Complete leading order analysis in chiral perturbation theory of the decays KL → γγ and KL → ℓ+ℓ−γ

The decays KL → γγ and KL → ℓ+ℓ−γ are studied at the leading order p6 in chiral perturbation theory. One-loop contributions stemming from the odd intrinsic parity |ΔS| = 1 effective Lagrangian of order p4 are included and shown to be of possible relevance. They affect the decay KL → γγ adding to the usual pole terms a piece free of counterterm uncertainties. In the case of the KL → ℓ+ℓ−γ decays the dependence of the form factor on the dilepton invariant mass requires a counterterm. The form factor may receive a sizeable contribution from chiral logarithms. Including considerations from the KL → π+π−γ direct emission amplitude, we obtain two consistent scenarios. In one scenario the long distance contributions from the one-loop terms are important, while in the other they are marginal. In both cases the counterterm is shown to be significant.