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Wavefunction-based operator optimization for two-hadron systems in lattice QCD

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Pith's one-line read Two-hadron states only 5 MeV apart can be cleanly disentangled in lattice QCD by building interpolating operators from the system's spatial wavefunctions.

desk verdict A genuinely useful lattice technique with a clever Z3-noise implementation, but the proof-of-principle's 5 MeV claim is softer than it appears because the gap is below the sensitivity of the diagnostics and the comparison energies are not independent. read the letter →

arxiv 2507.09933 v2 pith:NKENWQHR submitted 2025-07-14 hep-lat hep-phnucl-th

Yan Lyu , Sinya Aoki , Takumi Doi , Tetsuo Hatsuda , Kotaro Murakami , Takuya Sugiura This is my paper
classification hep-lathep-phnucl-th PACS 12.38.Gc
keywords latticeQCDtwo-hadronsystemsinterpolatingoperatorsNambu-Bethe-SalpeteramplitudeHALmethodZ3noisesmearingOmega_cccfinite-volumespectroscopy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper develops a systematic way to build interpolating operators for two-hadron systems that incorporate the spatial wavefunction of the pair, rather than using plane-wave or local operators. The claim is that such wavefunction-based operators, constructed from eigenfunctions of the leading-order potential obtained from the time-dependent HAL QCD method, make the correlation function for a selected state dominate at much shorter Euclidean time, so that nearby elastic scattering states no longer obscure the spectrum. As proof of principle, the method cleanly separates the ground and first excited $\Omega_{ccc}\Omega_{ccc}$ states around $2m_{\Omega_{ccc}}\simeq 9700$ MeV with an energy gap of about 5 MeV on a physical-point lattice with $La\simeq 8.1$ fm. A new quark smearing technique using $Z_3$ noise vectors is introduced to implement the wavefunction-weighted source without all-to-all quark propagators, and the method is shown to work as a sink filter for unoptimized sources as well.

What carries the argument

The load-bearing construction is the optimized two-hadron operator $O_n(t) = V^{-2}\sum_{\vec x,\vec r} B(\vec x+\vec r,t)B(\vec x,t)\Psi_n^*(\vec r)$, with dual functions $\Psi_n$ obtained from the inverse of the NBS-amplitude overlap matrix. In practice the paper takes $\Psi_n$ to be the finite-box eigenfunctions of the leading-order potential extracted by the time-dependent HAL QCD equation, so the wavefunction itself becomes the operator weight. At the source, the weight is implemented by a $Z_3$-noise smearing $F_n(\vec r)=\frac{1}{V_{\rm sub}^{1/3}}\sum_{\vec r_0} Z_3(\vec r_0)\Psi_n^{1/3}(\vec r_0)f(\vec r-\vec r_0)$, whose triple-product identity removes cross terms and lets the $\Psi_n$-weighted baryon pair be realized without all-to-all propagators.

What would settle it

Compute the same optimized operators for a two-hadron channel where the next-to-leading-order potential correction is known to be large, and test whether the one-pass correlation functions still have time-stable spatial profiles and whether their plateaus reproduce the finite-volume eigenvalues; a mismatch would show the leading-order-eigenfunction approximation is the point that fails.

Watch

Extended reading notes

Core claim

The central discovery is that the spatial profile of a two-hadron state, the Nambu-Bethe-Salpeter amplitude, can be used as the weight of an interpolating operator, and that this dramatically improves state isolation. The authors prove the concept by taking the eigenfunctions of the leading-order HAL QCD potential on a finite box as approximate NBS amplitudes, constructing dual functions $\Psi_n$ satisfying $\langle\Psi_n|\psi_m\rangle=\delta_{nm}$, and using them to define operators $O_n(t)$. With these operators, the correlation functions $R_n(\vec r,t)$ have spatial profiles that are essentially constant from $t/a=15$ to $30$, and the effective energies from $R_n(t)$ show plateaus consistent with the finite-volume eigenvalues $\varepsilon_0=-4.6(4)$ MeV and $\varepsilon_1=0.6(1)$ MeV. This disentangles the two states even though they lie within $\sim5$ MeV of each other near 9700 MeV, something the compact and wall sources fail to do.

