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

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 Schrödinger 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 Schrödinger 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.

13 pages, make minor corrections and add a reference, published version

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• Theoretical Foundation of the Nuclear Force in QCD and its applications to Central and Tensor Forces in Quenched Lattice QCD Simulations
• $K\to \pi \pi$ Decays in a Finite Volume
• I=2 Pion Scattering Length from Two-Pion Wave Functions
• Hadronic Interactions
• Reply to "Comment on `Relation between scattering amplitude and Bethe-Salpeter wave function in quantum field theory"'
• Scattering amplitude from Bethe-Salpeter wave function inside the interaction range

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• Scattering of charmed baryons on nucleons
• Pion-mass dependence of the nucleon-nucleon interaction
• $I=2$ $\pi \pi$ scattering phase shift from the HAL QCD method with the LapH smearing
• Comment on "Relation between scattering amplitude and Bethe-Salpeter wave function in quantum field theory"
• Reply to "Comment on `Relation between scattering amplitude and Bethe-Salpeter wave function in quantum field theory"'
• Scattering amplitude from Bethe-Salpeter wave function inside the interaction range
• Phenomenological view on baryon-baryon potentials from lattice QCD simulations
• Quark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range
• Two-pion scattering amplitude from Bethe-Salpeter wave function at the interaction boundary