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Investigating the internal structure of $X(6900)$ in the $2J/\psi$ decay channel

T0 review · 3 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read The X(6900) tetraquark's internal coupling is likely axial-vector–axial-vector, not vector–vector.

desk verdict Reasonable CQM calculation that prefers the A-A coupling for X(6900) based on the decay width; the preference is plausible but rests on an unproved cancellation of two crossed diagrams, and the abstract's 2-eta_c prediction is absent from the body. read the letter →

arxiv 2512.18569 v2 pith:DGBJGNJQ submitted 2025-12-21 hep-ph

Duo-Duo Lu , Shao-Zhou Jiang This is my paper · ORCID
classification hep-ph
keywords X(6900)fullycharmedtetraquarkdiquark-antidiquarkcovariantquarkmodeldecaywidthJ/psipairaxial-vectorcouplingstructure
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

This paper calculates the decay width of the fully charmed tetraquark candidate X(6900) into a pair of J/ψ mesons, assuming the state has a diquark–antidiquark [cc][c̄c̄] structure with spin-parity 2⁺⁺. Two possible Dirac couplings between the four constituent quarks are tested: vector–vector and axial-vector–axial-vector. The computed width for the axial-vector–axial-vector coupling lands in the 40–90 MeV range, matching the values observed by LHCb, ATLAS, and CMS, while the vector–vector coupling gives only 10–30 MeV, too small to explain the data. The paper therefore concludes that X(6900) is more likely built from axial-vector diquarks, and it predicts a decay width of 66–88 keV for the related X(6900)→2ηc channel.

What carries the argument

The central object is the nonlocal four-quark interpolating current for a diquark–antidiquark tetraquark, J^μν_X = [c_a^T C Γ₁ c_b][c̄_a Γ₂ C c̄_b^T] + (Γ₁↔Γ₂), with Γ₁,₂ chosen either as γ^μ⊗γ^ν (axial-vector–axial-vector) or γ^μγ⁵⊗γ^νγ⁵ (vector–vector). The vertex function is a Gaussian with scale Λ_X ≈ 6.8–7.2 GeV, and the coupling constants g_X and g_{J/ψ} are fixed by the compositeness condition. The decay amplitude is built from three Feynman diagrams, but two of them (the crossed quark-loop diagrams) are found to cancel exactly, leaving a single diagram whose scalar integrals N₁, N₂ (or R₁, R₂) determine the width.

What would settle it

Compute the crossed diagram contributions M_b + M_c explicitly (or replace the Gaussian vertex with a dipole form) and check whether the A-A width stays above the ~40 MeV lower edge of the data; if the cancellation fails or the shape change pulls the width below the measured values, the axial-vector–axial-vector conclusion would be weakened.

Watch

Extended reading notes

Core claim

In a covariant quark model with nonlocal four-quark currents, the decay X(6900)→2J/ψ is computed for two spin-2 tetraquark currents: the axial-vector–axial-vector (A-A) current Γ₁⊗Γ₂ = γ^μ⊗γ^ν and the vector–vector (V-V) current Γ₁⊗Γ₂ = γ^μγ⁵⊗γ^νγ⁵. The coupling constants are fixed by the compositeness condition, and the vertex is a Gaussian in the relative momenta. The A-A coupling yields a partial width that is clearly larger than the V-V result and consistent with the measured X(6900) widths across the three experiments, whereas the V-V coupling is too small unless the model parameters are pushed to extreme values. From this the paper argues that the internal structure of X(6900) is more

Load-bearing premise

The whole comparison with experiment rests on the assumed Gaussian vertex shape (with scale around the X mass) and on the asserted cancellation of two of the three Feynman diagrams; if either is wrong, the A-A decay width could shift outside the measured range.

