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We report a first, complete lattice QCD calculation of the long-distancecontribution to the $K^+\to\pi^+\nu\bar{\nu}$ decay within the standard model.This is a second-order weak process involving two four-Fermi operators that ishighly sensitive to new physics and being studied by the NA62 experiment atCERN. While much of this decay comes from perturbative, short-distance physicsthere is a long-distance part, perhaps as large as the planned experimentalerror, which involves nonperturbative phenomena. The calculation presentedhere, with unphysical quark masses, demonstrates that this contribution can becomputed using lattice methods by overcoming three technical difficulties: (i)a short-distance divergence that results when the two weak operators approacheach other, (ii) exponentially growing, unphysical terms that appear inEuclidean, second-order perturbation theory, and (iii) potentially largefinite-volume effects. A follow-on calculation with physical quark masses andcontrolled systematic errors will be possible with the next generation ofcomputers.

Exploratory Lattice QCD Study of the Rare Kaon Decay $K^+\to\pi^+\nu\bar{\nu}$