Abstract
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlation function and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlation function. Although the two methods should give the same results theoretically, there have been reported qualitative difference for observables from lattice QCD simulations. Recently, we pointed out in [1, 2] 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 for observables in the latter. In this paper, we identify the contribution of each state in the direct method by decomposing the two-baryon correlation functions into the finite-volume eigenmodes obtained from the HAL QCD method. As in our previous studies, we consider the ΞΞ system in the 1S0 channel at mπ = 0.51 GeV in (2+1)-flavor lattice QCD using the wall and smeared quark sources with spatial extents, La = 3.6, 4.3, 5.8 fm. 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 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 temporal correlation function with the optimized operators and (ii) the correct finite-volume 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 Lüscher’s finite volume method and the HAL QCD method for two baryons is now resolved at least for this particular system considered here: they are consistent with each other quantitatively only if the excited contamination is properly removed in the former.
Highlights
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlation function and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlation function
We demonstrate that the “pseudo-plateau” at early time slices (t = 1 ∼ 2 fm) from the smeared source in the direct method originates from the contamination of the excited states, and the plateau with the ground state saturation is realized only at t > 5 ∼ 15 fm corresponding to the inverse of the lowest excitation energy
Detailed studies in this channel were already performed with the direct method [1] as well as the HAL QCD method [20] in (2+1) flavor lattice QCD at mπ = 0.51 GeV and mK = 0.62 GeV, so that the main purpose of this paper is to present an in-depth analysis by combining both results: in particular, the excited state contaminations in the temporal correlation functions are quantitatively evaluated by decomposing them in terms of the finite-volume eigenmodes of Hamiltonian with the HAL QCD potential
Summary
There exist two methods to study two-baryon systems in lattice QCD: the direct method which extracts eigenenergies from the plateaux of the temporal correlation function and the HAL QCD method which extracts observables from the non-local potential associated with the tempo-spatial correlation function. The binding energies and phase shifts in the infinite volume are calculated by using the Schrodinger-type equation with the kernel as the potential, which has field theoretical derivation on the basis of the reduction formula for composite operators Both methods rely on the asymptotic behavior of the Nambu-Bethe-Salpeter (NBS) wave function, and should in principle give the same results for observables [10, 12, 13]. The difficulty of two-baryon systems compared to a single baryon originates from the existence of elastic scattering states Their typical excitation energies δE are one to two orders of magnitude smaller than O(ΛQCD), so that one needs to probe large Euclidean time t (δE)−1 to extract the genuine signal of the ground state in the direct method. Other systematic uncertainties such as the contaminations from the inelastic states and the finite volume effect for the potential are shown to be well under control [20]
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