# Quark mass dependence of on-shell and half off-shell scattering amplitudes from Bethe-Salpeter wave function inside the interaction range

## TL;DR

This study calculates on-shell and half off-shell scattering amplitudes for a two-pion system in quenched QCD, analyzing how these depend on quark mass and comparing different methods and phenomenological data.

## Contribution

It introduces a method to evaluate scattering amplitudes using Bethe-Salpeter wave functions inside the interaction range, focusing on quark mass dependence.

## Key findings

- Scattering length and effective range are extracted from the amplitudes.
- Results show consistency between the Bethe-Salpeter wave function method and the finite volume method.
- Quark mass dependence of scattering parameters is systematically analyzed.

## Abstract

We evaluate scattering amplitudes at on-shell and half off-shell for $I=2$ S-wave two-pion system using the Bethe-Salpeter wave function inside the interaction range in the quenched QCD. The scattering length and effective range are extracted from these scattering amplitudes. Quark mass dependence of them is investigated with the pion mass ranged in $0.52$--$0.86$~GeV. We examine consistency between a result by the conventional finite volume method and our estimate, as well as the phenomenological value.