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

Understand

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.