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

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The decays K L → γγ and K L → ℓ +ℓ − γ are studied at the leading order p 6 in chiral perturbation theory. One-loop contributions stemming from the odd intrinsic parity | ΔS| = 1 effective Lagrangian of order p 4 are included and shown to be of possible relevance. They affect the decay K L → γγ adding to the usual pole terms a piece free of counterterm uncertainties. In the case of the K L → 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 K L → 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.