Violation of|ΔI|=12Rule in theKπ30Decay and theK20→γ+γDecay

The experimental value of the branching ratio $\frac{\Gamma ({K_2}^0\to {\pi }^{+}+{\pi }^{-}+{\pi }^0)}{\Gamma (K^{+}\to {\pi }^{+}+{\pi }^{-}+{\pi }^0)}$ obtained by Alexander et al. differs from the prediction of the $\Delta I=\frac{1}{2}$ rule by 30%. We assume that the major part of the violation of the $\Delta I=\frac{1}{2}$ rule in $K_{\pi 3}$ decay comes from the chain of processes $K^0\to \eta \to {\pi }^{+}+{\pi }^{-}+{\pi }^0$, as pointed out by Bouchiat et al. With this assumption, we get a relation between the coupling constant $\frac{{f^2}_{K^0{\eta }^0}}{4\pi }$ and the width $\tau (\eta \to {\pi }^{+}+{\pi }^{-}+{\pi }^0)$. We use the experimental value of the branching ratio $\frac{\tau (\eta \to \mathrm{neutrals})}{\tau (\eta \to {\pi }^{+}+{\pi }^{-}+{\pi }^0)}\approx 3.0$ to infer the unknown branching ratio $\frac{\tau ({\eta }^0\to 2\gamma )}{\Gamma ({\eta }^0\to {\pi }^{+}+{\pi }^{-}+{\pi }^0)}$ and the theory of unitary symmetry to deduce the value of the width $\tau ({\eta }^0\to 2\gamma )$ from the experimental decay rate of ${\pi }^0\to 2\gamma$. Further, if only pion and $\eta$-meson pole terms are responsible for the mass difference $m\left({K_1}^0\right)-m\left({K_2}^0\right)$, we can also derive the value of the coupling constant $\frac{{f^2}_{K^0{\pi }^0}}{4\pi }$. These two coupling constants $\frac{{f^2}_{K^0{\eta }^0}}{4\pi }$ and $\frac{{f^2}_{K^0{\pi }^0}}{4\pi }$ are illustrated as a function of the deviation $x$ from the $\Delta I=\frac{1}{2}$ rule. Under the same assumption we obtain a result that $\Gamma ({K_2}^0\to 2\gamma )$ is comparable to $\Gamma ({K_2}^0\to {\pi }^{+}+{\pi }^{-}+{\pi }^0)$ or is negligibly small according as $f_{K^0{\pi }^0}$ and $f_{K^0{\eta }^0}$ are of the same sign or not. If the deviation $x=30\%$ is really correct, the decay rate of ${\Sigma }^{-}\to \eta +{\pi }^{-}$ can be explained with $\frac{{g^2}_{{\Sigma }^{0}n{K}^0}}{4\pi }\lesssim 1$. On the other hand, if the extended $\Delta I=\frac{1}{2}$ rule is valid for the couplings of $K-\pi$ and $K-\eta$, we obtain the following results: $x=15\%$, $m\left({K_1}^0\right)<m\left({K_2}^0\right)$, $\frac{{f^2}_{K^0{\pi }^0}}{4\pi }=2.8\mathrm{}\times {10}^{-16\mathrm{}}$, $\Gamma ({K_2}^0\to 2\gamma )=8\mathrm{}\times {10}^3$ ${\sec }^{-1\mathrm{}}$ and the decay rate ${\Sigma }^{-}\to n+{\pi }^{-}$ can be fitted with $\frac{{g^2}_{{\Sigma }^{0}n{K}^0}}{4\pi }\approx 6$. Precise experiments are desired on the decay ${K_2}^0\to 2\gamma$ as well as on the decay ${K_2}^0\to {\pi }^{+}+{\pi }^{-}+{\pi }^0$.