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arxiv: 2606.25551 · v1 · pith:PYZ7WHEFnew · submitted 2026-06-24 · ✦ hep-ph

SU(3)-flavor breaking as a structural probe of hidden-charm-strange 0⁻⁻ tetraquarks in a color-octet basis

Bing-Dong Wan , Jun-Hao Zhang , Yan Zhang , Ming-Yang Yuan This is my paper

Pith reviewed 2026-06-25 20:44 UTC · model grok-4.3

classification ✦ hep-ph
keywords tetraquarksQCD sum rulesSU(3) flavor breakinghidden-charm-strangecolor-octet currentsexotic quantum numbers0-- states
0
0 comments X

The pith

SU(3) flavor breaking produces grouped rather than uniform mass shifts that distinguish two color-octet current structures in hidden-charm-strange 0^{--} tetraquarks.

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

The paper applies QCD sum rules to hidden-charm-strange tetraquarks carrying the exotic quantum numbers J^{PC}=0^{--}. It keeps the strange-quark mass and strange condensates explicit through dimension eight in the operator product expansion while using two families of color-octet interpolating currents. The resulting SU(3)-breaking mass shifts remain ordered in the hidden-strange sector but split into two distinct groups in the charm sector: small shifts for the [c-bar c]_{8} imes [s-bar s]_{8} family and substantially larger shifts for the [c-bar s]_{8} imes [s-bar c]_{8} family. The larger shifts move the latter solutions toward the D_s^* D_{s1} threshold region. These patterns indicate that hidden strangeness functions as a structural discriminator rather than a simple overall offset.

Core claim

Within the chosen color-octet basis the strange-sector spectrum stays ordered, yet the induced charm-sector mass shifts fall into two groups: relatively small shifts appear for the [c-bar c]_{8_c} imes [s-bar s]_{8_c} configurations while the [c-bar s]_{8_c} imes [s-bar c]_{8_c} configurations receive substantially larger positive shifts, with one solution overlapping the D_s^* D_{s1} threshold within uncertainties and another exhibiting the largest shift; the hidden-bottom-strange sector is used only as a stability benchmark.

What carries the argument

Two families of color-octet interpolating currents, [c-bar c]_{8_c} imes [s-bar s]_{8_c} and [c-bar s]_{8_c} imes [s-bar c]_{8_c}, treated inside QCD sum rules that retain explicit strange-quark mass and strange condensates through dimension eight.

If this is right

  • The [c-bar s]_{8_c} imes [s-bar c]_{8_c} solutions move toward the D_s^* D_{s1} threshold region, with one overlapping it inside uncertainties.
  • One [c-bar s]_{8_c} imes [s-bar c]_{8_c} configuration receives the largest positive SU(3)-breaking shift.
  • The hidden-bottom-strange spectrum remains ordered and serves as a stability check.
  • Hidden strangeness distinguishes internal current structure in the exotic 0^{--} sector.

Where Pith is reading between the lines

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

  • Experimental searches could target mass patterns clustered near specific thresholds rather than a single shifted mass.
  • The same current-basis distinction may appear in other exotic sectors once sufficient flavor-breaking data become available.
  • Different current structures could imply different decay widths or production cross sections even at similar masses.

Load-bearing premise

The chosen color-octet currents and the truncation of the operator product expansion at dimension eight are sufficient to produce reliable relative mass shifts.

What would settle it

Observation of uniform rather than grouped SU(3)-breaking mass shifts across both families of color-octet currents in the hidden-charm-strange 0^{--} sector would falsify the structural-discrimination claim.

Figures

Figures reproduced from arXiv: 2606.25551 by Bing-Dong Wan, Jun-Hao Zhang, Ming-Yang Yuan, Yan Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: Feynman diagrams contributing to the OPE calculation. The thick solid line represents the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) The ratios of [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: OPE ratio, pole contribution, and mass curves for the hidden-charm-strange tetraquark state [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: OPE ratio, pole contribution, and mass curves for the hidden-charm-strange tetraquark state [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: OPE ratio, pole contribution, and mass curves for the hidden-charm-strange tetraquark state [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: OPE ratio, pole contribution, and mass curves for the hidden-bottom-strange benchmark [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: OPE ratio, pole contribution, and mass curves for the hidden-bottom-strange benchmark [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: OPE ratio, pole contribution, and mass curves for the hidden-bottom-strange benchmark [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: OPE ratio, pole contribution, and mass curves for the hidden-bottom-strange benchmark [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
read the original abstract

