Lattice QCD calculation of $K\to \ell\nu_\ell \ell'^+ \ell'^-$ decay width

TL;DR

This paper develops a lattice QCD methodology to compute the decay width of the rare kaon decay process $K o ext{lepton} u ext{lepton} ext{lepton}^+ ext{lepton}^-$, achieving results consistent with experiments and ChPT.

Contribution

It introduces a novel lattice QCD approach with scalar function and infinite volume reconstruction methods for this decay process.

Findings

Decay widths consistent with experimental data.

Method reduces systematic errors like finite-volume effects.

Demonstrates lattice QCD's potential for precise Standard Model predictions.

Abstract

We develop a methodology for the computation of the $K\to \ell\nu_\ell \ell'^+ \ell'^-$ decay width using lattice QCD and present an exploratory study here. We use a scalar function method to account for the momentum dependence of the decay amplitude and adopt the infinite volume reconstruction method to reduce the systematic errors such as the temporal truncation effects and the finite-volume effects. We then perform a four-body phase-space integral to obtain the decay width. The only remaining technical problem is the possible power-law finite-volume effects associated with the process of $K\to\pi\pi \ell\nu_\ell\to \ell\nu_\ell \ell'^+ \ell'^-$, where the intermediate state involves multiple hadrons. In this work, we use a gauge ensemble of twisted mass fermion with a pion mass $m_\pi=352$ MeV and a nearly-physical kaon mass. At this kinematics, the $\pi\pi$ in the intermediate state cannot be on shell simultaneously as $2m_\pi>m_K$ and the finite-volume effects associated with $\pi\pi$ state are exponentially suppressed. Using the developed methods mentioned above, we calculate the branching ratios for four channels of $K\to \ell\nu_\ell\ell'^+ \ell'^-$, and obtain the results comparable to the experimental measurements and ChPT predictions. Our work demonstrates the capability of lattice QCD to improve Standard Model prediction in $K\to \ell\nu_\ell \ell'^+ \ell'^-$ decay width.