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Register a new userSystematics of the HAL QCD Potential at Low Energies in Lattice QCD
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The $XiXi$ 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.
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In this article, we review the HAL QCD method to investigate baryon-baryon interactions such as nuclear forces in lattice QCD. We first explain our strategy in detail to investigate baryon-baryon interactions by defining potentials in field theories such as QCD. We introduce the Nambu-Bethe-Salpeter (NBS) wave functions in QCD for two baryons below the inelastic threshold. We then define the potential from NBS wave functions in terms of the derivative expansion, which is shown to reproduce the scattering phase shifts correctly below the inelastic threshold. Using this definition, we formulate a method to extract the potential in lattice QCD. Secondly, we discuss pros and cons of the HAL QCD method, by comparing it with the conventional method, where one directly extracts the scattering phase shifts from the finite volume energies through the Luschers formula. We give several theoretical and numerical evidences that the conventional method combined with the naive plateau fitting for the finite volume energies in the literature so far fails to work on baryon-baryon interactions due to contaminations of elastic excited states. On the other hand, we show that such a serious problem can be avoided in the HAL QCD method by defining the potential in an energy-independent way. We also discuss systematics of the HAL QCD method, in particular errors associated with a truncation of the derivative expansion. Thirdly, we present several results obtained from the HAL QCD method, which include (central) nuclear force, tensor force, spin-orbital force, and three nucleon force. We finally show the latest results calculated at the nearly physical pion mass, $m_pi simeq 146$ MeV, including hyperon forces which lead to form $OmegaOmega$ and $NOmega$ dibaryons.
We investigate the $I=1$ $pi pi$ interaction using the HAL QCD method in lattice QCD. We employ the (2+1)-flavor gauge configurations on $32^3 times 64$ lattice at the lattice spacing $a approx 0.0907$ fm and $m_{pi} approx 411$ MeV, in which the $rho$ meson appears as a resonance state. We find that all-to-all propagators necessary in this calculation can be obtained with reasonable precision by a combination of three techniques, the one-end trick, the sequential propagator, and the covariant approximation averaging (CAA). The non-local $I=1$ $pi pi$ potential is determined at the next-to-next-to-leading order (N$^2$LO) of the derivative expansion for the first time, and the resonance parameters of the $rho$ meson are extracted. The obtained $rho$ meson mass is found to be consistent with the value in the literature, while the value of the coupling $g_{rho pi pi}$ turns out to be somewhat larger. The latter observation is most likely attributed to the lack of low-energy information in our lattice setup with the center-of-mass frame. Such a limitation may appear in other P-wave resonant systems and we discuss possible improvement in future. With this caution in mind, we positively conclude that we can reasonably extract the N$^2$LO potential and resonance parameters even in the system requiring the all-to-all propagators in the HAL QCD method, which opens up new possibilities for the study of resonances in lattice QCD.
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlator and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlator. Although the two methods should give the same results theoretically, qualitatively different results have been reported. Recently, we pointed out that the separation of the ground state from the excited states is crucial to obtain sensible results in the former, while both states provide useful signals in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlators into the finite-volume eigenmodes obtained from the HAL QCD method. We consider the $XiXi$ system in the $^1$S$_0$ channel at $m_pi = 0.51$ GeV in 2+1 flavor lattice QCD using the wall and smeared quark sources. We demonstrate that the pseudo-plateau at early time slices (t = 1~2 fm) from the smeared source in the direct method indeed originates from the contamination of the excited states, and the true plateau with the ground state saturation is realized only at t > 5~15 fm corresponding to the inverse of the lowest excitation energy. We also demonstrate that the two-baryon operator can be optimized by utilizing the finite-volume eigenmodes, so that (i) the finite-volume energy spectra from the HAL QCD method agree with those from the optimized temporal correlator and (ii) the correct spectra would be accessed in the direct method only if highly optimized operators are employed. Thus we conclude that the long-standing issue on the consistency between the Luschers finite volume method and the HAL QCD method for two baryons is now resolved: They are consistent with each other quantitatively only if the excited contamination is properly removed in the former.
We make a detailed comparison between the direct method and the HAL QCD potential method for the baryon-baryon interactions, taking the $XiXi$ 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 $XiXi$ 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 $XiXi$($^1$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 $XiXi$($^1$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.
Excited state contamination remains one of the most challenging sources of systematic uncertainty to control in lattice QCD calculations of nucleon matrix elements and form factors. Most lattice QCD collaborations advocate for the use of high-statistics calculations at large time separations ($t_{rm sep}gtrsim1$ fm) to combat the signal-to-noise degradation. In this work we demonstrate that, for the nucleon axial charge, $g_A$, the alternative strategy of utilizing a large number of relatively low-statistics calculations at short to medium time separations ($0.2lesssim t_{rm sep}lesssim1$ fm), combined with a multi-state analysis, provides a more robust and economical method of quantifying and controlling the excited state systematic uncertainty, including correlated late-time fluctuations that may bias the ground state. We show that two classes of excited states largely cancel in the ratio of the three-point to two-point functions, leaving the third class, the transition matrix elements, as the dominant source of contamination. On an $m_piapprox310$ MeV ensemble, we observe the expected exponential suppression of excited state contamination in the Feynman-Hellmann correlation function relative to the standard three-point function; the excited states of the regular three-point function reduce to the 1% level for $t_{rm sep} >2$ fm while, for the Feynman-Hellmann correlation function, they are suppressed to 1% at $t_{rm sep}approx1$ fm. Independent analyses of the three-point and Feynman-Hellmann correlators yield consistent results for the ground state. However, a combined analysis allows for a more detailed and robust understanding of the excited state contamination, improving the demonstration that the ground state parameters are stable against variations in the excited state model, the number of excited states, and the truncation of early-time or late-time numerical data.
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