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

TL;DR

This paper clarifies the exact relationship between the Bethe-Salpeter wave function, scattering amplitude, and effective potential in quantum field theory, highlighting limitations of Schrödinger solutions at arbitrary momenta and implications for lattice QCD.

Contribution

It establishes an exact relation involving the reduced Bethe-Salpeter wave function and analyzes the limitations of effective potentials in predicting scattering amplitudes.

Findings

Exact relation between BS wave function and scattering amplitude

Schrödinger equation with effective potential is accurate only at specific momenta

Uncertainty in scattering amplitude due to choice of interpolating operator

Abstract

We reexamine the relations between the Bethe-Salpeter (BS) wave function of two particles, the on-shell scattering amplitude, and the effective potential in quantum filed theory. It is emphasized that there is an exact relation between the BS wave function inside the interaction range and the scattering amplitude, and the reduced BS wave function, which is defined in this article, plays an essential role in this relation. Based on the exact relation, we show that the solution of Schr\"odinger equation with the effective potential gives us a correct on-shell scattering amplitude only at the momentum where the effective potential is calculated, while wrong results are obtained from the Schr\"odinger equation at general momenta. We also discuss about a momentum expansion of the reduced BS wave function and an uncertainty of the scattering amplitude stemming from the choice of the interpolating operator in the BS wave function. The theoretical conclusion obtained in this article could give hints to understand the inconsistency observed in lattice QCD calculation of the two-nucleon channels with different approaches.