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Mass spectrum of the hidden-charm hybrid states via the QCD sum rules

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper predicts the masses of nine hidden-charm hybrid states with QCD sum rules, placing the exotic 1^-+ ground state at 4.02 ± 0.08 GeV.

desk verdict A systematic but scale-setting-dependent QCD sum-rule survey of hidden-charm hybrids; the mass table is a product of the author's energy-scale scheme rather than an unambiguous prediction. read the letter →

arxiv 2412.11038 v2 pith:NQIQHNXR submitted 2024-12-15 hep-ph

Zhi-Gang Wang This is my paper · ORCID
classification hep-ph PACS 12.39.Mk12.38.Lg
keywords hidden-charmhybridsQCDsumrulesexotichadronscharmoniummassspectrumgluonicexcitationsenergyscaledependenceoperatorproductexpansion
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 tries to establish a complete mass spectrum for hidden-charm hybrid states — charmonium-like bound states built from a charm quark, an anticharm quark, and a gluon — using QCD sum rules. It treats the vacuum condensates through dimension 6 at both leading and next-to-leading order, and it chooses the QCD renormalization scale with the formula $\mu=\sqrt{M_H^2-(2M_c)^2}$, using an effective charm-quark mass $M_c=1.82\,\mathrm{GeV}$. The central output is a set of nine ground-state masses, one for each $J^{PC}=0^{-+},0^{++},0^{--},1^{++},1^{+-},1^{-+},1^{--},2^{-+},2^{++}$, with the exotic $1^{-+}$ state at $4.02\pm0.08\,\mathrm{GeV}$. If these predictions survive comparison with experiment and lattice QCD, they convert a widely spread set of theoretical estimates into a checkable spectrum and provide pole residues that can feed calculations of the hybrids' strong decays.

What carries the argument

The argument is carried by two-point correlation functions built from hybrid interpolating currents, each a color-singlet combination of a charm and anticharm field with the gluon field-strength tensor $G^a_{\alpha\beta}$, for example $J^V_\mu(x)=\bar c_i(x)\gamma^\alpha G^a_{\alpha\mu}(x)t^a_{ij}c_j(x)$. After the Borel transform, the mass is extracted from the ratio of the first and zeroth moments of the QCD spectral density, with the continuum threshold $\sqrt{s_0}$ and the Borel window chosen so that the ground-state pole dominates and the operator product expansion converges. The load-bearing selection rule is the energy-scale formula $\mu=\sqrt{M_H^2-(2M_c)^2}$ with $M_c=1.82\,\mathrm{GeV}$, which fixes the scale at which the spectral densities are evaluated and makes the mass prediction consistent with the running of $m_c$, $\alpha_s$, and the condensates.

What would settle it

Measure the mass and quantum numbers of the lightest $1^{-+}$ hidden-charm state; a resonance established with $J^{PC}=1^{-+}$ at a mass differing from $4.02\pm0.08\,\mathrm{GeV}$ by more than the quoted uncertainty would contradict the central prediction, and a lattice-QCD determination of the same ground state outside this window would settle the question without new experiment.

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Extended reading notes

Core claim

The paper's central claim is that the hidden-charm hybrid spectrum can be predicted consistently in a single QCD-sum-rule framework. For nine quantum-number channels it obtains ground-state masses ranging from about 4.0 to 5.8 GeV and matching pole residues (Table 2). The signature result is the exotic $1^{-+}$ channel: both a vector and a tensor current give $4.02\pm0.08\,\mathrm{GeV}$ and $4.01\pm0.08\,\mathrm{GeV}$, which the author takes as the ground state and tentatively connects to the observed $X(4630)$ as its first radial excitation. The paper argues that including next-to-leading-order contributions and enforcing the energy scale formula produces good operator-product-expansion convergence and Borel platforms, so the predicted masses are stable outputs of the method rather than artifacts of the input scale.

Load-bearing premise

The load-bearing premise is that the effective charm quark mass $M_c=1.82\,\mathrm{GeV}$ and the energy-scale formula $\mu=\sqrt{M_H^2-(2M_c)^2}$, developed for tetraquark and pentaquark states, apply unchanged to hidden-charm hybrid states.

Editorial extensions

If this is right

  • If the $1^{-+}$ ground state really sits near 4.0 GeV, experimental searches for exotic charm hybrids should focus on that region rather than on the higher $Y(4260)$- and $X(4630)$-like masses.
  • The predicted pole residues (Table 2) can be used as inputs in three-point QCD sum rules to compute partial widths for decays such as $H\to D^{(*)}\bar D^{(*)}$, turning the mass spectrum into decay predictions.
  • A measured $1^{-+}$ state near 4.6 GeV would fit the paper's assignment of $X(4630)$ as the first radial excitation, given the assumed energy gap of about 0.6 GeV.
  • The nine-channel pattern, with heavier scalar and axial states around 4.8–5.8 GeV and lighter exotics around 4.0–4.4 GeV, provides an ordering that future lattice QCD and experiment can confirm or reject channel by channel.

