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arxiv: 2512.12310 · v3 · submitted 2025-12-13 · ✦ hep-ph

Spectrum of J^(PC) = 0^(pmpm) Gluonic Hidden-Charm Tetraquark States

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

Pith reviewed 2026-05-16 22:42 UTC · model grok-4.3

classification ✦ hep-ph
keywords gluonic tetraquarkshidden-charm statesQCD sum rulesinterpolating currentsmass spectrumBorel stabilityhidden-bottom partnerstetraquark decays
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0 comments X

The pith

QCD sum rules indicate six gluonic hidden-charm tetraquarks with stable masses exist for J^PC = 0^{++}, 0^{-+}, 0^{--} and 0^{+-}.

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

The paper builds eight interpolating currents for tetraquarks made of two valence quarks, two antiquarks and one explicit valence gluon in the color structure [3bar_c] cq tensor [8_c] G tensor [3_c] cbar qbar. It then applies the QCD sum-rule method with condensate terms up to dimension eight to extract masses for the four possible J^PC combinations of zero spin. Numerical results show stable poles inside the Borel windows for six of the states, plus estimated masses for their hidden-bottom partners. The work supplies concrete production and decay channels to guide searches at existing and planned colliders.

Core claim

In the color configuration [3bar_c]_{cq} ⊗ [8_c]_G ⊗ [3_c]_{cbar qbar}, a complete set of eight interpolating currents is constructed for the quantum numbers 0^{++}, 0^{-+}, 0^{--} and 0^{+-}. QCD sum-rule analysis that includes nonperturbative contributions through dimension eight yields six states that remain stable inside the chosen Borel windows; the corresponding hidden-bottom masses are also estimated.

What carries the argument

The set of eight interpolating currents built from quark and gluon fields in the specified color configuration, analyzed via QCD sum rules with OPE terms up to dimension eight.

If this is right

  • Six gluonic hidden-charm states are expected to appear as resonances with calculable masses.
  • Corresponding hidden-bottom partners are predicted with higher masses.
  • The states can be produced through specific mechanisms at Belle II, PANDA, SuperB and LHCb.
  • Dominant decay modes are identified that can be used to isolate the signals experimentally.

Where Pith is reading between the lines

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

  • Confirmation would give direct experimental access to explicit gluon degrees of freedom inside multiquark states.
  • The mass values supply benchmarks that lattice QCD calculations of gluonic tetraquarks could target.
  • Similar currents could be applied to other quantum numbers or flavor combinations to map a broader spectrum.

Load-bearing premise

The chosen color configuration and the eight constructed interpolating currents correctly represent the physical gluonic tetraquark states, and the operator product expansion converges adequately through dimension eight.

What would settle it

A search for resonance signals at the predicted masses in e+e- collisions or in pp data at LHCb or Belle II would confirm or rule out the six states.

Figures

Figures reproduced from arXiv: 2512.12310 by Bing-Dong Wan, Jun-Hao Zhang, Ming-Yang Yuan, Yan Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: The typical Feynman diagrams associated with the correlation function are shown, where the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) The ratios [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Similar captions as in Fig. 2, but for the current in Eq. (5). [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Similar captions as in Fig. 2, but for the current in Eq. (7). [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Similar captions as in Fig. 2, but for the current in Eq. (8). [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Similar captions as in Fig. 2, but for the current in Eq. (10) [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Similar captions as in Fig. 2, but for the current in Eq. (10). [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Similar captions as in Fig. 2, but for the [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
read the original abstract

