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arxiv: 2606.19840 · v1 · pith:K3UMHY5Inew · submitted 2026-06-18 · ✦ hep-lat · hep-ph· nucl-th

Scalar diquark mass and quark--diquark potential from lattice QCD using the potential method with a static quark

Kai-Wen Kelvin-Lee , Noriyoshi Ishii This is my paper

Pith reviewed 2026-06-26 15:23 UTC · model grok-4.3

classification ✦ hep-lat hep-phnucl-th
keywords lattice QCDscalar diquarkquark-diquark potentialHAL QCD methodCornell potentialstring tensionbaryon spectrumstatic quark
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The pith

Lattice QCD yields scalar diquark mass near (2/3) of nucleon mass with Cornell-type quark-diquark potential.

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

The paper applies a potential method to lattice QCD configurations of a scalar diquark paired with a static quark. The diquark mass is fixed by demanding that the derived potential reproduces the p-wave baryon spectrum seen in two-point correlators. On 2+1 flavor ensembles with pion mass near 702 MeV the extracted mass sits close to the naive constituent estimate (2/3)m_N. The resulting potential takes the Cornell form of Coulomb plus linear terms, and the linear coefficient matches the string tension from the ordinary static quark-antiquark Wilson loop to within roughly 5 percent.

Core claim

Applying the HAL QCD-inspired potential method to a scalar-diquark plus static-quark system on 2+1 flavor lattices, the diquark mass is fixed self-consistently so that the potential reproduces the p-wave baryonic spectrum; the mass obtained is close to (2/3)m_N and the potential is of Cornell type whose string tension agrees within about 5 percent with the Wilson-loop value.

What carries the argument

HAL QCD-inspired potential extracted from the diquark-static quark correlator, with diquark mass fixed by matching the p-wave baryon spectrum.

If this is right

  • The quark-diquark interaction is of the same Cornell form found for quark-antiquark pairs.
  • The confinement scale (string tension) is essentially unchanged when one participant is a diquark rather than an antiquark.
  • Baryons can be modeled as a static quark bound to a scalar diquark whose mass is set by the constituent estimate.
  • The same potential framework can be used to predict higher partial-wave baryon states once the mass and potential are fixed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to other diquark spin-flavor channels to extract their masses and potentials without full three-body simulations.
  • If the 5 percent agreement persists at physical pion mass, quark-diquark potentials might serve as a computationally cheaper route to baryon spectroscopy.
  • The close match to the naive mass estimate suggests that the diquark behaves as a compact object whose internal dynamics can be absorbed into a single effective mass parameter.

Load-bearing premise

The diquark mass can be chosen so that the potential model reproduces the p-wave spectrum obtained from two-point functions.

What would settle it

A repeat of the calculation at a lighter pion mass in which the self-consistent diquark mass deviates by more than 20 percent from (2/3)m_N while the spectrum match still holds.

Figures

Figures reproduced from arXiv: 2606.19840 by Kai-Wen Kelvin-Lee, Noriyoshi Ishii.

Figure 1
Figure 1. Figure 1: Effective mass difference, ∆mCΓ for various Γ. The resulting effective mass differences, ∆mCΓ are shown in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: The quark–diquark potential at t/a = 14 together with the QQ¯ po￾tential extracted from the Wilson loop. The quark-diquark data (circles) with its Cornell fit (red line) is compared against the QQ¯ data (triangles for without HYP, inverted triangles for with HYP) with their Cornell fit (green and blue dashed lines). To ease the comparison, the QQ¯ potentials are shifted vertically. ted parameters are estim… view at source ↗
read the original abstract

