  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.