Load-bearing premise

The method assumes that the wavefunctions obtained by diagonalizing the leading-order potential extracted from the initial wall-source measurement are close to the true two-hadron wavefunctions, so that a single construction step yields operators dominated by the target states.

Editorial extensions

If this is right

  • Optimized operators produce correlation functions whose spatial profiles are stable from $t/a=15$ to $30$, so effective energies can be read from plateaus instead of from early-time fits dominated by excited-state contamination.
  • The same construction resolves the ground and first excited $\Omega_{ccc}\Omega_{ccc}$ states about 5 MeV apart near 9700 MeV, with energies consistent with the finite-volume eigenvalues from the potential.
  • Using an optimized operator only at the sink still extracts the correct ground and excited energies from wall or compact sources, so the method works as a state filter even when the source is unoptimized.
  • Potentials extracted from the optimized sources are nearly time-independent, and the phase shifts they produce agree with those converted from the finite-volume spectrum.
  • Because the wavefunction input can come from the HAL QCD potential or from any model or effective-field-theory wavefunction, the construction generalizes to other two-hadron systems and to higher partial waves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to compare the leading-order-potential eigenfunctions with the true NBS amplitudes obtained from a large variational analysis in the same volume; strong disagreement would identify systems that need multiple iterations before the operators are trustworthy.
  • The near-orthogonality observed in this system hints that one-step construction may suffice for tightly bound heavy dibaryons, while loosely bound or resonant channels will be the stress test for the iterative loop.
  • The $Z_3$-noise smearing factors the inter-hadron wavefunction into independent quark smearings, so it can likely be combined with distillation-like single-hadron operators to reach moving frames or higher partial waves without all-to-all propagators.
  • The same state-filtering property should make optimized operators useful for computing matrix elements of currents between specific two-hadron states, since controlling the initial state is usually the main obstacle.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a method for constructing optimized two-hadron interpolating operators by incorporating inter-hadron spatial wavefunctions, which are obtained from eigenfunctions of the HAL QCD potential. The implementation uses a new Z3-noise-based quark smearing technique that encodes the wavefunction at the source without all-to-all propagators. The method is demonstrated on the Omega_ccc Omega_ccc system in the 1S0 channel using physical-point, 96^4 lattices with La ~ 8.1 fm. The authors report stable spatial profiles of optimized-source correlation functions and effective-energy plateaux that identify the ground and first excited states around 2m_Omega_ccc ~ 9700 MeV with an energy gap near 5 MeV, and they compare potentials and phase shifts with those obtained from wall and compact sources.

Significance. If the central claim is sustained, the paper offers a practical and possibly general strategy for two-hadron spectroscopy and matrix elements in dense-spectrum systems, and the Z3-noise smearing construction is itself a useful technical contribution. The paper contains several strong cross-checks: the LO potential is nearly independent of the source operator, the Laplacian term of the optimized correlators matches the finite-volume momentum in the asymptotic region (Fig. 16), the phase shifts from the optimized potential agree with the finite-volume conversion (Fig. 18), and the N2LO corrections are small (Fig. 20). The manuscript is also explicit about the assumption that the NBS amplitudes are nearly orthogonal, which is the condition for one-step convergence. However, the headline validation has a circular component: the optimized operators and the reference energies both come from the same HAL potential, and the stability diagnostics are not sensitive to the small-gap failure mode they are meant to rule out.