We study the scalar diquark mass and the quark--diquark potential by applying a HAL QCD-inspired potential method to a baryonic system composed of a scalar diquark and a static quark. The diquark mass is determined self-consistently by requiring that the p-wave baryonic spectrum obtained from two-point correlators be reproduced within the potential framework. Numerical calculations are performed using $2+1$ flavor QCD gauge configurations generated by the PACS-CS Collaboration on a $L^{3} \times T = 32^{3} \times 64$ lattice with $a^{-1} \approx 2.176$ GeV and the pion mass, $m_{\pi} \approx 702$ MeV. From the analysis, we obtain a scalar diquark mass which is close to the na\"{\i}ve constituent quark estimate $ (2/3)m_{N}$, together with a quark--diquark potential of the Cornell type (Coulomb + linear). The string tension extracted from the quark--diquark potential agrees within approximately 5% with that obtained from the static quark--antiquark potential (Wilson Loop).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper applies a HAL QCD-inspired potential method to a system consisting of a scalar diquark and a static quark on 2+1 flavor PACS-CS configurations (32^3×64, a^{-1}≈2.176 GeV, m_π≈702 MeV). The scalar diquark mass is fixed self-consistently by requiring that the Schrödinger equation with the extracted quark-diquark potential reproduces the p-wave baryon spectrum obtained from two-point correlators. The resulting mass is reported close to the naïve (2/3)m_N estimate, the potential is of Cornell form, and the string tension extracted from it agrees with the Wilson-loop static quark-antiquark value to within ~5%.

Significance. If the self-consistent matching is robust against systematics, the work supplies a lattice QCD determination of both the diquark mass and the quark-diquark interaction in a controlled heavy-light setting. The 5% agreement between the two string tensions constitutes a non-trivial internal consistency check that is not automatic in the potential-method framework.

major comments (2)
  1. [Numerical analysis / self-consistent mass determination] The self-consistent tuning of the diquark mass (described in the abstract and the numerical-analysis section) is load-bearing for both the reported mass value and the subsequent Cornell parameters. Because the target p-wave energies are taken from two-point correlators on a 32^3 lattice at m_π≈702 MeV, any residual excited-state contamination directly shifts the fitted mass and therefore the extracted string tension; the manuscript does not appear to quantify this systematic via multi-exponential fits or variational methods.
  2. [Results / string-tension comparison] The claim that the string tension agrees within approximately 5% with the Wilson-loop result rests on the same tuned mass; without an explicit propagation of the uncertainty from the p-wave spectrum fit into the potential parameters, it is unclear whether the 5% figure already incorporates this source of error or only statistical uncertainties.
minor comments (2)
  1. [Formalism] Notation for the diquark mass (m_D) and the reduced mass in the Schrödinger equation should be defined explicitly in the methods section to avoid ambiguity when the static quark is infinitely heavy.
  2. [Lattice setup] The manuscript would benefit from a brief statement of the fitting range and functional form used to extract the p-wave energies from the two-point functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the importance of controlling systematics in the self-consistent mass determination. We address each major comment below and will incorporate the suggested improvements in a revised version.

read point-by-point responses

  1. Referee: [Numerical analysis / self-consistent mass determination] The self-consistent tuning of the diquark mass (described in the abstract and the numerical-analysis section) is load-bearing for both the reported mass value and the subsequent Cornell parameters. Because the target p-wave energies are taken from two-point correlators on a 32^3 lattice at m_π≈702 MeV, any residual excited-state contamination directly shifts the fitted mass and therefore the extracted string tension; the manuscript does not appear to quantify this systematic via multi-exponential fits or variational methods.

    Authors: We agree that a quantitative assessment of excited-state contamination in the two-point correlators is necessary to support the robustness of the self-consistent diquark mass. Our current analysis relied on identifying a plateau in the effective-mass plot and performing single-exponential fits over a suitable time range. To address the referee's concern, we will add multi-exponential fits (and, where feasible, a variational analysis with multiple operators) to the revised manuscript, extract the associated systematic uncertainty on the p-wave energies, and propagate it through the self-consistent procedure to the diquark mass and the extracted potential parameters. revision: yes

  2. Referee: [Results / string-tension comparison] The claim that the string tension agrees within approximately 5% with the Wilson-loop result rests on the same tuned mass; without an explicit propagation of the uncertainty from the p-wave spectrum fit into the potential parameters, it is unclear whether the 5% figure already incorporates this source of error or only statistical uncertainties.

