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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 studyhere. We use a scalar function method to account for the momentum dependence ofthe decay amplitude and adopt the infinite volume reconstruction method toreduce the systematic errors such as the temporal truncation effects and thefinite-volume effects. We then perform a four-body phase-space integral toobtain the decay width. The only remaining technical problem is the possiblepower-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 stateinvolves multiple hadrons. In this work, we use a gauge ensemble of twistedmass 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 shellsimultaneously as $2m_\pi>m_K$ and the finite-volume effects associated with$\pi\pi$ state are exponentially suppressed. Using the developed methodsmentioned above, we calculate the branching ratios for four channels of $K\to\ell\nu_\ell\ell'^+ \ell'^-$, and obtain the results comparable to theexperimental measurements and ChPT predictions. Our work demonstrates thecapability of lattice QCD to improve Standard Model prediction in $K\to\ell\nu_\ell \ell'^+ \ell'^-$ decay width.

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