I = 2 ππ scattering phase shift from the HAL QCD method with the LapH smearing HAL QCD Collaboration

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Introduction
Understanding the spectra of hadrons including resonant states from the fundamental theory, quantum chromodynamics (QCD), is one of the major goals in particle and nuclear physics. Lattice QCD is a well-established approach to first-principles calculation of QCD: The calculation of the spectra of the ground state of the single hadron has been matured and it is an important next challenge to determine interactions between hadrons, which are essential to study hadronhadron scatterings and properties of resonances such as ρ and . In lattice QCD, hadron-hadron interactions are mainly investigated by two methods. The first is Lüscher's finite-volume method [1] and its extensions [2][3][4][5][6][7][8][9][10][11][12][13]. In this approach, the energies of the states on finite-volume lattices are extracted from temporal correlation functions and are converted to the scattering phase PTEP 2018, 043B04 D. Kawai et al.

The scheme in the HAL QCD method 2.1. Original HAL QCD method
The most important quantity in the HAL QCD method is the NBS wave function, which is defined for the I = 2 ππ system as where |0 is the QCD vacuum, |π − π − , W k is the ππ eigenstate in the (I , I z ) = (2, −2) channel, W k = 2 m 2 π + k 2 is the central mass energy with the momentum k and its magnitude k ≡ |k|, and π − n s (x) is the negatively charged pion operator, defined by π − n s (x) = aū a n s (x)γ 5 d a n s (x), where a is the index for color, and the label n s represents the smearing level, which will be explained in a later subsection. The crucial property of the NBS wave function (below the inelastic threshold W th = 4m π ) is that the phase shift δ(k) is encoded in the asymptotic behaviors of ψ W k n s (r) [33,35,36]. The non-local but energy-independent potential is defined from the NBS wave function as where H 0 ≡ −∇ 2 /m π , and m π is the pion mass. The potential U n s (r, r ) is faithful to the phase shift below the inelastic threshold, while U n s (r, r ) itself is not a physical observable and explicitly depends on the definition of the pion operator in the NBS wave function, such as the smearing level n s .

Time-dependent HAL QCD method
In this subsection, we explain the time-dependent HAL QCD method [34], which we employ to extract the potential reliably. We denote the two-point correlation functions as where n a (n b ) is the smearing level of the sink (source) operator, and π − =ūγ 5 d, π + =dγ 5 u. The potentials are extracted from the four-point correlation function with various combinations of n a and n b , which is given by n a ,n b (r, t; |P| , t 0 ) = x π − n a (x, t)π − n a (x + r, t)(π n b π n b ) with (π n s π n s ) ,μ I ,I z (|P| , t) = P |P|:fix I 1 ,I 2 x,y C ,μ (P)D I ,I z I 1 ,I 2 e −iP·x e iP·y π I 1 n s (x, t) π I 2 n s (y, t), where C ,μ (P) are the Clebsch-Gordan coefficients in the μth component of the irreducible representation of the cubic group, and the relative momentum P is an element of , P 0 = [n, 0, 0] in lattice unit with |P 0 | ≤ 2}, while D I ,I z I 1 ,I 2 is that from two pions with the z components of the isospin I 1 and I 2 to the two-pion system with total isospin I and its z component I z . Hereafter we exclusively take = A + 1 and μ = 1. The four-point correlation function in Eq. (5) can be decomposed into a product of the NBS wave function ψ W k n a (r) and the time-dependent part for each eigenstate as where 2,2 (|P| , 0) 0 is the overlap between the QCD eigenstate and the vacuum with the insertion of a two-pion operator, and the ellipsis represents inelastic contributions, which become negligible at moderately large t − t 0 and thus will be neglected in further discussions. Here, the indices for the irreducible representation and the isospin are omitted in A W k n b for simplicity.
The R-correlator, defined by where the scheme dependence of the potential on the sink operator is explicit as U n a [34]. It is essential that all elastic states can be used to extract the non-local potential U n a r, r , and the ground state saturation is not required any more in this method. We note that, by construction, U n a r, r does not depend on quantities in the source operator such as the relative momentum |P| and the source smearing level n b . In practice, we approximate the non-local potential U n a r, r by the first few orders of the derivative expansion as In this paper, we extract the potentials in the next-to-leading-order (NLO) decomposition, V n a (r) and V (1) n a (r), by combining R-correlators with |P| = 0, 1, neglecting O(∇ 4 ) terms, as explained in the next section. We also define the (effective) leading-order (LO) potential given by where we make the relative momentum (|P|) and source operator (n b ) dependence, introduced by the second term, explicit as V LO n a (r; |P|, n b ). Thus, if V LO n a (r, |P|, n b ) does not strongly depend on |P| or n b , we can conclude that n a (r) n a (r)) can be used to calculate the scattering phase shifts reliably around the energies probed by the R-correlators.

