From deep inelastic scattering to heavy-flavor semi-leptonic decays: Total rates into multi-hadron final states from lattice QCD

We present a new technique for extracting decay and transition rates into final states with any number of hadrons. The approach is only sensitive to total rates, in which all out-states with a given set of QCD quantum numbers are included. For processes involving photons or leptons, differential rates with respect to the non-hadronic kinematics may also be extracted. Our method involves constructing a finite-volume Euclidean four-point function, whose corresponding spectral function measures the decay and transition rates in the infinite-volume limit. This requires solving the inverse problem of extracting the spectral function from the correlator and also necessitates a smoothing procedure so that a well-defined infinite-volume limit exists. Both of these steps are accomplished by the Backus-Gilbert method and, as we show with a numerical example, reasonable precision can be expected in cases with multiple open decay channels. Potential applications include nucleon structure functions and the onset of the deep inelastic scattering regime, as well as semi-leptonic $D$ and $B$ decay rates.