Scattering amplitudes from finite-volume spectral functions

A novel proposal is outlined to determine scattering amplitudes from finite-volume spectral functions. The method requires extracting smeared spectral functions from finite-volume Euclidean correlation functions, with a particular complex smearing kernel of width $\epsilon$ which implements the standard $i\epsilon$-prescription. In the $L \to \infty$ limit these smeared spectral functions are therefore equivalent to Minkowskian correlators with a specific time ordering to which a modified LSZ reduction formalism can be applied. The approach is presented for general $m \to n$ scattering amplitudes (above arbitrary inelastic thresholds) for a single-species real scalar field, although generalization to arbitrary spins and multiple coupled channels is likely straightforward. Processes mediated by the single insertion of an external current are also considered. Numerical determination of the finite-volume smeared spectral function is discussed briefly and the interplay between the finite volume, Euclidean signature, and time-ordered $i\epsilon$-prescription is illustrated perturbatively in a toy example.