# Relation between scattering amplitude and Bethe-Salpeter wave function in quantum field theory

## TL;DR

This paper establishes an exact relation between the scattering amplitude and the Bethe-Salpeter wave function in quantum field theory, clarifying how effective potentials relate to scattering observables.

## Contribution

It introduces a precise relation involving the reduced Bethe-Salpeter wave function and discusses conditions for operator independence in the HALQCD method.

## Key findings

- The on-shell scattering amplitude can be computed from the BS wave function.
- The effective potential reproduces the correct scattering amplitude only at specific momenta.
- A derivative expansion of the BS wave function is discussed for operator independence.

## Abstract

We discuss an exact relation between the two-particle scattering amplitude and the Bethe-Salpeter (BS) wave function inside the interaction range in quantum field theory. In the relation the reduced BS wave function defined by the BS wave function plays an essential role. Through the relation the on-shell and half off-shell amplitudes can be calculated. We also show that the solution of Schr\"odinger equation with the effective potential determined from the BS wave function gives a correct on-shell scattering amplitude only at the momentum where the effective potential is determined. Furthermore we discuss a derivative expansion of the reduced BS wave function and a condition to obtain results independent of the interpolating operators in the time-dependent HALQCD method.