Exploratory lattice QCD study of the rare kaon decay $K^+\to\pi^+\nu\bar{\nu}$

TL;DR

This paper demonstrates the feasibility of lattice QCD calculations for the long-distance contribution to the rare kaon decay $K^+ o o u ar{ u}$, using novel techniques on a small lattice with unphysical quark masses.

Contribution

It introduces new computational techniques for lattice QCD calculations of rare kaon decay amplitudes, focusing on long-distance effects, and shows their applicability in future realistic studies.

Findings

Feasibility of lattice QCD for long-distance decay amplitude components.

Implementation of novel techniques for such calculations.

Potential for future realistic computations with physical quark masses.

Abstract

In Ref [1] we have presented the results of an exploratory lattice QCD computation of the long-distance contribution to the $K^+\to\pi^+\nu\bar{\nu}$ decay amplitude. In the present paper we describe the details of this calculation, which includes the implementation of a number of novel techniques. The $K^+\to\pi^+\nu\bar{\nu}$ decay amplitude is dominated by short-distance contributions which can be computed in perturbation theory with the only required non-perturbative input being the relatively well-known form factors of semileptonic kaon decays. The long-distance contributions, which are the target of this work, are expected to be of O(5%) in the branching ratio. Our study demonstrates the feasibility of lattice QCD computations of the $K^+\to\pi^+\nu\bar{\nu}$ decay amplitude, and in particular of the long-distance component. Though this calculation is performed on a small lattice ($16^3\times32$) and at unphysical pion, kaon and charm quark masses, $m_\pi=420$ MeV, $m_K=563$ MeV and $m_c^{\overline{\mathrm{MS}}}(\mbox{2 GeV})=863$ MeV, the techniques presented in this work can readily be applied to a future realistic calculation.