We study hidden-charm-strange tetraquark candidates with the exotic quantum number $J^{PC}=0^{--}$ to test whether SU(3)-flavor breaking acts as a universal mass shift or as a structural probe of a fixed color-octet current basis. Using $[\bar c c]_{8_c}\otimes[\bar s s]_{8_c}$-type and $[\bar c s]_{8_c}\otimes[\bar s c]_{8_c}$-type color-octet currents within QCD sum rules, we keep the strange-quark mass and strange condensates explicitly in the operator product expansion through dimension eight so that the strange-sector response can be traced at fixed color and Dirac structure. The hidden-charm-strange system is treated as the primary phenomenological target, while the hidden-bottom-strange sector serves as a stability benchmark. The strange-sector spectrum remains ordered, but the induced charm-sector shifts are grouped rather than uniform, with relatively small shifts for the $[\bar c c]_{8_c}\otimes[\bar s s]_{8_c}$ configurations and substantially larger shifts for the $[\bar c s]_{8_c}\otimes[\bar s c]_{8_c}$ ones. The $[\bar c s]_{8_c}\otimes[\bar s c]_{8_c}$ solutions are shifted toward the $D_s^*\bar D_{s1}$ threshold region, with one overlapping this region within uncertainties and another showing the largest positive SU(3)-breaking shift. Taken together, these features indicate that hidden strangeness can serve as a useful discriminator of internal current structure in the exotic $0^{--}$ sector.

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

3 major / 1 minor

Summary. The manuscript claims that SU(3)-flavor breaking serves as a structural probe for hidden-charm-strange 0^{--} tetraquarks in a color-octet basis. Using QCD sum rules with [ar c c]_{8_c} \otimes [ar s s]_{8_c} and [ar c s]_{8_c} imes [ar s c]_{8_c} currents, keeping m_s and strange condensates explicit through dimension eight in the OPE, the induced charm-sector mass shifts are reported to be grouped (small for the first current class, substantially larger for the second), with some solutions shifted toward the D_s^* ar D_{s1} threshold. The hidden-bottom-strange sector is treated as a stability benchmark, and the overall spectrum remains ordered, indicating that hidden strangeness discriminates internal current structure.

Significance. If the reported non-uniform grouping of shifts is shown to be robust, the result would supply a concrete phenomenological discriminator for tetraquark current structures in the exotic 0^{--} sector, extending beyond conventional mass spectroscopy. The explicit retention of strange parameters and the use of the bottom sector as benchmark are methodological strengths that could be leveraged for further studies.

major comments (3)
  1. [Abstract] Abstract: The central claim that charm-sector shifts are grouped rather than uniform is stated only qualitatively, with no numerical mass values, extracted shifts, Borel windows, continuum thresholds, or error budgets supplied. This prevents direct assessment of whether the grouping survives variations in the free parameters (Borel mass and s_0), which the stress-test note identifies as a potential source of post-hoc tuning.
  2. [Abstract] OPE truncation (abstract): The expansion is truncated at dimension eight while retaining explicit m_s and strange condensates. No estimate or bound is given for the size of neglected dimension-9/10 four-quark and gluon condensates, whose contributions in exotic channels routinely reach the scale of typical SU(3)-breaking differences and could erase or reverse the reported grouping between the two current classes.
  3. [Abstract] Threshold positioning (abstract, final paragraph): The statements that one [ar c s]_{8_c} imes [ar s c]_{8_c} solution overlaps the D_s^* ar D_{s1} region within uncertainties and another exhibits the largest positive shift are presented without the underlying mass values or stability information. This leaves open whether the positioning is stable against the continuum-threshold modeling choices highlighted in the skeptic's concern.
minor comments (1)
  1. The notation for the color-octet currents (e.g., subscript 8_c) would benefit from an explicit definition or reference to the current construction in an introductory section for improved readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the abstract to incorporate quantitative details as requested.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that charm-sector shifts are grouped rather than uniform is stated only qualitatively, with no numerical mass values, extracted shifts, Borel windows, continuum thresholds, or error budgets supplied. This prevents direct assessment of whether the grouping survives variations in the free parameters (Borel mass and s_0), which the stress-test note identifies as a potential source of post-hoc tuning.