Phase shift
Once the local potentials V (i) n a (r) are obtained, the scattering phase shifts can be calculated by solving the Schrödinger equation, Eq. (3). As noted before, the phase shifts should be independent of the sink operator scheme (n a ) as long as a sufficient number of local potentials V (i) n a (i = 0, 1, . . .) are employed to represent the non-local potential, though each term V (i) n a might have a sizable dependence on the sink operator scheme [36].
In this paper, we check this scheme independence of the I = 2 ππ scattering phase shifts at the NLO level. As has already been shown [59] and is confirmed in this paper (Sect. 4.1), the contribution from NLO or higher-order terms to the potential in the point-sink scheme is negligibly small in this channel, so that the effective LO potential gives the correct phase shifts below a certain energy. Therefore, we can use the scattering phase shifts from the potential in the point-sink scheme as the benchmark of our analysis. We compare the scattering phase shifts calculated from the potentials in the LapH smearing scheme with the benchmark, in order to see how good the NLO analysis in the derivative expansion is in this scheme.

Some remarks on Ref. [60]
As already stressed many times before [33,35,36], quantitative comparison between different schemes can be done only through physical observables but not through potentials. Comparison between potentials is analogous to the comparison of the running couplings among different schemes such as α MS (q) (the MS scheme), α V (q) (the potential scheme), or α SF (q) (the Schrödinger functional scheme) in QCD. We must compare physical quantities such as the scattering phase shifts in the case of the potential or the scattering amplitudes in the case of the running coupling. Although physical observables are scheme independent in principle, approximations introduced to calculate them bring scheme dependence into observables. An example for the approximations is the truncation of the derivative expansion for the potential or that of the perturbative expansion for the running coupling. In such cases, one scheme is better than others for the fast convergence of the approximation. In the case of the potential, the point-sink scheme is shown to be a good scheme for the fast convergence of the derivative expansion [40,59], so that even the local potential at the LO gives reasonable results at low energies. Analogous scheme dependence for the convergence of the perturbative expansion exists for the running coupling.
Here we make a few comments on a recent paper [60] whose discussions are trivially invalidated as shown below.
The first point discussed in Ref. [60] is the relation between the (local) energy-dependent potential and phase shifts, where it was claimed that the energy-dependent potential defined at a given energy can give the correct phase shift only at that energy, but gives incorrect phase shifts at different energies. We note that such a claim is nothing to do with the HAL QCD method, since the theoretical formulation of the HAL QCD method is based not on the energy-dependent potential, but the nonlocal energy-independent potential U (r, r ), where it has been proved that the latter is faithful to the phase shifts at all energies below the inelastic threshold [33,35,36]. As discussed above, as well as in our previous papers [33,35,36], the derivative expansion for the non-local potential, employed so far in our applications, gives some truncation errors at given orders of the expansion, which, however, should not be misunderstood as a theoretical limitation of the HAL QCD method. 5 The second point discussed in Ref. [60] is the derivative expansion of the non-local potential U (r, r ). We first note that the information of ψ W k n s (r) (in particular the phase shifts δ(k)) below the inelastic threshold are encoded in U (r, r ), where the degrees of freedom of r and k in the former are implicitly converted to those of r and r in the latter through Eq. (3). The derivative expansion of the non-local potential is given by where no symmetry is assumed. Reference [60] claimed that V n (r) cannot be k independent since U (r, r ) needs the same number of the degrees of freedom, r and r , to keep the k independence of U (r, r ). Clearly this statement is incorrect since n has enough degrees of freedom to describe r dependence. For instance, using the Taylor expansion as which are manifestly k independent. The third point discussed in Ref. [60] is a relation between the scattering phase shift and the smearing of the operator. Reference [60] considered the following relation (Eq. (11) in Ref. [60], and called the "fundamental relation" there): where δ 0 (k) is the S-wave scattering phase shift and ψ 0 is the S-wave NBS wave function. One can further consider the smeared NBS wave function defined bỹ where s(r) is a smearing function. In Ref. [60], it was claimed that the "fundamental relation" does not hold by this smearing as (Eq. (26) in Ref. [60]) and the correct phase shift δ 0 (k) is not obtained from the smeared NBS wave function, Eq. (17). This claim is also incorrect.

