Model analysis on the effectiveness of the HAL QCD method for hadron-hadron interactions

The HAL QCD method has been one of the powerful tools to extract hadron-hadron interactions directly from lattice QCD simulation data. In this paper, we aim at examining the effectiveness of the HAL QCD method by deriving a formula to calculate quantities in the HAL QCD method, such as the so-called R-correlators and HAL QCD local potentials, from the hadron-hadron scattering amplitudes within effective models. In this framework, we can judge whether the HAL QCD local potential, evaluated in the present formula, reproduces the properties of the original hadron-hadron interaction or not via the scattering amplitude, which is a solution of the Lippmann--Schwinger equation with the original hadron-hadron interaction as an input. In an analysis within a simple model of elastic scattering, we show that, when the original interaction is predominantly local, the HAL QCD local potentials quantitatively reproduce phase shifts of the hadron-hadron scatterings and correctly indicate the existence/absence of the bound state with its binding energy $\sim$ MeV. Lattice discretization of spacetime modifies the results only slightly. Furthermore, we consider the $\bar{K} N$ potential in a bare to $\bar{K} N$ transition amplitude, which shows singular behavior around the origin in the recent HAL QCD results of the lattice QCD simulation data, and discuss the cause of such singular behavior in the HAL QCD method through our model analysis of the $\bar{K} N$ scattering.