We propose a fully non-perturbative method to compute inelastic lepton-nucleon ($ell N$) scattering cross sections using lattice QCD. The method is applicable even at low energies, such as the energy region relevant for the recent and future neutrino-nucleon ($ u N$) scattering experiments, for which perturbative analysis is invalidated. The basic building block is the forward Compton-scattering amplitude, or the hadronic tensor, computed on a Euclidean lattice. Total cross section is constructed from the hadronic tensor by multiplying a phase space factor and integrating over the energy and momentum of final hadronic states. The energy integral that induces a sum over all possible final states is performed implicitly by promoting the phase space factor to an operator written in terms of the transfer matrix on the lattice. The formalism is imported from that of the inclusive semileptonic B meson decay [P. Gambino, S. Hashimoto, arXiv:2005.13730]. It can be generalized to compute the $ell N$ scattering cross sections and their moments, as well as the virtual correction to the nuclear $beta$-decay. Necessary quark-line contractions for two current insertions corresponding to the Compton amplitude to be computed on the lattice are summarized.
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