LapH smearing
The smeared pion operator at time t is constructed as from the smeared quark operator given by where q a,f (x, t) is a local quark field with a color index a and a flavor index f (f = 1, 2 for u, d quarks), and S is a smearing operator at time t with smearing level n s [52,53]. Note that the spinor indices of quarks are implicit and are summed over in the pion operator. Hereafter, a summation over repeated indices is assumed, unless otherwise stated.
In this paper, we employ the gauge-covariant smearing operator S, which is constructed from a gauge-covariant lattice Laplacian at t defined by where U ab i (x, t) represents a stout-smeared link variable [61]. This operator can be diagonalized as where λ n is the nth eigenvalue with |λ n | ≤ |λ m | for n < m, V a n (x, t) is the corresponding eigenvector, and n max = N color N x N y N z . Using eigenvectors, S with the smearing level n s is given by which is the projection operator to the space spanned by n s eigenvectors. We call this smearing the LapH smearing with level n s . Since the eigenmodes corresponding to larger values of |λ n | are absent in this subspace, the smearing operator S removes high-momentum components of quark fields in a gauge-covariant manner. Roughly speaking, n s = 8, 16, 32, and 64 in our setup correspond to momentum cutoffs of 680 MeV, 770 MeV, 900 MeV, and 1100 MeV, respectively. Note that the point quark with no smearing is given by n s = n max . We also study the wall quark operator (only at the source), for which we use the shorthand notation "n s = 0" even though it does not belong to the LapH smearing.

Numerical setup
Since the main purpose of this paper is to investigate the scheme independence of the scattering phase shifts, we perform a lattice QCD calculation at a single heavy pion mass on a small lattice. We employ the (2 + 1)-flavor gauge configuration on a 16 3 × 32 lattice, generated by the JLQCD and CP-PACS collaborations, with a renormalization-group improved gauge action at β = 1.83 and non-perturbatively improved clover action with c SW = 1.7610 at the hopping parameters k ud = 0.1376 and k s = 0.1371 [62,63]. These parameters correspond to the lattice spacing

Potentials and scattering phase shifts in the point-sink scheme
We first study the point-sink scheme. In Fig. 1, we show the effective LO potentials in the point-sink scheme, V LO n max (r; |P|, 16) with |P| = 0 (red up triangles), 1 (green squares), 2 (blue circles), together with the one calculated from the wall source, V LO n max (r; |P| = 0, 0) (orange down triangles). The temporal separation, t − t 0 = 12, is large enough for the potential to be stable against the change of t − t 0 . We first notice that the source operator dependence, the difference between the smeared with n b = 16 and the wall source, is negligible, by comparing V LO n max (r; |P| = 0, 16) (red up triangles) and V LO n max (r; |P| = 0, 0) (orange down triangles). In addition, it can be seen that V LO n max (r; |P|, 16) is almost independent of the source momentum |P| ≤ 2.
We obtain the scattering phase shifts by solving the Schrödinger equation with the potentials which are fit by the following functional form, The fit of potential works well for |P| = 0 and 2, while there exist some residual errors for |P| = 1, since V LO n max (r; |P| = 1, 16) at large r slightly deviates from zero. This introduces systematic uncertainties in results from |P| = 1, as discussed later.   Fig. 2 is that, at low energies, the phase shifts are independent of the source within statistical errors and the effective LO potential in the point-sink scheme describes the I = 2 ππ scattering rather precisely. We thus confirm that the conclusion of the previous quenched study for the I = 2 ππ scattering [59] that the potential in the point-sink scheme, the standard for the HAL QCD potential, is a also good scheme in the full QCD, so that the effective LO potential gives a reliable approximation for the non-local potential up to a certain energy. In order to estimate the energy above which NLO corrections would affect the results, we look at Fig. 2 in more detail. In the upper panel, we find that the results from |P| = 0 (red) and |P| = 1 (green) are consistent with each other at least up to k 2 0.4 GeV 2 . As discussed above, the results from |P| = 1 would suffer from additional systematic errors, and thus the results from |P| = 0 and 1 may be consistent even at higher energies. In the lower panel, we find that the results from |P| = 0 (red) and |P| = 2 (blue) are consistent with each other at least up to k 2 0.6 GeV 2 . As a conservative estimate, we conclude that effective LO analysis in the point-sink scheme is reliable at least up to k 2 0.4 GeV 2 . Among the results in the point-sink scheme, those from |P| = 0 give the smallest statistical fluctuations, and we therefore use the scattering phase shifts from V LO m max (r; |P| = 0, 16) as the benchmark in our investigation hereafter. 9

