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The rare kaon decays $K\to\pi\nu\bar{\nu}$ are strongly suppressed in thestandard model and widely regarded as processes in which new phenomena, notpredicted by the standard model, may be observed. Recognizing such newphenomena requires precise standard model prediction for the braching ratio of$K\to\pi\nu\bar{\nu}$ with controlled uncertainty for both short-distance andlong-distance contributions. In this work we demonstrate the feasibility oflattice QCD calculation of the long-distance contribution to rare kaon decayswith the emphasis on $K^+\to\pi^+\nu\bar{\nu}$. Our methodology covers thecalculation of both $W$-$W$ and $Z$-exchange diagrams. We discuss theestimation of the power-law, finite-volume corrections and two methods toconsistently combine the long distance contribution determined by the latticemethods outlined here with the short distance parts that can be reliablydetermined using perturbation theory. It is a subsequent work of our firstmethodology paper on $K\to\pi\ell^+\ell^-$, where the focus was made on the$\gamma$-exchange diagrams.

Prospects for a lattice computation of rare kaon decay amplitudes II $K\to\pi\nu\bar{\nu}$ decays