Two-baryon systems from HAL QCD method and the mirage in the temporal correlation of the direct method

Both direct and HAL QCD methods are currently used to study the hadron interactions in lattice QCD. In the direct method, the eigen-energy of two-particle is measured from the temporal correlation. Due to the contamination of excited states, however, the direct method suffers from the fake eigen-energy problem, which we call the"mirage problem,"while the HAL QCD method can extract information from all elastic states by using the spatial correlation. In this work, we further investigate systematic uncertainties of the HAL QCD method such as the quark source operator dependence, the convergence of the derivative expansion of the non-local interaction kernel, and the single baryon saturation, which are found to be well controlled. We also confirm the consistency between the HAL QCD method and the L\"uscher's finite volume formula. Based on the HAL QCD potential, we quantitatively confirm that the mirage plateau in the direct method is indeed caused by the contamination of excited states.

In this work, we investigate the reliability of the HAL QCD method, and show that systematic uncertainties are under control. We also reveal the origin of the fake plateau in the temporal correlator quantitatively, and demonstrate that correct plateaux emerge for both the ground and the 1st excited states if temporal correlation functions are projected to eigenstates of the HAL QCD potential.
Speaker, e-mail: takumi.iritani@riken.jp In the time-dependent HAL QCD method [13], one measures the Nambu-Bethe-Salpeter correlation function given by with a source operator J, n-th energy eigenvalue W n , the inelastic threshold W th , single baryon correlator G B (t) and the baryon mass m B . With elastic saturation, R( r, t) satisfies where U( r, r ) is the non-local interaction kernel. Using the velocity expansion in the spin-singlet S-wave channel, U( r, r ) V eff (r)δ( r − r ), the effective leading order (central) potential is defined by Considering the higher order term as U( r, r ) {V LO (r) + V NLO (r)∇ 2 }δ( r − r ), it leads to where the leading order (V LO (r)) and next leading order (V NLO (r)) potentials are obtained by solving linear equations with several R( r, t).

Source dependence of HAL QCD method and the next leading order potential
First, we discuss the quark source dependence of the HAL QCD method. We use 2+1 flavor QCD configurations in Ref. [4], which are the Iwasaki gauge action and O(a)-improved Wilson quark action at a = 0.08995(40) fm, where m π = 0.51 GeV, m N = 1.32 GeV, and m Ξ = 1.46 GeV. We employ both the wall source q wall (t) = y q( y, t) and the smeared source q smear ( are the same as those in Ref. [4]. The number of the configurations and simulation parameters are summarized in Table 1. In this work, we focus on ΞΞ( 1 S 0 ) channel, which has smaller statistical errors and belongs to the same representation as NN( 1 S 0 ) in the flavor SU(3) limit. The upper panels in Fig. 1 show the effective leading order potential V eff (r) from the wall and smeared sources at L = 64, respectively. For the wall source, the potentials are almost unchanged from t = 10 to t = 18, while the results from the smeared source show significant t dependence. The lower panels in Fig. 1 are comparisons between the two at t = 11 and 14. The results imply that V smear eff (r) tends to approach to V wall eff (r) as t increases, while there remain small discrepancies even at t = 14.
The small difference between V wall eff (r) and V smear eff (r) indicates the existence of the next leading order correction in the derivative expansion of the non-local kernel U( r, r ). Fig. 2 shows the (next) leading order potential V LO (r) (V NLO (r)), which are obtained by using R wall ( r, t) and R smear ( r, t). The effective leading potential from the wall source is almost identical with the leading order potential as shown in Fig. 2 (Left), while in the smeared source, the next leading order correction to the potential, [V NLO (r)∇ 2 R( r, t)]/R( r, t), cannot be neglected. Fig. 3 shows the scattering phase shifts using V wall eff (r), V LO (r), and V LO (r) + V NLO (r)∇ 2 . These phase shifts suggest that ΞΞ( 1 S 0 ) is an attractive but an unbound channel at m π = 0.51 GeV. As shown in Fig. 3 (Left), at lower energies, these potentials give the consistent results within statistical error. The NLO correction appears only at higher energies (see Fig. 3 (Right)). These results show that (i) the derivative expansion of the non-local kernel has good convergence and the corresponding systematic uncertainty can be controlled (ii) the effective leading order potential from the wall source is reliable at low energies in this system.
The smeared source is tuned to have a large overlap with a single baryon ground state, while the saturation of the single baryon state for the wall source is known to be relatively slower than that of the smeared source [15]. Recently, some concerns are expressed to the wall source for the study of the  two-baryon systems [20, 21] 1 . Fig. 4 (Left) shows the effective masses of the single baryon from the smeared and the wall sources. Although the ground state saturation of the wall source is slower than that of the smeared source, the results from both sources converge around t 16. (Even at a much earlier time, t = 10, the difference of the effective mass between the two is as small as 2%.) As shown in Fig. 4 (Right) (the same figure as in Fig. 1), we confirm that the wall source potentials at different t are consistent with each other including the time slices at t 16, and thus the systematic errors from the single baryon saturation are well under control. This also indicates that the contaminations from the single baryon excited states are almost canceled in the potential at early time slices.

