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Register a new userTowards fully non-perturbative computation of inelastic $ell N$ scattering cross sections from lattice QCD
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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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We develop a methodology for the computation of the $Kto ell u_ell ell^+ ell^-$ decay width using lattice QCD and present an exploratory study here. We use a scalar function method to account for the momentum dependence of the decay amplitude and adopt the infinite volume reconstruction (IVR) method to reduce the systematic errors such as the temporal truncation effects and the finite-volume effects. We then perform a four-body phase-space integral to obtain the decay width. The only remaining technical problem is the possible power-law finite-volume effects associated with the process of $Ktopipi ell u_ellto ell u_ell ell^+ ell^-$, where the intermediate state involves multiple hadrons. In this work, we use a gauge ensemble of twisted mass fermion with a pion mass $m_pi=352$ MeV and a nearly-physical kaon mass. At this kinematics, the $pipi$ in the intermediate state cannot be on shell simultaneously as $2m_pi>m_K$ and the finite-volume effects associate with $pipi$ state are exponentially suppressed. Using the developed methods mentioned above, we calculate the branching ratios for four channels of $Kto ell u_ellell^+ ell^-$, and obtain the results close to the experimental measurements and ChPT predictions. Our work demonstrates the capability of lattice QCD to improve Standard Model prediction in $Kto ell u_ell ell^+ ell^-$ decay width.
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