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

TL;DR

This study examines the convergence and systematic uncertainties of the HAL QCD potential in lattice QCD for the $ ext{Xi} ext{Xi}$ interaction, demonstrating that the leading order potential accurately reproduces low-energy scattering phase shifts with minimal corrections at N$^2$LO.

Contribution

It provides a detailed analysis of the derivative expansion convergence and systematic uncertainties of the HAL QCD potential at N$^2$LO for the $ ext{Xi} ext{Xi}$ system in lattice QCD.

Findings

Leading order potential accurately reproduces low-energy phase shifts.

N$^2$LO corrections are small below inelastic threshold.

Systematic uncertainties like inelastic contamination are well controlled.

Abstract

The $\Xi\Xi$ interaction in the $^1$S$_0$ channel is studied to examine the convergence of the derivative expansion of the non-local HAL QCD potential at the next-to-next-to-leading order (N$^2$LO). We find that (i) the leading order potential from the N$^2$LO analysis gives the scattering phase shifts accurately at low energies, (ii) the full N$^2$LO potential gives only 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 N$^2$LO 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 find that they are well under control for this particular system.