major comments (3)
  1. [§VI A 3, §VI B 1, Fig. 12] The validation of the central disentangling claim is partly circular. The optimized operators O0 and O1 are constructed from eigenfunctions of the LO potential V_LO obtained from the wall source (Sec. III B step (ii); Sec. VI A 3), and the reference energies epsilon_0 and epsilon_1 in Eqs. (41)-(42) are the eigenvalues of that same potential. The agreement of the plateaux in Fig. 12 with epsilon_0,1 therefore compares the operators with the input from which they were built. The Lüscher cross-check in Fig. 18 converts the same epsilon_0,1 into phase shifts, so it does not provide an independent energy determination. Please provide an energy extraction from an operator basis not derived from the HAL eigenfunctions, or give a direct quantitative bound on the excited-state contamination in R0(t) and R1(t).
  2. [§VI B 1, Eq. (44), Fig. 12] The stability diagnostics are insensitive to the specific failure mode they are intended to exclude. With epsilon_0 = -4.6(4) MeV and epsilon_1 = 0.6(1) MeV the gap is about 5 MeV; over the scan range t/a = 15-30 with a = 0.0844 fm one has Delta_E * t about 0.03-0.07. Under a two-state mixture the normalized profile R(r,t)/R(r,t_f) changes by only a few percent over this range, so the small residue factor L[R(r,t)] in Eq. (44) and the plateaux in Fig. 12 are consistent with a mixture of the two nearby levels rather than with isolation of either. Please provide a quantitative bound on the contamination amplitude, for example from a two-state fit to R0(t) and R1(t), or demonstrate insensitivity by extending t or by testing a system with a larger gap.
  3. [Abstract, §VI] The abstract states that the optimized operators outperform combinations of limited plane-wave operators in the variational analysis, but no such variational/GEVP comparison is actually presented. Section VI compares wall and compact sources with optimized sinks (Fig. 13) and optimized sources (Fig. 10), but it does not solve a GEVP in a plane-wave basis and compare the resulting energies with those from O0 and O1. Please either include that comparison or revise the claim to the form that is demonstrated in the paper.
minor comments (5)
  1. [§II, Fig. 12] The effective energy in Eq. (9) is defined by a ratio of Rn(t+1) and Rn(t), but the text does not explicitly state that t is in lattice units; please make the units explicit where the plateaux are discussed.
  2. [Abstract, §II] There are small language issues: 'enables clear identification' should be 'enable clear identification', and 'being lack of accurate Psi_n(r)' should be 'lacking accurate Psi_n(r)'.
  3. [Fig. 16] The vertical axis label in Fig. 16 omits units; please specify that V(r) and p^2/m_B are in MeV.
  4. [References] Reference [47] (Misner) has a formatting error, with 'journal =' appearing inside the citation; please correct the entry.
  5. [§VI B 1, Figs. 8-9] The units of the dual functions in Figs. 8 and 9 are given in fm^{-3/2}, while the normalization in Eq. (43) is a dimensionless sum over lattice points; please clarify the conversion to physical units.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the optimized-operator energies are extracted from lattice QCD correlation functions and are only compared with, not derived from, the HAL QCD potential eigenvalues.

full rationale

The paper's central chain is not circular by construction. The optimized operators O_n in Eq. (6) are built from dual functions obtained from eigenfunctions of the LO HAL QCD potential V_LO in Eq. (18), which itself is extracted from the wall-source correlation function. The correlation functions R_n(r,t) in Eq. (7) and R_n(t) in Eq. (8) are then computed as actual QCD four-point functions with these operators as sources and sinks. Their Euclidean time dependence is determined by the QCD Hamiltonian, not by the Schrödinger eigenvalue problem in Eq. (24). If the eigenfunctions of V_LO were poor approximations to the true NBS amplitudes, the projected correlators would retain admixtures of other states and the effective energies in Eq. (9) would not exhibit plateaus agreeing with the potential eigenvalues. The agreement in Fig. 12 between the measured effective energies and epsilon_0, epsilon_1 from Eqs. (41)-(42) is therefore a nontrivial consistency check, not an identity. Similarly, the residue-factor stability test in Eq. (44) and Fig. 11 uses only the measured R(r,t) profiles. The phase-shift comparison in Fig. 18 converts the measured finite-volume energies (not the potential eigenvalues alone) via the finite-volume formula and compares them with phase shifts from the potential; this is a self-consistency check of the HAL QCD framework rather than an external validation, but it is not a circular derivation. The self-citations to Refs. [22,35] for using HAL QCD eigenfunctions as initial NBS-amplitude approximations are methodological antecedents, and the HAL QCD potential framework itself is an established external formalism. The reader's concern that the 5 MeV gap makes the plateau and residue diagnostics insensitive to a small admixture of the nearby state is a legitimate statistical/correctness risk, not a circularity: it concerns whether the consistency check has enough resolving power, not whether the output is equal to the input by construction. No equation in the paper reduces to its own input; no fitted parameter is renamed as a prediction; and no load-bearing conclusion relies solely on a self-citation chain. Hence the circularity score is 0.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities. The central claim rests on the validity of the HAL QCD potential framework, the approximation that finite-box eigenfunctions of the LO potential represent the NBS amplitudes, and numerical choices (sublattice spacing l, smearing parameter B, compact radius rB) that are tuned rather than derived. These are domain assumptions rather than independently verified facts.

