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

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.