Potentials in the smeared-sink scheme
We now consider the potential in the smeared-sink scheme. Figure 3 shows the effective LO potentials in the smeared-sink scheme with n a = 64 from the smeared source with n b = 64, |P| = 0, 1, obtained at t − t 0 = 11. It turns out that potentials have sizable source momentum dependence, unlike the case of the point-sink scheme. It is also found that the potentials have non-negligible dependence on t − t 0 for |P| = 1. These observations indicate that the NLO contribution cannot be neglected in this scheme. We therefore determine V (0) and V (1) separately, by solving ⎛

Scattering phase shifts in the smeared-sink scheme
We obtain the scattering phase shifts by solving the Schrödinger equation with the potential including the NLO term. For the NLO analysis, V n a (r) is fitted with the form for all r, while V (1) n a (r) is fitted with for 0 ≤ r ≤ 0.8 fm, which is chosen to exclude singular behaviors of V (1) n a (r) at large r in the fit. In Fig. 4, fitted functions for V (0) (r) and V (1) (r) are also given by light-blue and yellow bands, respectively.
In order to see the effect of the NLO correction on the scattering phase shifts δ 0 (k) quantitatively, we show the scattering phase shifts as a function of k 2 in Fig. 5, where the benchmark result from V LO n max (r; 0, 16) (orange) in the point-sink scheme is compared with those from V LO 64 (r; 0, 64) (pink) and V (0) 64 (r) + V (1) 64 (r)∇ 2 (red) in the smeared-sink scheme. Let us first compare the scattering phase shifts between the point-sink scheme (orange) and the smeared-sink scheme (pink) at the LO. While both results agree with each other at very low energies, they start deviating as the energy increases. The difference in the phase shift becomes about 4 degrees at k 2 = 0.3 GeV 2 . If the NLO analysis is employed in the smeared-sink scheme, the agreement between the point-sink scheme (orange) and the smeared-sink scheme (red) is improved as expected. The difference in the phase shift is reduced to about 2 degrees ( 15%) at k 2 = 0.3 GeV 2 . This result indicates that use of the NLO analysis is almost mandatory in the smeared-sink scheme for quantitatively precise descriptions of scattering phenomena by the HAL QCD potential method. 11  In order to see the differences more precisely, k cot δ 0 (k) is plotted as a function of k 2 in Fig. 6, where the phase shifts obtained from the finite-volume energy by Lüscher's finite-volume method [1] (see Appendix B for details) are also shown at k 2 = 7.2 × 10 −3 ( +13 −5 ) and 0.43( +1 −0 ) GeV 2 (black bands). It is clearly observed that the NLO analysis significantly improves the agreement between different schemes/methods. The results from the effective LO potential in the point-sink scheme (orange) as well as those from the NLO potentials in the smeared-sink schemes with n a = 64 (red) and n a = 32 (blue) agree with results from the finite-volume energy shift not only at low energy (k 2 0 GeV 2 ) but also at higher energy (k 2 0.43 GeV 2 ) within their statistical errors, though the statistical errors of k cot δ 0 (k) from the finite-volume energy shift (black bands) are rather large. Among the different schemes in the potential method, we remind readers that the results from the effective LO potential in the point-sink scheme are the most reliable as the benchmark, since the phase shifts in the point-sink scheme are robust in the sense that the source momentum dependence (the NLO term) is sufficiently small. Therefore, the difference in results between the point-and smeared-sink schemes gives the remaining systematic errors in the smeared-sink scheme. The most plausible origin for the systematics at higher energies is the higher-order corrections in the derivative expansion in the smeared-sink scheme. In fact, the magnitude of the remaining systematics is compatible with the magnitude of the NNLO corrections, which can be roughly estimated by the difference between the LO and NLO results. Another possible origin is the systematics in the NLO terms, since there exist uncertainties in V (1) (r) at long range (r 0.8 fm in Fig. 4). Note that the NLO (and higher-order) corrections can be reduced by increasing the smearing level n a , as can be seen from the comparison between n a = 32 and 64.
At very low energies, k 2 0 GeV 2 , while the results between different schemes/methods achieve good agreement even at the LO analysis, the inclusion of the NLO contribution does not resolve the remaining (small) deviations between the point-and smeared-sink schemes. Considering that the NLO correction is small and that the derivative expansion is expected to be good at low energies, higher-order corrections would not be the origin of this systematic error. One possible reason is the uncertainties in V (1) (r) at long range, as mentioned above. Another possibility is the systematics associated with the long tail structure in the LapH smearing (Fig. A2), and studies to improve the PTEP 2018, 043B04 D. Kawai et al.  (41) locality of the LapH smearing are in progress. Again, we can reduce the systematics by increasing the smearing level n a , as is actually seen in the figure.
In Table 1, we present k cot δ 0 (k) at k 2 = 7.2 × 10 −3 and 0.43 GeV 2 , obtained from the effective LO potential in the point-sink scheme (n a = n max ) with |P| = 0, as the benchmark result. We also show the LO/NLO analyses in the smeared-sink scheme with n a = 64, together with results from the finite-volume method, where the errors are statistical only. The systematic errors in the smearedsink scheme can be estimated by the difference between the point-and smeared-sink schemes, as discussed above. We then confirm that the potential method with the smeared-sink scheme and finitevolume method give consistent results with similar sizes of uncertainties, which are dominated by systematic errors in the former but by statistical errors for the latter.