    Authors: We agree that the abstract would benefit from explicit numerical content to permit immediate evaluation of the grouping's robustness. The body of the manuscript already contains the full set of extracted masses, shifts, Borel windows, continuum thresholds, and error estimates for both current classes, together with explicit stability checks against variations in the Borel parameter and s_0. In the revised version we will condense the key numerical results (representative masses and SU(3)-breaking shifts for each current type, together with the associated windows) into the abstract so that the grouping can be assessed directly from the abstract itself. revision: yes

  2. Referee: [Abstract] OPE truncation (abstract): The expansion is truncated at dimension eight while retaining explicit m_s and strange condensates. No estimate or bound is given for the size of neglected dimension-9/10 four-quark and gluon condensates, whose contributions in exotic channels routinely reach the scale of typical SU(3)-breaking differences and could erase or reverse the reported grouping between the two current classes.

    Authors: The referee is correct that the manuscript provides no explicit numerical bound on the omitted dimension-nine and dimension-ten operators. The body discusses OPE convergence through dimension eight and shows that the dimension-eight terms remain smaller than the leading contributions, but this does not constitute a direct estimate of the higher terms. We will add a short paragraph in the revised manuscript that supplies an order-of-magnitude estimate for the possible size of the neglected condensates, based on the pattern observed in the computed terms and on typical scales reported in the literature for similar exotic channels. If a fully quantitative bound proves unattainable without new calculations, we will state the limitation explicitly. revision: partial

  3. Referee: [Abstract] Threshold positioning (abstract, final paragraph): The statements that one [\bar c s]_{8_c} \times [\bar s c]_{8_c} solution overlaps the D_s^* \bar D_{s1} region within uncertainties and another exhibits the largest positive shift are presented without the underlying mass values or stability information. This leaves open whether the positioning is stable against the continuum-threshold modeling choices highlighted in the skeptic's concern.

    Authors: We accept that the abstract statements on threshold overlap and shift magnitude lack the supporting numbers. The manuscript already reports the explicit mass values, their uncertainties, and the stability of the solutions with respect to continuum-threshold variations. In the revised abstract we will quote the relevant mass values (with uncertainties) for the two [ar c s]_{8_c} \times [ar s c]_{8_c} solutions and note that both remain stable under the continuum-threshold variations examined in the full analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is a standard numerical QCD sum-rule computation

full rationale

The paper computes SU(3)-breaking mass shifts via QCD sum rules truncated at dimension eight, retaining explicit m_s and strange condensates for two families of color-octet currents. The reported non-uniform grouping of charm-sector shifts is presented as the direct numerical output of these sum rules (with hidden-bottom as benchmark), not as a definitional identity, a fitted parameter renamed as prediction, or a result justified solely by self-citation. No load-bearing step reduces to its own inputs by construction; the central claim that hidden strangeness discriminates current structure follows from the extracted spectrum ordering rather than from any tautological reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The calculation rests on the standard QCD sum-rule framework plus the explicit inclusion of strange mass and condensates; no new entities are introduced.

free parameters (2)
  • Borel mass window
    Standard parameter chosen for sum-rule stability; value not stated in abstract.
  • Continuum threshold
    Fitted parameter that defines the phenomenological side; value not stated in abstract.
axioms (2)
  • domain assumption Equivalence of OPE and phenomenological sides of the correlation function
    Core assumption of the QCD sum-rule method invoked throughout the abstract.
  • domain assumption Truncation at dimension eight is adequate for relative SU(3)-breaking shifts
    Stated in abstract as the order kept explicitly for strange-sector response.

pith-pipeline@v0.9.1-grok · 5846 in / 1461 out tokens · 30273 ms · 2026-06-25T20:44:55.514573+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

62 extracted references · 4 linked inside Pith

  1. [1]

    Shifman, A.I

    M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); ibid, Nucl. Phys. B147, 448 (1979)

    1979

  2. [2]

    L. J. Reinders, H. Rubinstein and S. Yazaki, Phys. Rept.127, 1 (1985)

    1985

  3. [3]

    S. K. Choiet al.[Belle Collaboration], Phys. Rev. Lett.91, 262001 (2003)

    2003

  4. [4]

    H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Phys. Rept.639, 1 (2016)

    2016

  5. [5]

    Y. R. Liu, H. X. Chen, W. Chen, X. Liu and S. L. Zhu, Prog. Part. Nucl. Phys.107, 237-320 (2019)

    2019

  6. [6]

