neglecting the continuation, the decay rate of the $K_{3\pi }$ is related to that of the $K_{2\pi }$. The ratio of the decay rates is estimated and a reasonable agreement with experiment is obtained. Only the charge independence is assumed for the strong interactions.