K→ππγ and chiral perturbation theory

Weak radiative decays K_L,S→π⁺π^-γ and K⁺→π⁺π⁰γ are reexamined. The electromagnetic form factors and long-distance contributions to the direct photon emission are evaluated using the higher order effective chiral Lagrangian. We find that (1) the naive soft-pion theorem cannot be applied to the magnetic-type transition amplitude, (2) the shortdistance contribution to K_L→π⁺π^-γ is comparable to or even bigger than the long-distance one, (3) the ΔI=1/2 enhancement persists in the decay K⁺→π⁺π⁰γ, (4) to the order of 1/Λ_χ² (Λ_χ being the chiral-symmetry breaking scale) the direct photon emission amplitude does not receive a contribution from penguin operators, and (5) the 1/N_c expansion improves the discrepancy between theory and experiment.