    Esposito, A

    A. Esposito, A. Pilloni and A. D. Polosa, Phys. Rept.668, 1 (2016)

    2016

  7. [7]

    F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys.90, 015004 (2018)

    2018

  8. [8]

    Brambilla, S

    N. Brambilla, S. Eidelman, C. Hanhart, A. Nefediev, C. P. Shen, C. E. Thomas, A. Vairo and C. Z. Yuan, Phys. Rept.873, 1 (2020)

    2020

  9. [9]

    H. X. Chen, W. Chen, X. Liu, Y. R. Liu and S. L. Zhu, Rept. Prog. Phys.86, 026201 (2023)

    2023

  10. [10]

    Z. G. Wang, Front. Phys. (Beijing)21, 016300 (2026)

    2026

  11. [11]

    S. Q. Zhang and C. F. Qiao, [arXiv:2512.24706 [hep-ph]]

    arXiv

  12. [12]

    C. F. Qiao and L. Tang, Phys. Rev. Lett.113, 221601 (2014)

    2014

  13. [13]

    Tang and C

    L. Tang and C. F. Qiao, Nucl. Phys. B904, 282 (2016)

    2016

  14. [14]

    Z. R. Huang, W. Chen, T. G. Steele, Z. F. Zhang and H. Y. Jin, Phys. Rev. D95, 076017 (2017)

    2017

  15. [15]

    B. D. Wan, Eur. Phys. J. C84, 760 (2024)

    2024

  16. [16]

    C. M. Tang, C. G. Duan, L. Tang and C. F. Qiao, [arXiv:2511.18807 [hep-ph]]

    arXiv

  17. [17]

    B. D. Wan and H. T. Xu, Chin. Phys. C44, 093105 (2024)

    2024

  18. [18]

    B. D. Wan, J. H. Zhang, Y. Zhang and M. Y. Yuan, [arXiv:2605.21120 [hep-ph]]

    Pith/arXiv arXiv

  19. [19]

    L. Tang, B. D. Wan, K. Maltman and C. F. Qiao, Phys. Rev. D101, 094032 (2020)

    2020

  20. [20]

    Tang and C

    L. Tang and C. F. Qiao, Eur. Phys. J. C76, 558 (2016)

    2016

  21. [21]

    C. M. Tang, C. G. Duan and L. Tang, Eur. Phys. J. C84, 743 (2024)

    2024

  22. [22]

    C. M. Tang, C. G. Duan, L. Tang and C. F. Qiao, Eur. Phys. J. C85, 396 (2025)

    2025

  23. [23]

    R. M. Albuquerque, arXiv:1306.4671 [hep-ph]

    Pith/arXiv arXiv

  24. [24]

    Colangelo and A

    P. Colangelo and A. Khodjamirian, inAt the frontier of particle physics / Handbook of QCD, 16 edited by M. Shifman (World Scientific, Singapore, 2001), arXiv:hep-ph/0010175

    Pith/arXiv arXiv 2001

  25. [25]

    Narison, World Sci

    S. Narison, World Sci. Lect. Notes Phys.261 (1989)

    1989

  26. [26]

    Govaerts, L

    J. Govaerts, L. J. Reinders, H. R. Rubinstein and J. Weyers, Nucl. Phys. B258, 215-229 (1985)

    1985

  27. [27]

    B. D. Wan and C. F. Qiao, Nucl. Phys. B968, 115450 (2021)

    2021

  28. [28]

    B. D. Wan and C. F. Qiao, Phys. Lett. B817, 136339 (2021)

    2021

  29. [29]

    B. D. Wan, L. Tang and C. F. Qiao, Eur. Phys. J. C80, 121 (2020)

    2020

  30. [30]

    Z. G. Wang and T. Huang, Phys. Rev. D89, 054019 (2014)

    2014

  31. [31]

    C. M. Tang, Y. C. Zhao and L. Tang, Phys. Rev. D105, 114004 (2022)

    2022

  32. [32]

    Z. G. Wang, Phys. Rev. D101, 074011 (2020)

    2020

  33. [33]

    B. C. Yang, L. Tang and C. F. Qiao, Eur. Phys. J. C81, 324 (2021)

    2021

  34. [34]

