Rare kaon decays $K^- \to \ell^- \bar{\nu}_\ell \ell'^{+} \ell'^{-}$: Standard Model predictions from lattice QCD

TL;DR

This paper provides Standard Model predictions for rare charged kaon decays involving lepton pairs, based on lattice QCD calculations, and discusses their potential for precision tests of the SM and determination of the Cabibbo angle.

Contribution

First complete lattice QCD calculation of structure-dependent form factors for rare kaon decays, enabling precise SM predictions and extraction of CKM matrix element |V_{us}|.

Findings

Predicted branching fractions with 2-7% uncertainties.

Results agree with existing measurements for three decay modes.

Predicted decay rate for $K_{\mu2\mu}$ matches preliminary experimental data.

Abstract

Weak decays of charged kaons with an additional lepton-antilepton pair, $K^- \to \ell^- \bar{\nu}_\ell \ell'^{+} \ell'^{-}$ ($K_{\ell2\ell'}$), are suppressed at order $O(G_{F}^{2}\alpha_{\rm em}^{2})$ in the Standard Model (SM) and provide sensitive probes of its flavour structure, as well as independent determinations of the Cabibbo angle $|V_{us}|$. In this Letter we present the SM predictions for all four channels with $\ell,\ell' =e,\mu$, based on the first complete lattice QCD calculation of the structure-dependent form factors reported in a companion paper [1]. Using the PDG value [2] $|V_{us}|^{\rm PDG}=0.22431(85)$, we obtain branching fractions with controlled uncertainties and precisions ranging from $2\%$ to $7\%$, depending on the channel. For the three modes with published measurements, our results agree with experiment. For the $K_{\mu2\mu}$ mode, for which no published experimental result is available, we compare our prediction with the preliminary NA62 result, finding agreement at the $1.4\sigma$ level. Conversely, the measured decay rates can be used together with our results to extract $|V_{us}|$ from these modes. A weighted average over the two most precise channels, $K_{\mu2e}$ and $K_{\mu2\mu}$, yields $|V_{us}|=0.2283(42)$, corresponding to a $1.8\%$ determination. These results pave the way for using $K_{\ell2\ell'}$ decays as precision probes of the SM.