Kaon radiative leptonic decays with a real or virtual photon from lattice QCD

We present a lattice-QCD determination of the form factors describing the kaon radiative leptonic decay $K^- \to \ell^- \bar{\nu}_\ell \gamma$ and the kaon four-lepton decay mediated by an off-shell photon ($\gamma^*$), $K^- \to \ell^- \bar{\nu}_\ell (\ell'^+ \ell'^-)_{\gamma^*}$. These form factors enter the structure-dependent part of the decay amplitudes and, together with the kaon decay constant, constitute the non-perturbative inputs necessary for the calculation of experimental observables. The form factors are computed from first principles using lattice QCD simulations with physical light and strange quark masses. Our calculation includes quark-disconnected contributions, going beyond the electroquenched approximation, and is accompanied by a comprehensive study of systematic uncertainties. Finite-volume effects are investigated using lattice spatial extents in the range $L \in [3.8,\,7.6]~\mathrm{fm}$, while discretization effects are estimated by employing lattice spacings in the interval $a \in [0.08, 0.057]~\mathrm{fm}$. For the radiative decay $K^- \to \ell^- \bar{\nu}_\ell \gamma$, we achieve a state-of-the-art precision for the form factors. For the four-lepton decay $K^- \to \ell^- \bar{\nu}_\ell \ell'^+ \ell'^-$, we present the first lattice determination at the physical point that includes all relevant systematic effects. Using the form factors obtained in this work, we compute the corresponding experimental observables and compare our results with available measurements.