A phenomenological description of $K\to\pi\pi\gamma$ magnetic   transitions
  A phenomenological analysis of $K\to \pi\pi\gamma$ ($K_L\to\pi^+\pi^-\gamma$ and $K^+\to\pi^+\pi^0\gamma$) with the direct emission photon is carried out beyond the leading order in the chiral perturbation theory. We show that the experimental evidence for the large photon energy dependence of the magnetic amplitude in $K_L\to\pi^+\pi^-\gamma$ seems to indicate an interesting consequence: vector meson dominance must be implemented at $O(p^4)$, which is not a general feature of the chiral perturbation theory. The phenomenology of $K^+\to\pi^+\pi^0\gamma$ is also analyzed using the same scheme. 
Giancarlo D'Ambrosio and Dao-Neng Gao