Complete Leading Order Analysis in Chiral Perturbation Theory of the Decays $K_L \to \gamma\gamma$ and $K_L \to \ell^+ \ell^- \gamma$

The decays $K_L\to \gamma\gamma$ and $K_L \to \ell^+ \ell^- \gamma$ are studied at the leading order p^6 in Chiral Perturbation Theory. One-loop contributions stemming from the odd intrinsic parity $\mid \Delta S\mid =1$ effective Lagrangian of order p^4 are included and shown to be of possible relevance. They affect the decay $K_L \to \gamma\gamma$ adding to the usual pole terms a piece free of counterterm uncertainties. In the case of the $K_L dilepton invariant mass requires a counterterm. The form factor may receive a sizeable contribution from chiral logarithms. Including considerations from the $K_L \to \pi^+ \pi^- \gamma$ 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