Light-quark loops in K->πνν-bar

We present a comprehensive analysis of the contributions to K->πνν-bar decays not described by the leading dimension-six effective Hamiltonian. These include both dimension-eight four-fermion operators generated at the charm scale, and genuine long-distance contributions which can be described within the framework of chiral perturbation theory. We show that a consistent treatment of the latter contributions, which turn out to be the dominant effect, requires the introduction of new chiral operators already at O(GF2p2). Using this new chiral Lagrangian, we analyze the long-distance structure of K->πνν-bar amplitudes at the one-loop level, and discuss the role of the dimension-eight operators in the matching between short- and long-distance components. From the numerical point of view, we find that these O(GF2ΛQCD2) corrections enhance the SM prediction of B(K+->π+νν-bar ) by about 6%