Analysis of K_L -> pi^+ pi^- gamma in Chiral Perturbation Theory

We study the long-distance dominated K_L -> pi^+ pi^- gamma decay at O(p^6) in Chiral Perturbation Theory. The complete calculation of the O(p^6) loop magnetic amplitude is carried out. At this chiral order the model dependent part of the vector meson exchange contribution to the magnetic amplitude is evaluated in the two models : FM (Factorization Model) and FMV (Factorization Model in the Vector Couplings). We predict, in an almost model independent way, a slope of K_L -> pi^+ pi^- gamma in the range c = -1.6,-1.8, consistently with the experimental value. We find that the experimental result for the width of K_L -> pi^+ pi^- gamma is compatible with a bigger breaking of the nonet symmetry in the weak vertices than previously stated. Thus we conclude that our analysis does not exclude an opposite sign to the one given by the dominance of the pion pole in the poorly known K_L -> gamma gamma amplitude. A complete analysis of the O(p^3) weak Vector-Pseudoscalar-Pseudoscalar (VPP) vertex is also performed.