Relation Between Scattering Amplitude and Bethe-Salpeter Wave Function Inside Interaction Range

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. The reduced BS wave function, which is defined by the BS wave function inside the interaction range, plays an essential role in the relation. The half off-shell scattering amplitude can be obtained from the relation as well as the on-shell scattering amplitude. The relation can be regarded as the LSZ reduction formula using the BS wave function. The corresponding relation on the lattice can be derived with the condition that the interaction range is less than the half of the spatial extent. In order to apply the relation on the lattice to the S-wave isospin $I=2$ two-pion scattering channel, we calculate the BS wave function in the quenched lattice QCD. It is found that the scattering length obtained from the on-shell scattering amplitude agrees with that obtained from the finite volume method. We also calculate the half off-shell amplitude for the first time, and observe a clear signal of the amplitude in a wide range of the momentum. From the slope of the half off-shell amplitude, the effective range is estimated using two assumptions. Although the half off-shell amplitude is not an observable in experiment, it could be useful to understand property of hadron scatterings.