free parameters (4)
  • exponential smearing parameter B = 0.475 a^-1
    Tuned to achieve ground-state saturation for single Omega_ccc at early Euclidean time (Sec V, Fig 2).
  • compact sink radius rB = 4 a
    Cutoff in Eq (40) set by the single-hadron smearing size; plateau energy shifts with rB (footnote 3).
  • sub-lattice spacing l = 8 a
    Choice balancing Z3 variance and high-momentum contamination from sublattice sum (Sec IV A, Sec V).
  • three-range Gaussian fit parameters a_i, b_i = not quoted
    Fit of V_LO(r) in Eq (47) to compute infinite-volume phase shifts (Sec VI C).
assumptions (4)
  • domain assumption Existence of an energy-independent nonlocal HAL QCD potential U(r,r') with derivative expansion to LO/N2LO.
    Invoked in Sec III A (Eqs 14-15); standard HAL QCD framework.
  • domain assumption Eigenfunctions of the finite-box LO potential approximate the true NBS amplitudes psi_n(r).
    Used in Sec III B step (ii) to define dual functions and optimized operators; only approximately true if the LO potential is accurate and truncation errors small.
  • standard math Z3 noise vectors satisfy the triple-product identity Eq (29) so that cross-support terms vanish on average.
    Stochastic estimator property used in Sec IV A Eq (30).
  • domain assumption The sublattice sum approximates the full-lattice sum at sufficiently large Euclidean time t >> m_B l^2/(4 pi^2).
    Stated in Sec IV A (i), used to justify the sparsened source.

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Cite this review

Pith. "Pith review of Wavefunction-based operator optimization for two-hadron systems in lattice QCD." pith.science (2026). https://pith.science/paper/NKENWQHR

@misc{pith2026250709933,
  author       = {Pith},
  title        = {Pith review of: Wavefunction-based operator optimization for two-hadron systems in lattice QCD},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NKENWQHR}},
  note         = {Machine review of arXiv:2507.09933}
}
abstract

A systematic way to constructing optimized interpolating operators for two-hadron systems is developed by incorporating inter-hadron spatial wavefunctions. The wavefunctions can be obtained from an iterative process with an appropriate initial guess. To implement these operators, a novel quark smearing technique utilizing $Z_3$ noise vectors is proposed, which allows for effectively incorporating inter-hadron spatial wavefunctions at the source without using all-to-all quark propagators. Proof-of-principle application to the $\Omega_{ccc}\Omega_{ccc}$ system using physical-point lattice configurations with a large size $La\simeq8.1$~fm demonstrates that optimized operators outperform combinations of limited plane-wave operators in the variational analysis, enabling clear identification of states around $2m_{\Omega_{ccc}}\simeq 9700$ MeV with the energy gap as narrow as $\sim 5$ MeV. A comparison on correlation functions, effective energies, and HAL QCD potentials between unoptimized operators and optimized operators is given, with a special emphasis on the effects from nearby elastic scattering states. Potential applicability of the optimized operator to various two-hadron systems and its relation to the variational method are also discussed.