    Authors: The quoted 5% agreement refers to the central values of the string tension obtained with the self-consistently tuned diquark mass. We did not propagate the uncertainty arising from the p-wave spectrum fit into the final potential parameters in the submitted version. In the revision we will include this systematic uncertainty (obtained from the multi-exponential analysis mentioned above) and present the string-tension comparison with the combined statistical and systematic errors, thereby clarifying the significance of the agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent lattice extractions and explicit fitting

full rationale

The paper explicitly determines the scalar diquark mass via self-consistent tuning to reproduce the p-wave spectrum measured in two-point correlators, while the quark-diquark potential (and its Cornell parameters including string tension) is extracted from the lattice baryonic system using the HAL QCD-inspired method and compared directly to an independent Wilson-loop static potential. This matching step is transparent and does not rename a fitted quantity as an independent prediction; the reported closeness to (2/3)m_N and the ~5% string-tension agreement are outcomes of the lattice data rather than reductions by construction. No load-bearing self-citation chain, ansatz smuggling, or uniqueness theorem is invoked to force the central results. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim depends on the self-consistent fitting procedure for the diquark mass and the applicability of the potential method, with no new entities postulated.

free parameters (1)
  • diquark mass
    Determined self-consistently by matching the spectrum, so it is fitted to the data within the model.
axioms (2)
  • domain assumption The HAL QCD-inspired potential method accurately extracts the effective potential from the lattice correlators in this system.
    Invoked in the application of the method to the baryonic system.
  • domain assumption The lattice configurations with m_π ≈ 702 MeV adequately represent the physics for this extraction.
    Used in the numerical calculations.

pith-pipeline@v0.9.1-grok · 5751 in / 1471 out tokens · 38889 ms · 2026-06-26T15:23:35.244836+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Scalar diquarks in the QCD vacuum

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    A first-principles FRG approach to two-flavor QCD derives low-energy constants for the pion, sigma-meson and scalar diquark without parameters beyond QCD itself, including new diquark properties for color-superconduct...

Reference graph

Works this paper leans on

22 extracted references · 8 canonical work pages · cited by 1 Pith paper

  1. [1]

    R. L. Jaffe, Exotica, Nucl. Phys. B Proc. Suppl. 142 (2005) 343

    2005

  2. [2]

    ’t Hooft, Physical Review D 14 (12) (1976) 3432–3450

    G. ’t Hooft, Physical Review D 14 (12) (1976) 3432–3450. doi:10.1103/physrevd.14.3432

    work page doi:10.1103/physrevd.14.3432 1976

  3. [3]

    Shuryak, The role of instantons in quantum chromo- dynamics, Nuclear Physics B 203 (1) (1982) 93–115

    E. Shuryak, The role of instantons in quantum chromo- dynamics, Nuclear Physics B 203 (1) (1982) 93–115. doi:10.1016/0550-3213(82)90478-3. 5

    work page doi:10.1016/0550-3213(82)90478-3 1982

  4. [4]

    Schäfer, E

    T. Schäfer, E. V . Shuryak, Instantons in qcd 70 (2) (1998) 323–425. doi:10.1103/revmodphys.70.323

    work page doi:10.1103/revmodphys.70.323 1998

  5. [5]

    Shuryak, I

    E. Shuryak, I. Zahed, A schematic model for pentaquarks based on diquarks, Physics Letters B 589 (1–2) (2004) 21–27. doi:10.1016/j.physletb.2004.03.019

    work page doi:10.1016/j.physletb.2004.03.019 2004

  6. [6]

    Workman, et al., Review of particle physics, Progress of Theoretical and Experimental Physics 2022 (8) (2022) 083C01

    R. Workman, et al., Review of particle physics, Progress of Theoretical and Experimental Physics 2022 (8) (2022) 083C01. doi:https://doi.org/10.1093/ptep/ptac097

    work page doi:10.1093/ptep/ptac097 2022

  7. [7]

    M. Hess, F. Karsch, E. Laermann, I. Wetzorke, Diquark masses from lattice qcd, Phys. Rev. D 58 (1997) 111502. arXiv:hep-lat/9804023

    Pith/arXiv arXiv 1997

  8. [8]

