Abstract
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.
Highlights
The determination of real-time scattering amplitudes from Euclidean lattice field theory simulations is challenging
This work details a novel approach for determining scattering amplitudes from finite-volume Euclidean lattice field theory simulations
It is based on a relationship derived using the LSZ formalism between finite-volume spectral functions and arbitrary real-time infinite-volume scattering amplitudes
Summary
The determination of real-time scattering amplitudes from Euclidean lattice field theory simulations is challenging. The residue at this pole gives the purely hadronic inclusive rate πð p1Þ þ πð p2Þ → X Reference [85] advocates avoiding the inverse problem by integrating experimental data against a multipole function to extract moments that can be directly compared to lattice QCD data It is worth considering the relation of the present work to Ref. VI together with some brief remarks on the straightforward generalization to multiple species of complex arbitrary spin fields
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