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

Abstract

The decays $K_L \to γγ$ and $K_L \to \ell^+ \ell^- γ$ 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 $K_L \to γγ$ adding to the usual pole terms a piece free of counterterm uncertainties. In the case of the $K_L \to \ell^+ \ell^- γ$ 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 $K_L \to π^+ π^- γ$ 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.

10 pages, one table, two figures. One reference has been added and one typo in reference to Enagonio et al. has been corrected