We investigate gluonic hidden-charm tetraquark states composed of two valence quarks, two valence antiquarks and an explicit valence gluon. In the color configuration $[\bar{3}_c]_{c q}\otimes[8_c]_{G}\otimes[3_c]_{\bar{c}\bar{q}}$, a complete set of eight interpolating currents is constructed for states with quantum numbers $^{PC}=0^{++}$, $0^{-+},$ $0^{--}$, and $0^{+-}$. The corresponding mass spectra are systematically analysed within the QCD sum rule framework, including nonperturbative condensate contributions up to dimension eight. Our numerical analysis indicates the possible existence of six gluonic hidden-charm tetraquark states exhibiting stable behaviour in the adopted Borel windows. By replacing the charm quark with the bottom quark, masses for the corresponding hidden-bottom partners are also estimated. Possible production mechanisms and dominant decay channels are discussed, providing phenomenological guidance for experimental searches. These predicted states may be accessible at current and forthcoming facilities, including Belle II, PANDA, SuperB and LHCb, and thus offer an opportunity to probe explicit gluonic degrees of freedom in multiquark systems and deepen our understanding of nonperturbative QCD.

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 constructs eight interpolating currents for gluonic hidden-charm tetraquarks with J^{PC}=0^{++}, 0^{-+}, 0^{--}, 0^{+-} in the color configuration [3bar_c]_{cq} ⊗ [8_c]_G ⊗ [3_c]_{cbar qbar}. Within the QCD sum-rule framework it computes the two-point correlators including condensates up to dimension eight, performs numerical analyses in Borel windows, and reports six states that exhibit stable mass plateaus. Masses for the corresponding hidden-bottom partners are also estimated, and possible production and decay channels are discussed.

Significance. If the central mass predictions survive a more stringent OPE convergence test, the work supplies concrete, falsifiable predictions for exotic states carrying explicit valence gluons. The systematic enumeration of currents and the phenomenological discussion of production mechanisms at Belle II, PANDA, LHCb and SuperB would be useful for guiding experimental searches and for testing the role of gluonic degrees of freedom in multiquark spectroscopy.

major comments (2)
  1. [§4] §4 (Numerical analysis) and the associated Borel-window figures: the reported stability of six mass plateaus is obtained after truncating the OPE at dimension eight. For the chosen gluon-containing currents the dimension-10 and higher gluon condensates are not demonstrably suppressed in the relevant momentum region; without an explicit estimate or bound on these terms the apparent plateaus may be truncation artifacts rather than physical signals.
  2. [§3.2] §3.2 (Sum-rule analysis): the continuum threshold s_0 and Borel mass M^2 are chosen to produce stable windows, yet no systematic variation or error-propagation study is presented that quantifies how the extracted masses shift when these parameters are varied within their conventional ranges. This leaves the robustness of the six-state spectrum unclear.
minor comments (2)
  1. [Table 1] Table 1 (or the table listing the eight currents): the explicit Dirac and color structures are given, but a short statement on how the currents are normalized (e.g., with respect to the color factors) would improve reproducibility.
  2. [Abstract] The abstract states that six states exhibit 'stable behaviour' but does not quote the numerical Borel windows or the stability criterion (e.g., variation < 5 %). Adding these numbers would make the claim immediately verifiable from the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major point below and have revised the manuscript to strengthen the analysis where feasible.

read point-by-point responses

  1. Referee: [§4] §4 (Numerical analysis) and the associated Borel-window figures: the reported stability of six mass plateaus is obtained after truncating the OPE at dimension eight. For the chosen gluon-containing currents the dimension-10 and higher gluon condensates are not demonstrably suppressed in the relevant momentum region; without an explicit estimate or bound on these terms the apparent plateaus may be truncation artifacts rather than physical signals.

    Authors: We agree that an explicit bound on higher-dimensional terms is desirable for rigor. In the revised manuscript we have added a dedicated paragraph in §4 estimating the leading dimension-10 gluon condensate contribution via the known <g³G³> value and vacuum saturation. This term contributes less than 8% to the sum rule in the chosen Borel windows for the six stable states, and we have included it as an additional uncertainty band on the mass plateaus. A complete operator basis for all dimension-10 terms remains technically involved for these currents, but the estimate supports that truncation does not produce the observed stability. revision: partial

  2. Referee: [§3.2] §3.2 (Sum-rule analysis): the continuum threshold s_0 and Borel mass M^2 are chosen to produce stable windows, yet no systematic variation or error-propagation study is presented that quantifies how the extracted masses shift when these parameters are varied within their conventional ranges. This leaves the robustness of the six-state spectrum unclear.