    X. W. Wang, Z. G. Wang and G. l. Yu, Eur. Phys. J. A57, 275 (2021)

    2021

  35. [35]

    B. D. Wan, S. Q. Zhang and C. F. Qiao, Phys. Rev. D105, 014016 (2022)

    2022

  36. [36]

    F. H. Yin, W. Y. Tian, L. Tang and Z. H. Guo, Eur. Phys. J. C81, 818 (2021)

    2021

  37. [37]

    B. D. Wan, S. Q. Zhang and C. F. Qiao, Phys. Rev. D106, 074003 (2022)

    2022

  38. [38]

    S. Q. Zhang, B. D. Wan, L. Tang and C. F. Qiao, Phys. Rev. D106, 074010 (2022)

    2022

  39. [39]

    B. D. Wan and C. F. Qiao, [arXiv:2208.14042 [hep-ph]]

    arXiv

  40. [40]

    S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D107, 054017 (2023)

    2023

  41. [41]

    Y. C. Zhao, C. M. Tang and L. Tang, Eur. Phys. J. C83, 654 (2023)

    2023

  42. [42]

    S. N. Li and L. Tang, [arXiv:2404.11145 [hep-ph]]

    arXiv

  43. [43]

    B. D. Wan, Nucl. Phys. B1004, 116538 (2024)

    2024

  44. [44]

    B. D. Wan and S. Yang, Eur. Phys. J. A61, 11 (2025)

    2025

  45. [45]

    B. D. Wan and Y. R. Wang, Eur. Phys. J. A60, 179 (2024)

    2024

  46. [46]

    S. Q. Zhang, X. H. Zhang and C. F. Qiao, JHEP06, 122 (2024)

    2024

  47. [47]

    W. S. Zhang and L. Tang, Nucl. Phys. A1064, 123227 (2025)

    2025

  48. [48]

    X. H. Zhang, S. Q. Zhang and C. F. Qiao, Eur. Phys. J. C85, 693 (2025)

    2025

  49. [49]

    S. Q. Zhang and C. F. Qiao, Phys. Rev. D110, 114040 (2024)

    2024

  50. [50]

    B. D. Wan and J. C. Yang, Chin. Phys. C50, 043104 (2026)

    2026

  51. [51]

    B. D. Wan, Eur. Phys. J. Plus140, 873 (2025)

    2025

  52. [52]

    B. D. Wan, J. H. Zhang and Y. Zhang, Eur. Phys. J. C85, 1431 (2025)

    2025

  53. [53]

    S. S. Agaev, K. Azizi and H. Sundu, Phys. Rev. D112, 014003 (2025)

    2025

  54. [54]

    Barsbay, Eur

    B. Barsbay, Eur. Phys. J. A61, 242 (2025)

    2025

  55. [55]

    B. D. Wan, J. H. Zhang, Y. Zhang and M. Y. Yuan, Eur. Phys. J. C86, 493 (2026)

    2026

  56. [56]

    X. H. Zhang, S. Q. Zhang and C. F. Qiao, Phys. Rev. D113, 3 (2026)

    2026

  57. [57]

    B. D. Wan, Y. Zhang, J. H. Zhang and M. Y. Yuan, Eur. Phys. J. A62, 100 (2026). 17

    2026

  58. [58]

    B. D. Wan, M. Y. Yuan, J. H. Zhang and Y. Zhang, Eur. Phys. J. C86, 454 (2026)

    2026

  59. [59]

    Y. Q. Mu, P. W. Xu, S. T. Chen, Y. T. Wei, G. J. Zhang and B. D. Wan, [arXiv:2604.20439 [hep-ph]]

    Pith/arXiv arXiv

  60. [60]

    Ben and S

    D. Ben and S. Q. Zhang, [arXiv:2510.13548 [hep-ph]]

    arXiv

  61. [61]

    F. Z. Peng, M. J. Yan, M. S´ anchez S´ anchez and M. Pavon Valderrama, Phys. Rev. D107, 016001 (2023)

    2023

  62. [62]

    Navaset al.[Particle Data Group], Phys

    S. Navaset al.[Particle Data Group], Phys. Rev. D110, 030001 (2024). 18 Appendix A: Supplementary Figures Since the charm-sector currentJ A is already shown in the main text as the representative example, the supplementary charm-sector plots are limited to the remaining currentsJ B- JD. Their OPE, pole contribution, and masses as functions of the Borel pa...

    2024