Baryon interactions in lattice QCD: the direct method vs. the HAL QCD potential method

We make a detailed comparison between the direct method and the HAL QCD potential method for the baryon-baryon interactions, taking the $\Xi\Xi$ system at $m_\pi= 0.51$ GeV in 2+1 flavor QCD and using both smeared and wall quark sources. The energy shift $\Delta E_\mathrm{eff}(t)$ in the direct method shows the strong dependence on the choice of quark source operators, which means that the results with either (or both) source are false. The time-dependent HAL QCD method, on the other hand, gives the quark source independent $\Xi\Xi$ potential, thanks to the derivative expansion of the potential, which absorbs the source dependence to the next leading order correction. The HAL QCD potential predicts the absence of the bound state in the $\Xi\Xi$($^1$S$_0$) channel at $m_\pi= 0.51$ GeV, which is also confirmed by the volume dependence of finite volume energy from the potential. We also demonstrate that the origin of the fake plateau in the effective energy shift $\Delta E_\mathrm{eff}(t)$ at $t \sim 1$ fm can be clarified by a few low-lying eigenfunctions and eigenvalues on the finite volume derived from the HAL QCD potential, which implies that the ground state saturation of $\Xi\Xi$($^1$S$_0$) requires $t \sim 10$ fm in the direct method for the smeared source on $(4.3 \ \mathrm{fm})^3$ lattice, while the HAL QCD method does not suffer from such a problem.