Editorial extensions

If this is right

  • If the A-A structure is correct, the two charm quarks in each diquark pair into an axial-vector state, which constrains the quantum numbers of the diquark and the overall tetraquark wavefunction.
  • The same coupling structure predicts X(6900)→2ηc with a width of 66–88 keV, offering an independent testable channel.
  • The vector–vector interpretation is disfavored, meaning any future model or lattice calculation should focus on the axial-vector diquark configuration.
  • The covariant quark model, with its compositeness-condition normalization, proves able to describe at least one fully heavy exotic hadron, encouraging its extension to other tetraquark candidates.
  • The computed widths are only weakly sensitive to the vertex size parameters, so the ordering between A-A and V-V is a robust feature within this framework.

Reading between the lines

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

  • The absolute-width comparison hinges on the Gaussian vertex ansatz: a different functional form (e.g., a dipole) could rescale both A-A and V-V widths and potentially shift the verdict, though the A-A-over-V-V hierarchy might survive if the cancellation of crossed diagrams persists.
  • The asserted exact cancellation of the two crossed diagrams (M_b = M_c = 0) is a strong simplification; verifying it explicitly with momentum-dependent vertices would test whether the single-diagram dominance holds beyond the present setup.
  • The predicted 2ηc width could be measured at the LHC or a future collider; if it lands outside 66–88 keV, the common coupling constants inferred from the 2J/ψ channel would need revision.
  • A direct lattice QCD calculation of X(6900) hadronic decay widths, using the same spin-2 currents, could independently decide between A-A and V-V without relying on the model's vertex shape.
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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 paper calculates the decay width of the presumed tetraquark state X(6900) into J/psi J/psi in a covariant quark model with a diquark-antidiquark [cc][cbar cbar] current. Two possible Dirac structures are compared: vector-vector (V-V) and axial-vector-axial-vector (A-A) coupling. The coupling constants g_X and g_J/psi are fixed through compositeness conditions, not fitted to the observed width. The paper finds that the A-A coupling yields a width of roughly 40–90 MeV, which it claims is consistent with the measured X(6900)->2J/psi width, while the V-V coupling yields only about 10–30 MeV and is too small. The authors conclude that X(6900) is more likely to have the A-A internal structure. An additional prediction for X(6900)->2 eta_c, 66–88 keV, is stated in the abstract.

Significance. If the result holds, it provides a concrete, model-based discrimination between two plausible four-quark Dirac currents for the J^PC=2++ X(6900), going beyond mass-only calculations. The compositeness-condition treatment of g_X and g_J/psi is a strength: it avoids fitting the decay width, and the parameter scan over Lambda_X and Lambda_J/psi is transparent. The analytic expressions for the vertex functions and widths are given in sufficient detail that the calculation could, in principle, be reproduced. However, the central claim rests on two fragile pillars: the asserted cancellation of the crossed Feynman diagrams, and the absolute normalization set by an ad hoc Gaussian vertex ansatz. The significance is therefore conditional on these two points being properly justified.

