Symmetries and Dynamics of Nonleptonic Decays

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It is shown that the strength of the ${K}_{1}\ensuremath{\rightarrow}\mathrm{vacuum}\mathrm{vertex}$, arising solely through medium-strong $\mathrm{SU}(3)$-breaking interactions, is large and can provide a natural explanation for the octet enhancement of the parity-violating nonleptonic amplitudes. Furthermore, the sum rule $\ensuremath{-}A(\ensuremath{\Lambda}\ensuremath{\rightarrow}p+{\ensuremath{\pi}}^{\ensuremath{-}})+2A({\ensuremath{\Xi}}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\Lambda}+{\ensuremath{\pi}}^{\ensuremath{-}})=\sqrt{3}A({\ensuremath{\Sigma}}^{+}\ensuremath{\rightarrow}p+{\ensuremath{\pi}}^{0})$ for parity-violating hyperon decays, derived on the basis of the ${\ensuremath{\lambda}}_{6}$ 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 ${\ensuremath{\lambda}}_{7}$. Thus it is concluded that neither the forbiddenness of the ${K}_{1}$ tadpole in the limit of $\mathrm{SU}(3)$ nor the existence of the sum rule for the parity-violating decays on the basis of the ${\ensuremath{\lambda}}_{6}$ 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 ${K}_{1}\ensuremath{\rightarrow}2\ensuremath{\pi}$ decay), (ii) the meaning of enhancement of nonleptonic rates compared with leptonic ones, and (iii) the ratio of $\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{0})$ to $\ensuremath{\Gamma}({K}_{1}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}+{\ensuremath{\pi}}^{\ensuremath{-}})$.