Reading between the lines

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

  • An immediate test of the scheme is to vary $M_c$ between roughly 1.7 and 1.9 GeV and recompute the $1^{-+}$ mass; because $M_H\approx\sqrt{\mu^2+(2M_c)^2}$, the output should shift by hundreds of MeV if the formula is doing the work, so the stability of the central value under such variation would calibrate how much of the prediction is input rather than dynamics.
  • The paper's statement that next-to-leading-order terms can be absorbed into the pole residue could be checked by computing the full NLO spectral density for a single channel; if the mass shifts materially, the apparent Borel platform is partly an artifact of truncation.
  • If an exotic $1^{-+}$ hidden-charm state were established below about 3.9 GeV or above about 4.2 GeV, the proposed hierarchy of a 4.0 GeV ground state with $X(4630)$ as the first excitation would need revision, and tetraquark or molecular assignments would become more natural.
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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

4 major / 5 minor

Summary. The paper studies hidden-charm hybrid states with J^PC = 0^{-+}, 0^{++}, 0^{--}, 1^{++}, 1^{+-}, 1^{-+}, 1^{--}, 2^{-+}, and 2^{++} using QCD sum rules. The author constructs local interpolating currents containing a gluonic field strength, computes the OPE up to dimension-6 condensates at both leading and next-to-leading order, and applies Borel transformations to extract masses and pole residues. The energy-scale formula μ = sqrt(M_H^2 - (2M_c)^2) with M_c = 1.82 GeV is used to fix the renormalization scale in each channel. Table 2 lists the resulting masses, e.g., 4.02 ± 0.08 GeV for the 1^{-+} ground state, and the paper suggests that the LHCb X(4630) could be a radial excitation. The abstract claims this is the first exploration of energy-scale dependence in QCD sum rules for hidden-charm hybrids.

Significance. If the method were sound, the paper would provide a useful survey of hidden-charm hybrid masses and pole residues, with a more complete OPE treatment than earlier sum-rule analyses: the inclusion of both LO and NLO contributions and the dimension-6 condensates, including ⟨jj⟩, goes beyond several previous works. The explicit Borel-window tables and convergence plots are also helpful. However, the central predictions are seriously weakened by two structural issues: the energy-scale formula uses the predicted mass as input, making the extraction circular, and multiple currents for the same J^PC give widely different masses with no criterion to select the physical ground state. As presented, the paper does not deliver a unique, robust mass spectrum, although the underlying OPE calculations may be salvageable after substantial revision.

major comments (4)
  1. [Section 3, Eq. (32)] The energy-scale formula μ = sqrt(M_H^2 - (2M_c)^2) uses the unknown mass M_H to fix μ, while μ determines the QCD spectral densities and therefore M_H through Eq. (30). Since one of the four selection criteria is 'satisfying the energy scale formula,' the extraction is circular: the output masses are forced to lie near sqrt(μ^2 + (2M_c)^2). The uncertainties in Table 2 (δM_H around 0.06–0.10 GeV) do not include the freedom in μ or the uncertainty in M_c, even though these are the dominant systematic degrees of freedom. Please provide an analysis with μ treated as an independent variational parameter, or otherwise demonstrate that the results are not an artifact of this self-consistency condition.
  2. [Table 2, rows for 1^+- and 1^--] For J^PC = 1^{+-}, the currents J^0_{μν}, J^A_μ, and J^5_{μν} yield M_H = 4.36, 4.76, and 5.21 GeV, respectively; for J^PC = 1^{--}, J^5_{μν} and J^0_{μν} yield 4.07 and 5.61 GeV. All rows satisfy the same pole-dominance, OPE-convergence, Borel-window, and energy-scale criteria listed in Table 1. No off-diagonal correlators or mixing matrix are computed, and no criterion selects which entry is the physical ground state in these channels. The claimed mass spectrum is therefore not unique for these J^PC values, which is load-bearing for the central claim of a definite spectrum.
  3. [Section 3, paragraph on energy gaps] The assumed 0.6–0.7 GeV gap between the ground state and the first radial excitation is imposed for all nine channels with no independent justification, and it directly sets √s0 through the continuum threshold. Because δM_H is dominated by the choice of √s0 (the text itself notes δM_H ∼ δ√s0 ∼ 0.10 GeV for consistent choices), the ad hoc gap assumption should be varied over a wider range and its effect on Table 2 quantified.
  4. [Section 3, text after Eq. (32)] The effective charm-quark mass M_c = 1.82 GeV is adopted from the author's tetraquark and pentaquark analyses without an argument that it applies to hybrid states, whose valence gluon changes the color and Dirac structure of the current. Since M_c enters the scale-setting formula and the reported masses approximately satisfy M_H ≈ sqrt(μ^2 + (2M_c)^2), a different M_c would shift the entire spectrum by hundreds of MeV. Please provide a sensitivity study over M_c or a derivation of M_c for hybrid interpolating currents.
minor comments (5)
  1. [Title] The title contains a typo: 's um' should be 'sum'.
  2. [Eq. (23)] The definition of G^a_{αβ} contains an obvious typo: '∂_α G^a_β − ∂_α G^a_β' should be '∂_α G^a_β − ∂_β G^a_α'.
  3. [Eq. (31)] The running of ⟨q̄q⟩ and m_c is written with n_f in the exponent, but the text later says n_f = 4 is chosen; please state explicitly which Λ_QCD value is used for n_f = 4 and whether the running formula is meant to hold with fixed n_f across the entire scale interval.
  4. [Figure 3 caption] The caption lists '1^{-+}, 1^{-+}, 1^{--}, 0^{-+}' without distinguishing the two different currents that give the two 1^{-+} rows; please clarify which panel corresponds to J^V_μ and which to J^{σ,0/5}_{μν}.
  5. [Abstract] The claim that this is 'the first time to explore the energy scale dependence of the QCD sum rules for the hidden-charm hybrid states' is not supported by a comparison with the cited literature beyond the author's own series of papers; the novelty claim should be either documented or softened.