major comments (3)
  1. [Sec. III, Eqs. (41), (44); Sec. IV] The vanishing of the crossed diagrams, M_b = M_c = 0, is asserted without a derivation. The only explanation in Sec. IV is a garbled sentence about quarks cancelling 'such as c_b(x2), bar c_a(x3) and c_a(x1), bar c_b(x4)'. No color factor, trace identity, or integral argument is supplied. This cancellation is load-bearing: it reduces the full amplitude to the single box diagram of Eqs. (40)/(43), and the difference between the A-A and V-V widths—the sign of the interference term in Eqs. (47)/(48)—is entirely carried by that box. If the crossed diagrams are nonzero, the width formulas and the A-A/V-V hierarchy can change. Please provide an explicit demonstration, or a numerical evaluation showing that the crossed diagrams are negligible.
  2. [Eq. (16), Eq. (31), Sec. IV, Fig. 3] The absolute decay width—and hence the statement that A-A is 'consistent with the experiments'—is controlled by the assumed Gaussian vertex functions, with Lambda_X chosen by hand as 6.8–7.2 GeV and Lambda_J/psi scanned over 2.8–3.2 GeV. These functional forms and scales are taken from the model literature, not derived from QCD. The paper concedes that V-V could become consistent if Lambda_X is very small and/or Lambda_J/psi is very large, but the same caveat applies to the A-A conclusion: a different vertex shape or a Lambda_X outside the quoted band could change the normalization significantly. The parameter scan alone does not bound this model dependence. Please either quantify the sensitivity to the vertex ansatz or temper the claim that the A-A result is uniquely consistent with experiment.
  3. [Abstract] The abstract states an additional prediction, Gamma(X(6900)->2 eta_c) = 66–88 keV, but nowhere in the manuscript is this decay calculated, nor are any formulas, inputs, or numerical results for eta_c given. This is not a presentation detail: it is a claim of a quantitative result that cannot be checked. Either add the calculation and results, or remove the eta_c prediction from the abstract.
minor comments (5)
  1. [Sec. I, paragraph 2] Typo: 'its mass is inconsistent with the the above experimental results' should read 'with the above experimental results'.
  2. [Sec. II A, Eq. (16)] The Gaussian form is written as phi_X(k^2)=exp(k^2/Lambda_X^2). Since the text says it should decrease rapidly in the ultraviolet in Euclidean space, please clarify the sign convention: as written, with a Euclidean k^2, the exponential grows. Presumably k^2 is the Minkowski argument or the Euclidean k^2 enters with a minus sign.
  3. [Sec. IV, last paragraph] The sentence 'Figures 1b and 1c is zero' is grammatically unclear; also the explanation of the cancellation is hard to follow. Rewrite to state precisely which quark lines are contracted and why the color/trace factors cancel.
  4. [Appendix A, Eq. (A12)] The statement R1=2N1, R2=2N2 'but with different g_X' is confusing: the definitions of N1,N2 already include g_X. Please clarify the relation between the scalar integrals used in Eqs. (39) and (42) and the explicit forms in the appendix.
  5. [Fig. 3] The vertical axis label is missing; the caption says 'Decay width' but the quantity plotted (presumably Gamma in MeV) should be stated on the axis or in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the X(6900)→2J/ψ width is computed from fixed model inputs and compared with external data; the main model choices are acknowledged ansätze, not parameters fitted to the predicted width.

full rationale

The paper's derivation chain is not circular. The couplings g_X and g_J/ψ are fixed by compositeness conditions using only masses and vertex parameters, e.g. Eq. (21): Z_X = 1 - g_X^2 dΠ_X^2(s)/ds|_{s=m_X^2} = 0, and Eq. (34) analogously for g_J/ψ. The decay width is then a genuine prediction, not a refit of the same observable. The Gaussian vertex form is explicitly introduced as a model ansatz: the paper states 'the strict form of φ_X is hard to calculate' and 'This paper chooses the Gaussian form, φ_X(k²)=exp(k²/Λ_X²)', with Λ_X chosen 'around the mass of X(6900)' (Λ_X=6.8∼7.2 GeV). This is an input assumption, not a masked prediction: the width is computed after the choice, and the comparison with experiment is made afterward. The A-A vs V-V discrimination arises from the sign of the interference term in Eqs. (47) and (48), which is a result of the trace algebra and the structure of the currents, not from adjusting any constant to reproduce the width. The unproved assertion M_b = M_c = 0 in Eqs. (41) and (44) is a potential correctness risk — if the crossed diagrams contribute, the numerical hierarchy could change — but it is not circular: the paper does not define X or its width in terms of that cancellation, nor is the claim equivalent to an input. There are also no load-bearing self-citations by the present authors; the covariant quark model formalism is cited from an external literature [31–38]. The caveat 'Unless Λ_X is very small and/or Λ_J/ψ is very large, the V-V coupling cannot be consistent with the experiments' is an honest statement of model dependence, again not an indication that the competing conclusion was encoded in the inputs. Overall, the calculation is self-contained and its central comparison is externally falsifiable, so no significant circularity is found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model contributes the full nonlocal-current machinery and the compositeness normalization, but the predictive content rests on hand-picked vertex scales (Λ_X, Λ_J/ψ), a constituent mass, and the assumption of a pure diquark–antidiquark composition. No entity is invented, and the target width is not used as an input.