Figures

Figures reproduced from arXiv: 2507.09933 by the authors.

Figure 1
Figure 1. FIG. 1. Setup for the novel quark smearing in Eq. (27). [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Determination of the smearing parameter in Eq. (36). [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The effective mass of Ω [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The correlation function [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The effective energies extracted from temporal corre [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The LO potentials calculated using the wall source (left) and the compact source (right) at Euclidean time [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The time dependence of the total potential (top), of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: tions Rn(⃗r, t) (proportional to the NBS amplitude), and {Ψn(⃗r)} would not be orthogonal to each other 6 . Having hadronic correlation functions R1,0(⃗r, t) with least nearby state contamination, we should be able to identify each state in their spectra. Shown in [PI…
Figure 9
Figure 9. Figure 9: FIG. 9. The radial projection of the dual functions Ψ [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The effective energies ∆ [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The effective energies for the ground state (red symbols) and the first excited state (green symbols) derived by [PITH_FULL_IMAGE:figures/full_fig_p014_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The LO potentials calculated using the optimized sources [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]
Figure 16
Figure 16. Figure 16: FIG. 16. Numerical confirmation of Eq. (46). The red and [PITH_FULL_IMAGE:figures/full_fig_p015_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17. A comparison of the LO potentials calculated using each source at [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 18
Figure 18. Figure 18: FIG. 18. The scattering phase shifts calculated using the [PITH_FULL_IMAGE:figures/full_fig_p016_18.png]
Figure 19
Figure 19. Figure 19: FIG. 19. The N [PITH_FULL_IMAGE:figures/full_fig_p017_19.png]
Figure 21
Figure 21. Figure 21: FIG. 21. A comparison on the relative error (defined as [PITH_FULL_IMAGE:figures/full_fig_p017_21.png]
Figure 22
Figure 22. Figure 22: FIG. 22. The scattering phase shifts calculated using the LO potentials for each source at multiple Euclidean time [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]

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    Hadronic correlation functions We first examine the spatial profile of the hadronic cor- relation functions R(⃗ r, t) defined in Eq. (16). Shown in Fig. 4 are R(⃗ r, t) with the normalization P ⃗ r∈Λ R2(⃗ r, t) = 1 at Euclidean time t/a = 15, 20, 25, and 30 calculated us- ing the wall source and the compact source, respectively. The spatial profiles exhib...

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    Lap” and“ Dt

    Hadronic correlation functions Using the dual functions in Fig. 8, we construct opti- mized two-baryon operators defined in Eq. (6) and use them as source operators according to Sec. IV A to com- pute hadronic correlation functions given in Eq. (7). 5 These eigen functions are orthogonal to each other, as they are eigen modes with different eigen values o...

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    Potentials from the time-dependent HAL QCD method Let us now perform the time-dependent HAL QCD analysis using the R0,1(⃗ r, t) in Fig. 10. In Fig. 14, we show the LO potentials extracted at Euclidean time t/a = 25 by using Eq. (18). The total potentials in both cases are dominated by the Laplacian terms, with small contributions from the Dt terms, which ...

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    17 is a comparison of the local poten- tials extracted from R0(⃗ r, t), R1(⃗ r, t) and those calcu- lated using the wall and compact sources at Euclidean time t/a = 25

    Results from LO potentials Shown in Fig. 17 is a comparison of the local poten- tials extracted from R0(⃗ r, t), R1(⃗ r, t) and those calcu- lated using the wall and compact sources at Euclidean time t/a = 25. These potentials show almost identical behaviors except only a slight deviation at short distances for the compact source from others. To determine...

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    Results from N 2LO potentials To see the effect of high-order terms in the derivative expansion to the potential, we derive the N 2LO poten- tials according to Sec. III A by using two hadronic cor- relation functions R0(⃗ r, t) and Rwall(⃗ r, t). This is be- cause the former is dominated by the ground state, while the latter includes scattering states mod...

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