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Analysis of the hidden-charm tetraquark mass spectrum with the QCD sum rules

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A twenty-current QCD sum-rule analysis assigns a dozen observed X and Z states to hidden-charm tetraquarks, and argues the method is valid from zeroth order in the strong coupling.

desk verdict The paper's value is the new OPE work for eleven tetraquark currents; the mass table is plausible but not an independent prediction because the scale and threshold are fitted to the experimental states. read the letter →

arxiv 1908.07914 v4 pith:7ZHQNIKD submitted 2019-08-20 hep-ph

Zhi-Gang Wang This is my paper · ORCID
classification hep-ph PACS 12.39.Mk12.38.Lg
keywords TetraquarkstateQCDsumruleshidden-charmstatesX(3872)Zc(3900)diquark-antidiquarkcurrentsenergy-scaleformulaoperatorproductexpansion
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 that the observed hidden-charm X and Z states can be understood, in one consistent QCD sum-rule treatment, as ground-state tetraquarks built from diquark-antidiquark currents. It constructs twenty scalar, axialvector, and tensor tetraquark currents, carries the operator product expansion through vacuum condensates of dimension 10, and fixes the Borel scale with the formula $\mu = \sqrt{M^2 - (2M_c)^2}$ using a universal effective charm-quark mass $M_c = 1.82$ GeV. The resulting masses match X(3860), X(3872), X(3915), Zc(3900), Zc(4020), Zc(4050), Zc(4055), Zc(4430), and Zc(4600) with specific diquark spin structures, while leaving no pure-tetraquark room for X(3940), X(4160), Zc(4100), and Zc(4200) without extra mixing. It also argues that tetraquark QCD sum rules are feasible and that color-singlet tetraquark currents begin contributing at order $\mathcal{O}(\alpha_s^0)$, not $\mathcal{O}(\alpha_s^2)$. A sympathetic reader would care because this offers a unified and testable assignment scheme for exotic states that the two-quark model cannot place.

What carries the argument

The carrying object is the interpolating tetraquark current: a local four-quark operator built from two color-antitriplet diquarks in scalar, pseudoscalar, axialvector, vector, or tensor Dirac structures and their antidiquark counterparts, giving twenty currents with $0^{++}$, $1^{+-}$, $1^{++}$, and $2^{++}$ quantum numbers. Its two-point correlation function is expanded on the quark side in vacuum condensates up to dimension 10 and matched, after a Borel transform (a Laplace-like operation that suppresses excited states), to a single ground-state pole on the hadron side. The carrying identity is the energy-scale formula $\mu = \sqrt{M^2 - (2M_c)^2}$ with $M_c = 1.82$ GeV, which sets the renormalization scale for each channel and is checked together with pole dominance, OPE convergence, and the appearance of Borel platforms.

What would settle it

Measure the spin-parity of Zc(4050) and Zc(4055): the scheme requires Zc(4050) to be $1^{++}$ or $2^{++}$ and Zc(4055) to be $1^{+-}$, so a confirmed $1^{+-}$ for Zc(4050) or a $1^{++}$ for Zc(4055) would rule out Table 5. A purely calculational check would be to repeat the twenty sum rules with $M_c = 1.275$ GeV instead of $1.82$ GeV and see whether the mass agreement survives.

Watch

Extended reading notes

Core claim

The paper's central claim is that a QCD sum-rule calculation using twenty local diquark-antidiquark currents predicts ground-state hidden-charm tetraquark masses that coincide with a specific set of experimental X and Z states. The calculation takes scalar, pseudoscalar, axialvector, vector, and tensor diquark operators as building blocks, evaluates two-point correlators with the operator product expansion carried through dimension-10 condensates, and selects Borel windows and continuum thresholds with the energy-scale formula $\mu = \sqrt{M^2 - (2M_c)^2}$, $M_c = 1.82$ GeV. In this scheme X(3860) is a $[uc]_S[dc]_S$ scalar, X(3915) a $[uc]_A[dc]_A$ scalar, X(3872) the symmetric $[uc]_S[dc]_A + [uc]_A[dc]_S$ axialvector, Zc(3900) the antisymmetric $[uc]_S[dc]_A - [uc]_A[dc]_S$ axialvector, while Zc(4020)/Zc(4055), Zc(4050), Zc(4430), and Zc(4600) are assigned to axialvector and tensor currents or to radial excitations. The paper further claims that four of the observed states, X(3940), X(4160), Zc(4100), and Zc(4200), cannot be accommodated as pure tetraquarks without fine-tuning, and that the sum-rule treatment of both diquark-antidiquark and color-singlet tetraquark currents is valid because these states receive leading contributions at order $\mathcal{O}(\alpha_s^0)$, not $\mathcal{O}(\alpha_s^2)$.

