Symmetries and Dynamics of Nonleptonic Decays

It is shown that the strength of the $K_1\to \mathrm{vacuum}\mathrm{vertex}$, arising solely through medium-strong $\mathrm{SU}\left(3\right)\mathrm{}$-breaking interactions, is large and can provide a natural explanation for the octet enhancement of the parity-violating nonleptonic amplitudes. Furthermore, the sum rule $-A(\Lambda \to p+{\pi }^{-})+2A({\Xi }^{-}\to \Lambda +{\pi }^{-})=\sqrt{3}A({\Sigma }^{+}\to p+{\pi }^0)$ for parity-violating hyperon decays, derived on the basis of the ${\lambda }_6$ transformation, also holds if the amplitudes are dominated by any or all of the following: the baryon octet, baryon decuplet, or scalar- or vector-octet pole terms, arising through the $K_1$ tadpole. This is true even though the latter transforms like ${\lambda }_7$. Thus it is concluded that neither the forbiddenness of the $K_1$ tadpole in the limit of $\mathrm{SU}\left(3\right)\mathrm{}$ nor the existence of the sum rule for the parity-violating decays on the basis of the ${\lambda }_6$ transformation provide any argument against a possible dynamical picture of octet enhancement in nonleptonic transitions. The dynamics considered in the present note should be important regardless of whether the octet transformation has a dynamical or primary origin. Comments are also made on some other directly related problems: (i) the possible effect of symmetry violation on an otherwise forbidden transition (especially $K_1\to 2\pi$ decay), (ii) the meaning of enhancement of nonleptonic rates compared with leptonic ones, and (iii) the ratio of $\Gamma (K^{+}\to {\pi }^{+}+{\pi }^0)$ to $\Gamma (K_1\to {\pi }^{+}+{\pi }^{-})$.