Systematics of the HAL QCD Potential at Low Energies in Lattice QCD

The Xi Xi interaction in the S-1(0) channel is studied to examine the convergence of the derivative expansion of the nonlocal HAL QCD potential at the next-to-next-to-leading order ((NLO)-L-2). We find that (i) the leading order potential from the (NLO)-L-2 analysis provides the scattering phase shifts accurately at low energies, (ii) the full (NLO)-L-2 potential provides only a small correction to the phase shifts even at higher energies below the inelastic threshold, and (iii) the potential determined from the wall quark source at the leading order analysis agrees with the one at the (NLO)-L-2 analysis except at short distances, and thus, it gives correct phase shifts at low energies. We also study the possible systematic uncertainties in the HAL QCD potential, such as the inelastic state contaminations and the finite volume artifact for the potential, and we find that they are well under control for this particular system.