free parameters (3)
  • Λ_X (Gaussian size parameter of X(6900) vertex) = 6.8–7.2 GeV (scan)
    Chosen 'around the mass of X(6900)' (Sec. II A, Eq. 16); not fitted to the decay width but varied for sensitivity; the absolute width scale depends on it.
  • Λ_J/ψ (Gaussian size parameter of J/ψ vertex) = 2.8, 3.0, 3.2 GeV (scan)
    Chosen by hand in Sec. II C (Eq. 31); affects absolute normalization of the width.
  • m_c (constituent charm quark mass) = 1.67 GeV
    Standard CQM input from prior literature (Sec. IV), not fitted here; enters all loop integrals and propagators. Fixed, with no variation shown.
assumptions (5)
  • domain assumption The compositeness condition Z_X = 1 − g_X² dΠ₂/ds|_{s=m_X²} = 0 (Eq. 21) fixes the coupling by requiring vanishing wave-function renormalization.
    Standard Weinberg compositeness prescription adopted from the CQM literature (Ref [40]); it anchors the absolute normalization of all widths. Unproved in the paper, but an established model postulate.
  • domain assumption X(6900) is a pure [cc][c̄c̄] diquark–antidiquark tetraquark with no molecular or mixed component.
    Stated in the abstract and Sec. I; if X(6900) is a hadronic molecule or a threshold effect, the A-A conclusion does not follow.
  • ad hoc to paper The nonlocal vertex functions are single Gaussians, ϕ_X(k²)=exp(k²/Λ_X²), ϕ_J/ψ(k²)=exp(k²/Λ_J/ψ²), with Λ_X around m_X.
    Eqs. (16) and (31). The functional form is borrowed from the CQM literature, but the scale range Λ_X=6.8–7.2 GeV is chosen by hand 'around the mass of X(6900)'; the absolute width scale depends on this choice, and only the chosen band is tested.
  • domain assumption The crossed decay diagrams vanish: M_b = M_c = 0 (Eqs. 41, 44).
    Secs. III and IV justify this by a one-sentence cancellation argument among the symmetrized terms of Eq. (5); no trace-level demonstration is given, and the result drives the quoted widths.
  • domain assumption Charm propagator uses constituent mass m_c=1.67 GeV with no confinement scale.
    Eq. (20) and Sec. IV; standard CQM input, but the specific value and the neglect of confinement/IR cutoffs are model choices.

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Pith. "Pith review of Investigating the internal structure of $X(6900)$ in the $2J/\psi$ decay channel." pith.science (2026). https://pith.science/paper/DGBJGNJQ

@misc{pith2026251218569,
  author       = {Pith},
  title        = {Pith review of: Investigating the internal structure of $X(6900)$ in the $2J/\psi$ decay channel},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGBJGNJQ}},
  note         = {Machine review of arXiv:2512.18569}
}
abstract

Assuming $X(6900)$ is a tetraquark state, the decay width of $X(6900)\to 2J/\psi$ is calculated in a covariant quark model, with the diquark-antidiquark $[cc][\bar{c}\bar{c}]$ picture. Two possible structures, vector-vector and axial-vector--axial-vector coupling, are investigated. The result indicates that the axial-vector--axial-vector coupling is consistent with the experiments. Additionally, as another application of the covariant quark model, the decay width of $X(6900)\to 2\eta_c$ is predicted to be $66\sim88$ keV.

Figures

Figures reproduced from arXiv: 2512.18569 by the authors.

Figure 1
Figure 1. FIG. 1: Feynman diagrams in this paper. (a), (b) and (c) are three [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Numerical results of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. also shows that ΓX is weakly dependent on both ΛX and ΛJ/ψ, for both couplings. Unless ΛX is very small and/or ΛJ/ψ is very large, the V-V coupling cannot consistent with the experiments. Table II and [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.