  We present a comprehensive analysis of the contributions to K->pi nu nu
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(GF^2
p^2). Using this new chiral Lagrangian, we analyze the long-distance structure
of K->pi nu nu 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(GF^2
LambdaQCD^2) corrections enhance the SM prediction of Br(K+->pi+ nu nu) by
about 6%