Consistency between the Lüscher's finite volume method and the HAL QCD method
Next, we discuss the consistency between the HAL QCD potential and the Lüscher's finite volume formula [18], which extracts the scattering phase shift from the energy shift in the finite box. Fig. 5 (Left) shows the volume dependence of the lowest eigenvalue of the finite volume Hamiltonian H = H 0 + V wall eff (r). These spectra are proportional to 1/L 3 and converge to zero within error in the infinite volume limit. This volume dependence of the lowest energy strongly supports an absence of the bound state, which is consistent with the phase shift analysis by the HAL QCD method in Fig. 3.
We next calculate the scattering phase shift using the Lüscher's finite volume formula where k is given by Fig. 5 (Right) shows k cot δ 0 (k) as a function of k 2 using the ground state energy on three volumes and the 1st excited state energy on L = 64, which are compared with k cot δ 0 (k) in the infinite volume calculated from the HAL QCD potential (pink band). We here confirm not only a consistency between the two methods but also a smooth behavior of the finite volume energy: k cot δ 0 (k) for k 2 > 0 from the finite volume energy agrees with the pink band from the potential, and k cot δ 0 (k) for k 2 < 0 by the Lüscher's formula smoothly converges to the positive intersect at k 2 = 0, consistent with the pink band.

Diagnosis of the direct method form the HAL QCD potential
Finally, we reveal the origin of the fake plateau in the direct method. Using the low-lying eigenfunctions Ψ n ( r) and eigenvalues ∆E n , which are obtained by solving H = H 0 + V wall eff (r) in the finite box, the R-correlator in Eq. (1) can be decomposed as where a wall/smear n is determined by the orthogonality of Ψ n ( r). Fig. 6 (Left) shows the magnitude of the ratio |b n /b 0 | for both wall and smeared sources as a function of ∆E n , where the filled (open) symbol  represents a positive (negative) value. This quantity represents the magnitude of the contamination of the excited states in R-correlator. For example, the contamination of the 1st excited state is smaller than 1% in the wall source, while it is as large as 10% (with a negative sign) in the smeared source.
In Fig. 6 (Right), we show the reconstructed effective energy shift using three low-lying modes for both wall and smeared sources, which is given by It well reproduces the (fake) plateaux-like behavior in the direct method around t = 15. We can estimate that about t ∼ 100a ∼ 10 fm is required for the smeared source to reach the correct ground state. Finally, we demonstrate reliability of eigenstates of the HAL QCD potential, using the projected effective energy shift defined by where R (n) (t) ≡ r Ψ n ( r)R( r, t) with the eigenfunction Ψ n ( r). Fig. 7 shows projected effective energy shifts for the ground and 1st excited states, which give source independent plateaux consistent with eigenenergies ∆E 0,1 within statistical errors. This demonstration establishes the correctness of the HAL QCD potential in this case, since its (finite volume) eingenenergies are faithful to the finite volume energies. Moreover, once correct eigenstates are obtained from the potential, we can construct correlation functions projected to these eigenstates, whose plateaux agree with correct eigenenergies even at rather small t.

Summary
In this paper, we have established reliability of the HAL QCD method by checking systematic uncertainties. Unlike the direct method, the time-dependent HAL QCD method is free from the elastic state contamination in two-baryon systems, and source dependence can be controlled in the derivative expansion. We have also shown the convergence of the derivative expansion in the non-local kernel, and the next leading order correction is negligible at low energies. We have revealed that the fake plateau in the direct method is caused by the contaminations from low-lying elastic excited states, and established that finite volume eigenenergies from the HAL QCD potential agree with effective energies of projected correlation functions.