Conclusion
In this paper, the scheme independence in the HAL QCD method has been investigated for the I = 2 ππ scattering phase shifts at m π 870 MeV. We have considered two schemes, the point-sink scheme and the smeared-sink scheme with various smearing levels n a , where we newly introduce the LapH smearing in the latter. We have found in the point-sink scheme, which is the standard scheme in the HAL QCD method, that the NLO contributions are sufficiently small below k 2 ∼ 0.4 GeV 2 . This means that the (effective) LO potential in the point-sink scheme is a good description of the I = 2 ππ interaction at least up to k 2 ∼ 0.4 GeV 2 (2.15 GeV in terms of the central mass energy). 1 This result in full QCD confirms the conclusion in previous quenched QCD studies that the point-sink scheme is a good scheme for the HAL QCD potential for the NN system [40] and the I = 2 ππ system [59].
In the case of the smeared-sink scheme, the effective LO potential shows a sizable dependence on the momentum at the source, so that the NLO contribution is non-negligible. In fact, at the effective LO analysis, the scattering phase shifts in the smeared-sink scheme show some deviation from the benchmark result given by the point-sink scheme, which increases as the energy increases. 2 We thus calculated the phase shifts using the NLO analysis in the smeared sink scheme. The NLO analysis improves the agreement with the benchmark result at both low and high energies, and results in both the point-sink scheme and the smeared-sink scheme with n a = 64 being consistent with the scattering phase shifts from the finite-volume energy shift calculated by Lüscher's formula at two energies, within large statistical errors in the finite-volume method. The systematic errors of the potential method in the smearing sink scheme with n a = 64 can be estimated from the difference between the point-sink scheme and the smeared-sink scheme, and are found to be a similar size compared to the statistical errors of the finite-volume method. Our study in this paper gives the following lessons for future studies of resonant states such as I = 0, 1 ππ scatterings by the HAL QCD potential method in the smeared-sink scheme: (1) It is better to increase the smearing level n a as long as computational resources (CPU time, memory, disk space, etc.) allow. (2) It is better to observe the n a dependence carefully. (3) It is important to employ NLO analysis if possible.
Studies of resonant states in the HAL QCD potential method are ongoing. The results will be reported in near future.