    Babich, N

    R. Babich, N. Garron, C. Hoelbling, J. Howard, L. Lel- louch, C. Rebbi, Diquark correlations in baryons on the lattice with overlap quarks, Phys. Rev. D 76 (2007) 074021. arXiv:hep-lat/0701023

    Pith/arXiv arXiv 2007

  9. [9]

    Y . Bi, H. Cai, Y . Chen, M. Gong, Z. Liu, H.-X. Qiao, Y .-B. Yang, Diquark mass differences from unquenched lattice qcd, Chin. Phys. C 40 (2016) 073106. arXiv:1510.07354

    Pith/arXiv arXiv 2016

  10. [10]

    Alexandrou, P

    C. Alexandrou, P. de Forcrand, B. Lucini, Evidence for diquarks in lattice qcd, Phys. Rev. Lett. 97 (2006) 222002. arXiv:hep-lat/0609004

    Pith/arXiv arXiv 2006

  11. [11]

    Francis, P

    A. Francis, P. de Forcrand, R. Lewis, K. Maltman, Di- quark properties from full qcd lattice simulations, JHEP 05 (2022) 062. arXiv:2106.09080

    arXiv 2022

  12. [12]

    Orginos, Diquark properties from lattice qcd, in: PoS LAT2005, 2006, p

    K. Orginos, Diquark properties from lattice qcd, in: PoS LAT2005, 2006, p. 054. arXiv:hep-lat/0510082

    Pith/arXiv arXiv 2006

  13. [13]

    Green, J

    J. Green, J. Negele, M. Engelhardt, P. Varilly, Spatial di- quark correlations in a hadron, in: PoS LATTICE2010, 2010, p. 140. arXiv:1012.2353

    Pith/arXiv arXiv 2010

  14. [14]

    Watanabe, N

    K. Watanabe, N. Ishii, Building diquark model from lat- tice qcd, Few Body Syst. 23 (3) (2021) 45

    2021

  15. [15]

    Watanabe, Quark-diquark potential and diquark mass from lattice qcd, Phys

    K. Watanabe, Quark-diquark potential and diquark mass from lattice qcd, Phys. Rev. D 105 (2022) 074510. arXiv:2111.15167

    arXiv 2022

  16. [16]

    Nishioka, N

    S. Nishioka, N. Ishii, Axialvector diquark mass and quark- diquark potential inΣ c, PoS (LATTICE2024) 305 (2025). arXiv:2502.06260

    arXiv 2025

  17. [17]

    Nishioka, N

    S. Nishioka, N. Ishii, Axialvector diquark Mass and quark-diquark potential inΣ c, PoS HADRON2025 (2026)

    2026

  18. [18]

    doi:10.22323/1.500.0013

    work page doi:10.22323/1.500.0013

  19. [19]

    Aoki, K.-I

    S. Aoki, K.-I. Ishikawa, N. Ishizuka, T. Izubuchi, D. Kadoh, K. Kanaya, Y . Kuramashi, Y . Namekawa, M. Okawa, Y . Taniguchi, A. Ukawa, N. Ukita, T. Yoshie, 2+1 flavor lattice qcd toward the physical point, Phys. Rev. D 79 (2009) 034503

    2009

  20. [20]

    Hasenfratz, F

    A. Hasenfratz, F. Knechtli, Flavor symmetry and the static potential with hypercubic blocking, Physical Review D 64 (3) (2001). doi:10.1103/physrevd.64.034504

    work page doi:10.1103/physrevd.64.034504 2001

  21. [21]

    Nagahiro, S

    H. Nagahiro, S. Yasui, A. Hosaka, M. Oka, H. Noumi, Structure of charmed baryons studied by pionic decays, Physical Review D 95 (1) (2017). doi:10.1103/physrevd.95.014023

    work page doi:10.1103/physrevd.95.014023 2017

  22. [22]

    S. Ueda, S. Aoki, T. Aoyama, K. Kanaya, H. Matsu- furu, S. Motoki, Y . Namekawa, H. Nemura, Y . Taniguchi, N. Ukita, Development of an object oriented lattice qcd code ’bridge++’, J. Phys. Conf. Ser. 523 012046. (523) (2014). 6

    2014