Zero energy scattering calculation in Euclidean space

TL;DR

This paper demonstrates that zero-energy scattering amplitudes can be obtained directly from Euclidean Bethe-Salpeter equations, simplifying calculations and avoiding the need for Minkowski space analysis or Luscher formalism.

Contribution

It introduces a method to extract scattering lengths from Euclidean Bethe-Salpeter amplitudes at zero energy, bypassing complex Minkowski space computations.

Findings

Euclidean Bethe-Salpeter equation decouples at zero energy

Scattering length can be derived directly from Euclidean amplitudes

Method applicable to scalar particles and extendable to fermions

Abstract

We show that the Bethe-Salpeter equation for the scattering amplitude in the limit of zero incident energy can be transformed into a purely Euclidean form, as it is the case for the bound states. The decoupling between Euclidean and Minkowski amplitudes is only possible for zero energy scattering observables and allows determining the scattering length from the Euclidean Bethe-Salpeter amplitude. Such a possibility strongly simplifies the numerical solution of the Bethe-Salpeter equation and suggests an alternative way to compute the scattering length in Lattice Euclidean calculations without using the Luscher formalism. The derivations contained in this work were performed for scalar particles and one-boson exchange kernel. They can be generalized to the fermion case and more involved interactions.