Funding
Open Access funding: SCOAP 3 . Figure A1 shows the maximum eigenvalue of the lattice-covariant Laplacian in Eq. (22), |˜ | [GeV 2 ], as a function of the smearing level n s .

Appendix A. The maximum eigenvalue and the spatial distribution vs. the smearing level
In Ref. [52], a measure of the spatial distribution for the LapH smearing is defined by Tr S n s (x, x + r, t)S n s (x + r, x, t) , which is gauge invariant and represents the extent to which the quark field is smeared. Figure A2 gives this quantity for n s = 16, 32, and 64. The figure tells us that the smeared quark is more localized as n s increases.

Appendix B. Finite-volume energies from the variational method
In the calculation of the finite-volume energy, the unwanted constant contributions from thermal quark loops to the two-pion correlation function are removed as in Ref. [58], where we set t = 1 in this paper.
In the variational method [30,31], we consider the matrix where C(P, P ; t) is given by the Fourier-transformed four-point correlator We denote λ n (t, t 0 ) as the eigenvalue of the matrix in Eq. (B2). The eigenenergy of the system E n is extracted as where we assume E 0 ≤ E 1 ≤ E 2 ≤ · · · and E n (t) is called the effective energy. In this study, we employ a 3 × 3 matrix from P, P = 0, 1, 2 correlation functions with n a = n b = 64. Figure B1 shows the effective energy E 0,1 (t − t 0 ) as a function of t − t 0 .
Once the E n are obtained, phase shifts can be extracted by Lüscher's finite-volume method [1]. We calculate the momentum k 2 , which corresponds to the energy difference from the two-pion mass, k 2 n = E 2 n 4 − m 2 π . As long as l ≥ 4 partial waves are negligible, Lüscher's formula for the A + 1 representation relates k 2 n to the scattering phase shift as  where L is the spatial lattice extension, q n is the dimensionless momentum defined as q n ≡ k n L/2π , and Z 00 is a generalized zeta function, Z 00 (s, q 2 ) ≡ 1 √ 4π

Appendix C. Effective LO analysis in the smeared-sink scheme
As discussed in the main text (Sect. 4.2), the NLO correction is large in the smeared-sink scheme, in particular at high energies, and the effective LO analysis is not sufficient. In this appendix, we nonetheless present the effective LO analysis in the smeared-sink scheme and demonstrate how the truncation error in the derivative expansion appears in the lattice QCD results. Figure C1 shows the effective LO potential in the smeared-sink scheme V LO n a (r; 0, n b ) at P = [0, 0, 0], together with V LO n max (r; 0, n b ) for comparison. As seen from the figure, the effective LO potentials in the smeared-sink scheme have a much reduced repulsive core than those in the point-sink scheme, while the long-distant part of the potential looks similar for all cases, including the one in the pointsink scheme. We remind readers that the potential is expected to be dependent on the scheme. In the inset of Fig. C1, a short-distance part of the potentials in the smeared-sink scheme is also shown. The magnitude of the potential at short distance increases monotonically as n a increases (equivalently, the size of the smearing range decreases, as shown in Appendix A). 16

C.2. Scattering phase shift
We study the S-wave I = 2 ππ scattering phase shifts δ 0 (k) in the effective LO analysis. We extract the scattering phase shifts by solving the Schrödinger equation as described in Sect. 2. We fit the potentials in Fig. C1 with for all r. Figure C2 shows the phase shifts obtained from V LO 16 (r; 0, 16) (purple), V LO 32 (r; 0, 32) (green), and V LO 64 (r; 0, 64) (orange), together with the benchmark result from V LO n max (r; 0, 16) (red), as a function of k 2 , where k is the magnitude of the scattering momentum.
The phase shifts in Fig. C2 show non-negligible dependence on the sink operator scheme. As the smearing level n a increases, the phase shifts show more repulsive behavior and approach the benchmark result in the point-sink scheme. The difference of the phase shifts between the smearedsink scheme and the point-sink scheme is small at low energies but gradually becomes larger as 17 k 2 increases. The discrepancy in phase shifts between the smeared sink and the point sink originates from sizable NLO contributions in the smeared-sink scheme, which, however, decrease as n a increases.