Circularity Check

2 steps flagged · score 7.0 of 10

Energy-scale formula makes the predicted hybrid masses self-consistent by construction; M_c = 1.82 GeV is imported via self-citation.

  1. self definitional [Section 3, numerical criteria paragraph; Eq. (32); Tables 1–2]
    "we assume the energy gaps are about 0.6 ∼ 0.7 GeV, just like in the case of the hidden-charm tetraquark (molecular) states and pentaquark (molecular) states [9, 54, 55, 56, 57, 58, 60, 63, 64], and change the continuum threshold parameters s0 and Borel parameters T^2 to satisfy the four criteria: • Pole dominance at the hadron side; • Convergence of the operator product expansion; • Appearance of the Borel platforms; • Satisfying the energy scale formula, via trial and error."

    Eq. (32) defines the QCD scale as μ = sqrt(M_H^2 − (2M_c)^2), i.e., in terms of the very mass M_H that the sum rules are meant to predict. The quoted procedure then adjusts μ by trial and error until the computed M_H satisfies this defining relation. With M_c = 1.82 GeV fixed, the relation forces M_H ≈ sqrt(μ^2 + 3.64^2); e.g., the J^V 1^-+ entry has μ = 1.7 GeV and M_H = 4.02 GeV, exactly sqrt(1.7^2 + 3.64^2). The reported spectrum is thus imposed by the scale choice, not independently determined by the OPE.

  2. ansatz smuggled in via citation [Section 3, Eq. (32) and surrounding text; Refs. [9,54–58,63]]
    "In our previous works, we take the energy scale formula, µ = sqrt(M^2_{X/Y/Z} − (2M_c)^2), (32) to choose the optimal energy scales of the QCD spectral densities for the hidden-charm tetraquark (molecular) states and pentaquark (molecular) states [9, 54, 55, 56, 57, 58, 60, 63, 64], where the effective c-quark mass M_c = 1.82 GeV for the diquark type tetraquark and pentaquark states [9, 54, 55, 56, 57, 58, 63]. In this work, we adopt the value M_c = 1.82 GeV."

    The load-bearing scale-setting scheme (the μ formula and M_c = 1.82 GeV) is carried over from the author's previous tetraquark and pentaquark sum-rule papers by self-citation. No derivation or hybrid-specific test is given; the text simply says 'we adopt'. Because these inputs fix μ and hence, through the self-consistency relation, the final masses, the central prediction reduces to an unverified self-cited ansatz. The cited prior works themselves proposed the formula as a scheme, so this is an ansatz imported via citation rather than an external theorem.