Load-bearing premise

The load-bearing premise is that the QCD energy scale in every channel is set by the formula $\mu = \sqrt{M^2 - (2M_c)^2}$ with a single effective charm-quark mass $M_c = 1.82$ GeV, and that each continuum threshold can be placed about $0.58$--$0.59$ GeV above the target mass; both choices are tuned so the extracted masses land on the experimental states, so if either prescription is changed the Table 5 assignments do not follow.

Editorial extensions

If this is right

  • If the assignments are right, X(3860), X(3915), and X(3872) are compact tetraquarks with definite diquark spin couplings: scalar-scalar, axialvector-axialvector, and symmetric scalar-axialvector, respectively.
  • Zc(3900) and Zc(4430) form a ground-state and first-radial-excitation pair in the same antisymmetric scalar-axialvector channel, with the mass gap mirroring the $J/\psi$--$\psi'$ splitting.
  • Zc(4020), Zc(4055), and Zc(4600) are assigned to nearly degenerate $1^{+-}$ axialvector or tensor diquark currents, with Zc(4600) possibly the first radial excitation of Zc(4020).
  • The pure-tetraquark scenario is disfavored for X(3940), X(4160), Zc(4100), and Zc(4200), leaning instead toward conventional $\eta_c(3S)$/$\eta_c(4S)$ assignments for the X states and toward mixing or color-octet-octet currents for the Z states.
  • Diquark-antidiquark and color-singlet tetraquark currents become legitimate objects for QCD sum rules, with leading contributions starting at $\mathcal{O}(\alpha_s^0)$ rather than $\mathcal{O}(\alpha_s^2)$.

Reading between the lines

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

  • Inference: The paper's stated rule that $M_c$ is simply replaced by $M_b$ for bottom quarks implies the same twenty-current machinery predicts a hidden-bottom analogue spectrum; comparing those predictions with the observed $Z_b$ states would test whether the effective heavy-quark mass is universal.
  • Inference: The threshold choice $\sqrt{s_0} = M_Z + 0.58/0.59$ GeV acts as an implicit prediction that each assigned ground state has a radial partner roughly $0.5$--$0.6$ GeV higher, so the same final states should show additional peaks at those energies.
  • Inference: The paper notes that hand-picked scales $\mu = 1.2$ GeV and $1.4$ GeV reproduce Zc(4200) and Zc(4100); this flexibility suggests the central assignment table would be more decisive if the energy scale were fixed by an independent observable, such as a conventional charmonium mass or decay width, rather than by the target mass itself.
  • Inference: The pole residues computed here are not yet used, although the decay channels are listed; feeding them into three-point sum rules would turn each mass assignment into a width prediction that experiments can check.
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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 / 4 minor

Summary. The manuscript constructs twenty local diquark-antidiquark interpolating currents for hidden-charm tetraquark states with quantum numbers 0++, 1+-, 1++, and 2++, performs QCD sum-rule analyses with the operator product expansion carried to dimension 10, and extracts ground-state masses and pole residues. It then assigns the known X and Z states to the currents whose predicted masses are closest, and argues that tetraquark and color-singlet-color-singlet (molecular) currents receive contributions at O(alpha_s^0) rather than O(alpha_s^2).

