Short-distance contribution toK→ππγand theΔI=12rule

We analyze $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$ and $K_{L,S}^0\to {\pi }^{+}{\pi }^{-}\gamma$ decays in connection with the explanation of the $\Delta I=\frac{1}{2}$ rule proposed by Shifman, Vainshtein, and Zakharov. We ask whether the mechanism which enhances the $\Delta I=\frac{1}{2}$ transition can enhance $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$, thus contradicting the experimental observation $\Gamma (K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma ) \sim \left(\frac{\alpha }{\pi }\right)\Gamma (K^{\pm }\to {\pi }^{\pm }{\pi }^0)$. We show, within the same approximation scheme used in the $\Delta I=\frac{1}{2}$ rule analysis, that the $\Delta I=\frac{1}{2}$ rule enhancement is absent in the decay $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$.