Systematics of the HAL QCD potential at low energies in lattice QCD

The $\mathrm{\Xi }\mathrm{\Xi }$ interaction in the ${}^1{\mathrm{S}}_0$ 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 (${\mathrm{N}}^2\mathrm{LO}$). We find that (i) the leading order potential from the ${\mathrm{N}}^2\mathrm{LO}$ analysis provides the scattering phase shifts accurately at low energies, (ii) the full ${\mathrm{N}}^2\mathrm{LO}$ 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 ${\mathrm{N}}^2\mathrm{LO}$ 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 we find that they are well under control for this particular system.

Figure 1

The $R$ correlator at $t=10–16$ from the wall source (Left) and the smeared source (Right).

Figure 2

The normalized $R$ correlator at $t=10–16$ from the wall source (Left) and the smeared source (Right).

Figure 3

The potential at the leading order analysis, $V_0^{\mathrm{LO}}(r)$, (red circles) for the wall source (Left) and the smeared source (Right) at $t=13$. The blue squares, green triangles, and black diamonds denote 1st, 2nd, and 3rd terms in Eq. (9), respectively.

Figure 4

The potential at the leading order analysis, $V_0^{\mathrm{LO}}(r)$, for the wall source (Left) and the smeared source (Right) at $t=10–16$.

Figure 5

A comparison of the potential at the leading order analysis, $V_0^{\mathrm{LO}}(r)$, between the wall source (red circles) and the smeared source (blue squares) at $t=10$, 12, 14, 16.

Figure 6

(Left) The LO potential at the ${\mathrm{N}}^2\mathrm{LO}$ analysis, $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)$ (red circles), together with the potential at the LO analysis for the wall source, $V_0^{\mathrm{LO}(\mathrm{wall})}(r)$ (blue diamonds) at $t=13$. (Right) The ${\mathrm{N}}^2\mathrm{LO}$ potential at the ${\mathrm{N}}^2\mathrm{LO}$ analysis, $V_2^{{\mathrm{N}}^2\mathrm{LO}}(r)$, multiplied by $m_{\pi }^2$.

Figure 7

The LO (Left) and ${\mathrm{N}}^2\mathrm{LO}$ (Right) potentials at the ${\mathrm{N}}^2\mathrm{LO}$ analysis in the range of $t=13–16$.

Figure 8

(Left) The effective mass of a single baryon $\mathrm{\Xi }$ for the wall source (red circles) and the smeared source (blue squares). (Right) The potential at the LO analysis, $V_0^{\mathrm{LO}}(r)$, for the wall source at $t=9–17$.

Figure 9

The potential at the LO analysis, $V_0^{\mathrm{LO}}(r)$, for the wall source on $L=40$, 48, and 64 at $t=13$.

Figure 10

The scattering phase shifts in the form of $k\cot {\delta }_0(k)/m_{\pi }$ (Left) and ${\delta }_0(k)$ (Right) from $V_0^{\mathrm{LO}(\mathrm{wall})}(r)$ (black diamonds), $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)$ (blue squares) and $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)+V_2^{{\mathrm{N}}^2\mathrm{LO}}(r){\nabla }^2$ (red circles) at $t=13$.

Figure 11

The scattering length $a_0$ in the form of $(a_0{m}_{\pi }{)}^{-1}$ from $V_0^{\mathrm{LO}(\mathrm{wall})}(r)$ (black diamonds), $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)$ (blue squares), and $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)+V_2^{{\mathrm{N}}^2\mathrm{LO}}(r){\nabla }^2$ (red circles) at $t=13–16$.

Figure 12

(Left) The lowest eigenenergies on finite volumes from the HAL QCD potential. The red line corresponds to the fit by the asymptotic Lüscher’s finite volume formula in the large $L$, Eq. (17). (Right) The scattering phase shifts from finite volume eigenenergies using Lüscher’s finite volume formula (green triangle, blue square, red diamonds), together with those in the infinite volume from the Schrödinger equation (blue band). The black dotted lines denote the constraints by Lüscher’s finite volume formula, and the black solid line represents the bound state condition in the infinite volume. The red dashed line with the pink band corresponds to the NLO ERE analysis to the finite volume data.

Figure 13

The energy dependence of the effective potential, $V^{\mathrm{eff}}(r;E)$, (red circles) compared with the LO potential at the ${\mathrm{N}}^2\mathrm{LO}$ analysis, $V_0^{{\mathrm{N}}^2\mathrm{LO}}(r)$ (blue squares) at $E=10\mathrm{MeV}$ (Top Left), $E=50\mathrm{MeV}$ (Top Right), $E=100\mathrm{MeV}$ (Bottom Left), and $E=200\mathrm{MeV}$ (Bottom Right).