Significance. A reliable comprehensive QCD sum-rule survey of twenty tetraquark currents would be a useful contribution, and the paper does check the four standard sum-rule criteria, reporting pole contributions around 40-60%, |D(10)| mostly below 1%, and Borel windows. The OPE to dimension 10 and the systematic treatment of the current set are positive technical features. However, the predictive content of the mass extraction is compromised by the way the energy scale and continuum threshold are fixed, and the spectral densities are not displayed, so the central assignment claims are not supported as they stand.

major comments (4)
  1. [Sec. 3, Eqs. (2) and (15), Tables 3 and 5] The mass extraction is not independent of the states it is used to assign. The energy scale is fixed by mu = sqrt(M^2 - (2M_c)^2) with M_c = 1.82 GeV, where M is the mass of the experimental candidate being assigned, and the continuum threshold is chosen as sqrt(s0) = M + 0.58/0.59 GeV 'via trial and error.' The Borel window is then selected so that the pole contribution is 40-60%. Because the same experimental mass enters through mu and s0 and only then leaves through Eq. (15), the agreement in Table 5 is largely a restatement of inputs. The four criteria constrain the window but not the mean value. To support the assignments, the paper must show that the masses in Table 4 are stable under independent variation of mu and s0, for example by scanning these parameters without reference to the target masses or by adopting a fixed threshold prescription.
  2. [Sec. 2, after Eq. (14)] The QCD spectral densities rho(s) are not given; the text states that they are 'available upon request.' Since Eq. (15) is determined by these densities, the central numerical results cannot be checked or reproduced from the paper. The manuscript should include the explicit spectral functions in an appendix or as supplementary material.
  3. [Sec. 3, final paragraph, and Sec. 4] The paper itself demonstrates the sensitivity of the assignments to the energy-scale parameter by noting that hand-setting mu = 1.2 GeV reproduces Zc(4200) and mu = 1.4 GeV reproduces Zc(4100). This shows that the energy-scale prescription, not the OPE alone, controls which experimental states are selected. Combined with the first comment, it also makes the 'no room' conclusions for Zc(4100) and Zc(4200) conditional on the same fitted input, so those conclusions cannot be presented as robust predictions of the method.
  4. [Sec. 2, digression on O(alpha_s^0) contributions] The claim that color-singlet-color-singlet type tetraquark currents begin to receive contributions at O(alpha_s^0), which appears in the abstract, is not derived in this paper but is asserted on the basis of Refs. [37,39]. Since this is one of the advertised conclusions, it should either be substantiated with the relevant calculation here or explicitly presented as a review of previous work rather than a new result of this analysis.
minor comments (4)
  1. [Fig. 1] The horizontal-axis labels in Fig. 1 appear garbled in the manuscript; the figure should be regenerated so that the Borel-parameter values are legible.
  2. [Table 3] The column header 'pole' is not defined in the table; the pole contribution PC is defined in Eq. (21), but the table should identify the column as the pole contribution for clarity.
  3. [Sec. 2, around Eq. (13)] The digression on Fierz rearrangement and the O(alpha_s) counting is long and uses the auxiliary notation gamma^t_mu, gamma^v_mu, sigma^t, and sigma^v that is not needed in the main sum-rule derivation; consider moving this discussion to an appendix or substantially shortening it.
  4. [Sec. 3, paragraph after Eq. (16)] The effective charm-quark mass M_c = 1.82 GeV is introduced as 'updated' with reference [41], but no uncertainty or derivation is given; since this value is a central input to the energy-scale formula, its origin and uncertainty should be stated.

Circularity Check

3 steps flagged · score 7.0 of 10

The mass extraction is not an independent prediction: Eq. (2) sets μ from the experimental mass and √s0 = M + 0.58/0.59 GeV, so the Table 5 agreement is largely a restatement of input choices; the secondary feasibility claim rests on self-citation.

  1. self definitional [Introduction, Eq. (2); Section 3, paragraph following Eq. (17)]
    "In Ref.[19], we suggest a formula, µ = sqrt(M^2_{X/Y/Z} − (2M_c)^2), with the effective heavy mass M_c to determine the optimal energy scales ... In the present work, we use the energy scale formula µ = sqrt(M^2_{X/Y/Z} − (2M_c)^2) with the updated effective c-quark mass M_c = 1.82 GeV to determine the ideal energy scales for the QCD sum rules [41]."

