K → ππγ and chiral perturbation theory

Weak radiative decays KL,S → π+π-γ and K+ → π+π0γ are reexamined. The electro-magnetic 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 short-distance contribution to KL → 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/Λ2χ (Λχ being the chiral-symmetry breaking scale) the direct photon emission amplitude does not receive a contribution from penguin operators, and (5) the 1/Nc expansion improves the discrepancy between theory and experiment. (orig.)