  We report a first, complete lattice QCD calculation of the long-distance
contribution 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 is
highly sensitive to new physics and being studied by the NA62 experiment at
CERN. While much of this decay comes from perturbative, short-distance physics
there is a long-distance part, perhaps as large as the planned experimental
error, which involves nonperturbative phenomena. The calculation presented
here, with unphysical quark masses, demonstrates that this contribution can be
computed using lattice methods by overcoming three technical difficulties: (i)
a short-distance divergence that results when the two weak operators approach
each other, (ii) exponentially growing, unphysical terms that appear in
Euclidean, second-order perturbation theory, and (iii) potentially large
finite-volume effects. A follow-on calculation with physical quark masses and
controlled systematic errors will be possible with the next generation of
computers.
