# Scattering amplitudes from finite-volume spectral functions

## TL;DR

This paper proposes a new method to extract Minkowskian scattering amplitudes from finite-volume Euclidean spectral functions using a specific smearing kernel that implements the $i\,\epsilon$-prescription, applicable to scalar fields and potentially generalizable.

## Contribution

It introduces a novel approach to determine scattering amplitudes from finite-volume spectral functions with a specific complex smearing kernel, extending the LSZ formalism to Euclidean lattice data.

## Key findings

- The method connects smeared spectral functions to Minkowskian correlators in the infinite-volume limit.
- Perturbative analysis demonstrates the interplay between finite volume, Euclidean signature, and the $i\epsilon$-prescription.
- Discussion includes numerical considerations for extracting spectral functions from finite-volume data.

## 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.