First exploratory calculation of the long-distance contributions to the rare kaon decays $K\to\pi\ell^+\ell^-$

TL;DR

This paper presents the first lattice QCD calculations of long-distance contributions to rare kaon decays into pions and lepton pairs, reducing theoretical uncertainties in these processes sensitive to new physics.

Contribution

It introduces a novel lattice QCD methodology to compute long-distance effects in $K o\pi\, ext{l}^+ ext{l}^-$ decays, providing initial numerical results.

Findings

Determined the form factor $V(z)$ at small $z$ values from lattice data.

Used domain wall fermion ensembles with specific pion and kaon masses.

First exploratory lattice study evaluating all but disconnected diagrams.

Abstract

The rare decays of a kaon into a pion and a charged lepton/antilepton pair proceed via a flavour changing neutral current and therefore may only be induced beyond tree level in the Standard Model. This natural suppression makes these decays sensitive to the effects of potential New Physics. The CP conserving $K\to\pi \ell^+\ell^-$ decay channels however are dominated by a single photon exchange; this involves a sizeable long-distance hadronic contribution which represents the current major source of theoretical uncertainty. Here we outline our methodology for the computation of the long-distance contributions to these rare decay amplitudes using lattice QCD and present the numerical results of the first exploratory studies of these decays in which all but the disconnected diagrams are evaluated. The domain wall fermion ensembles of the RBC and UKQCD collaborations are used, with a pion mass of $M_{\pi}\sim 430\,\mathrm{MeV}$ and a kaon mass of $M_{K}\sim 625\,\mathrm{MeV}$. In particular we determine the form factor, $V(z)$, of the $K^+\to\pi^+\ell^+\ell^-$ decay from the lattice at small values of $z=q^2/M_{K}^{2}$, obtaining $V(z)=1.37(36),\, 0.68(39),\, 0.96(64)$ for the three values of $z=-0.5594(12),\, -1.0530(34),\, -1.4653(82)$ respectively.