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

We report a first, complete lattice QCD calculation of the long-distance contribution to the $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ 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.

Figure 1

From top to bottom: quark and lepton contractions for $W\text{−}W$, connected and disconnected $Z$-exchange diagrams. The four dotted arrows point to possible locations for the $Z$-exchange vertex. The operator labels are defined in Ref. [6]. A few, illustrative gluon lines are also shown.

Figure 2

$W\text{−}W$ and $Z$-exchange diagram results, and their total, shown from left to the right. The gray bands show the amplitude, normalized as in Eq. (2), from the unrenormalized, bilocal operator. The red circles indicate the RI-renormalized, bilocal contribution. The blue diamonds give the total charm contribution $P_c$ while the green squares show the difference between the lattice and perturbative results, $P_c-P_c^{\mathrm{PT}}$.