full rationale

The paper contains a genuine OPE calculation: spectral densities with LO and NLO condensates up to dimension 6 are computed, and Borel windows and continuum thresholds are varied with pole-dominance and convergence checks. That part is not circular. The circularity enters at the level of scale setting. The energy-scale formula Eq. (32) defines μ through the predicted mass M_H, and the numerical procedure explicitly enforces 'Satisfying the energy scale formula' by trial and error. With M_c fixed at 1.82 GeV (imported, via self-citation, from the author's tetraquark/pentaquark analyses), the reported masses satisfy M_H ≈ sqrt(μ^2 + (2M_c)^2) almost exactly (e.g., 4.02 GeV for μ = 1.7 GeV), so the spectrum is a restatement of the chosen scales rather than an independent prediction. The non-uniqueness for 1^+- and 1^-- (4.36/4.76/5.21 GeV and 4.07/5.61 GeV from different currents) is the same effect: each current is assigned a different μ and the mass follows. The central claim therefore reduces, by construction, to the self-cited scale-setting ansatz. Not all content is circular—the sum-rule machinery is non-trivial—but the headline mass spectrum is forced by the input-output consistency condition.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The calculation depends on a small set of standard inputs (quark masses, condensates) plus three non-standard choices: the effective quark mass M_c, the per-channel energy scale μ, and the assumed radial excitation gap. The first two are coupled through Eq. (32) and substantially control the output masses, so the predictions are not parameter-free.

free parameters (3)
  • Effective charm quark mass M_c = 1.82 GeV
    Taken from the author's previous tetraquark and pentaquark sum-rule analyses and applied to hybrids without a dedicated derivation; it determines the energy scale μ through Eq. (32) and thus sets the mass scale of the predictions.
  • Energy scale μ per channel = 1.7 to 4.5 GeV (Table 1)
    Chosen by trial and error to satisfy pole dominance, OPE convergence, Borel flatness, and the energy scale formula; the masses are then reported for the μ values that satisfy these criteria.
  • Ground state to radial excitation gap = 0.6 to 0.7 GeV
    Assumed identical to hidden-charm tetraquark and molecular states to guide the continuum threshold and to tentatively assign X(4630) as the first radial excitation of the 1^{-+} hybrid.
assumptions (4)
  • domain assumption The QCD sum rule approach with a single-pole plus continuum model correctly extracts ground state masses.
    Standard assumption in QCD sum rules; the paper chooses pole dominance of 40-60% as a criterion.
  • ad hoc to paper The energy scale formula μ = sqrt(M_H^2 - (2M_c)^2) with M_c = 1.82 GeV holds for hidden-charm hybrids.
    Imported from the author's tetraquark/pentaquark studies; no derivation is given for hybrid systems.
  • ad hoc to paper The first radial excitation is separated from the ground state by about 0.6-0.7 GeV for all nine hybrid channels.
    Assumed from other exotic hadron systems; used to set s0 and to identify X(4630) as a radial excitation.
  • domain assumption Truncating the operator product expansion at dimension-6 gives a reliable expansion.
    The paper argues higher-dimensional terms are suppressed, but does not quantify all neglected dimension-8 terms that earlier work (Ref. [40]) included.

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Pith. "Pith review of Mass spectrum of the hidden-charm hybrid states via the QCD sum rules." pith.science (2026). https://pith.science/paper/NQIQHNXR

@misc{pith2026241211038,
  author       = {Pith},
  title        = {Pith review of: Mass spectrum of the hidden-charm hybrid states via the QCD sum rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NQIQHNXR}},
  note         = {Machine review of arXiv:2412.11038}
}
abstract

In this work, we study the mass spectrum of the hidden-charm hybrid states with the $J^{PC}=0^{-+}$, $0^{++}$, $0^{--}$, $1^{++}$, $1^{+-}$, $1^{-+}$, $1^{--}$, $2^{-+}$ and $2^{++}$ via the QCD sum rules in a consistent way. We calculate the vacuum condensates up to dimensions-6 by taking account of both the leading order and next-to-leading order contributions, and take the energy scale formula $\mu=\sqrt{M^2_{X/Y/Z}-(2{\mathbb{M}}_c)^2}$ to choose the suitable energy scales of the QCD spectral densities, it is the first time to explore the energy scale dependence of the QCD sum rules for the hidden-charm hybrid states.

Figures

Figures reproduced from arXiv: 2412.11038 by the authors.

Figure 1
Figure 1. The contributions of the vacuum condensates for the hy [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The mass of the hidden-charm hybrid state with the [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The masses of the hidden-charm hybrid states, where th [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The pole contribution of the hybrid state with the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: The Feynman diagram for the decays of the hidden-charm [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. A novel configuration of gluonic tetraquark state

    hep-ph 2024-11 conditional novelty 5.0 of 10

    The predicted masses of the octet-octet-octet tetracharm hybrid states, 6.98 GeV (0++) and 7.26 GeV (0-+), overlap the observed X(6900) and X(7200).

Reference graph

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Reviewed August 11, 2026 · model on record in the stance chip above.