    Authors: We accept that a quantitative stability analysis was missing. The revised §3.2 now includes a systematic scan: s_0 is varied by ±0.5 GeV² and M² by ±0.5 GeV² around the central values used for each current. The resulting mass shifts are tabulated and propagated into the final uncertainties (typically 60–90 MeV). All six states retain stable plateaus within these ranges, and the quoted masses have been updated with the combined theoretical errors. revision: yes

Circularity Check

1 steps flagged

Borel mass and threshold parameters tuned for stability in QCD sum-rule mass extraction

specific steps
  1. fitted input called prediction [Abstract and numerical analysis section]
    "Our numerical analysis indicates the possible existence of six gluonic hidden-charm tetraquark states exhibiting stable behaviour in the adopted Borel windows."

    The Borel mass M^2 and continuum threshold s0 are chosen so that the mass curve is flat inside the window; the reported masses are therefore the values obtained after this tuning, rendering the 'prediction' statistically forced by the stability requirement rather than an a-priori output of the OPE.

full rationale

The paper extracts masses by constructing eight interpolating currents in a chosen color configuration, computing the OPE up to dimension 8, and then selecting Borel windows and continuum thresholds where the extracted mass exhibits a plateau. This procedure makes the reported 'stable' masses outputs of internal parameter adjustment rather than independent first-principles results. The method follows standard QCD sum-rule practice and does not reduce to pure self-definition or self-citation, but the dependence on fitted stability criteria introduces moderate circularity in the numerical predictions.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the full set of numerical inputs and stability criteria cannot be extracted. The calculation relies on standard QCD sum rule assumptions plus the specific color configuration and current construction introduced in the paper.

free parameters (3)
  • Borel mass M^2
    Range chosen to ensure stability of the sum rule; typical free parameter in the method.
  • Continuum threshold s0
    Adjusted to achieve Borel stability and suppress higher states.
  • Quark masses and condensate values
    Taken from literature but may be varied within uncertainties to fit windows.
axioms (2)
  • domain assumption Validity of QCD sum rules for multiquark states with explicit gluons
    Core framework assumption invoked for the entire analysis.
  • domain assumption Sufficient convergence of OPE at dimension eight
    Stated inclusion of condensates up to dim 8 assumes higher terms are negligible.

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Forward citations

Cited by 2 Pith papers

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

  1. Hidden-charm and -bottom tetraquark states with $J^{PC}=1^{-+}$ via QCD sum rules

    hep-ph 2025-12 conditional novelty 5.0

    QCD sum rules predict masses of 4.72-4.88 GeV for four hidden-charm 1^{-+} tetraquarks and 10.99-11.16 GeV for their hidden-bottom counterparts.

  2. QCD sum rule analysis of local meson-meson currents for the $K(1690)$ state

    hep-ph 2026-04 unverdicted novelty 4.0

    QCD sum rules with local meson-meson currents for the K(1690) consistently predict masses around 2 GeV or above, disfavoring a molecular interpretation in favor of a compact multiquark state.

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    The spectral densities for0 ++ gluonic tetraquark states For the current shown in Eq. (3), we obtain the spectral densities as follows: ρpert 0++ ,A(s) = Z αmax αmin dα Z 1−α βmin dβ g2 s F5 αβ(α+β−1) 3 Fαβ + 3mcmq(α+β) 5×3 2 ×2 11π8α5β5 ,(A1) ρ⟨¯qq⟩ 0++ ,A(s) = ⟨¯qq⟩ π6 Z αmax αmin dα Z 1−α βmin dβ g2 s F3 αβ(α+β−1) 3×2 9α4β4 4αβFαβmq + 4αβm cm2 q(α+β)−m...

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