    The mass extracted from Eq. (15) depends on the spectral density ρ(s, μ), while μ is fixed by the same experimental mass M_{X/Y/Z} that the extracted mass is later compared with. With M_c fixed, M^2 = μ^2 + (2M_c)^2 is imposed as an input, not obtained from the OPE. Thus the agreement between Table 4 and Table 5 for the assigned states is built into the scale choice. The paper's own admission that hand-setting μ = 1.2 GeV reproduces Zc(4200) and μ = 1.4 GeV reproduces Zc(4100) shows how much assignment power this one parameter carries.

  2. fitted input called prediction [Section 3, paragraph on continuum thresholds; Tables 3–5]
    "In the present work, we tentatively choose the continuum threshold parameters as √s0 = M_Z + 0.58/0.59 GeV and vary the continuum threshold parameters s0 and Borel parameters T^2 to satisfy the following 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."

    Equation (15) is a ratio of truncated Laplace integrals over ρ(s) from 4m_c^2 to s0. With √s0 chosen as M_target + 0.58/0.59 GeV, the integration interval is anchored just above the target state, and the Borel window is then selected by trial and error to give 40–60% pole contribution. The output mass is therefore a moment of a spectral density cut off by the very mass it is supposed to predict. Calling the Table 4 results 'predictions' that 'support' the Table 5 assignments is misleading: the assigned mass entered through s0 (and μ), so the comparison is largely a consistency check on the input choices.

1 more flagged steps
  1. self citation load bearing [Section 2, digression on the feasibility of QCD sum rules for tetraquark states]
    "In Ref.[37], I refute the assertion of Lucha, Melikhov and Sazdjian in order and in details, and use two examples to illustrate that the two-meson scattering states cannot saturate the QCD sum rules, while the 1c1c-type tetraquark states can saturate the QCD sum rules, the 1c1c-type tetraquark states begin to receive contributions at the order O(α_s^0/α_s^1) rather than at the order O(α_s^2)."

    This is the paper's answer to the external objection in Ref. [36] that tetraquark contributions start at O(α_s^2). The rebuttal is not derived in the present paper; it is delegated to Ref. [37], a paper by the same author, and to Ref. [39] (also the same author). The abstract and Section 2 then present the O(α_s^0) conclusion as a result. Absent an independent, machine-checked or externally reproduced derivation, the load-bearing support for this methodological claim is a self-citation chain.

full rationale

The paper contains a substantive OPE calculation: twenty tetraquark currents, vacuum condensates through dimension 10, and a real QCD sum-rule machinery. That is not vacuous. However, the extraction is not an independent prediction. The two parameters that most directly control the mass ratio in Eq. (15) are fixed by the experimental mass of the state being assigned: μ by μ = sqrt(M^2 − (2M_c)^2) with M_c = 1.82 GeV, and √s0 by M + 0.58/0.59 GeV; the Borel windows are then selected by trial and error. The output mass is thus a self-consistent solution of an equation whose external inputs include the very mass it is compared with. For unassigned channels the same procedure ties s0 and μ to the output mass, making the 'prediction' a fixed point rather than a test. Tables 4 and 5 largely restate the input masses. The additional feasibility claim (O(α_s^0) rather than O(α_s^2)) is supported chiefly by the author's own Refs. [37] and [39]. The OPE, the currents, and the condensate calculations provide independent content, but the headline agreement is engineered by input choice; score 7.

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

The calculation rests on standard QCD sum rule postulates plus two paper-specific inputs: the energy scale formula with M_c = 1.82 GeV and per-channel continuum thresholds informed by the experimental masses. These inputs carry much of the predictive power, so the ledger is dominated by domain assumptions and the ad hoc scale formula.

free parameters (3)
  • Effective charm quark mass M_c = 1.82 GeV
    Appears in the energy scale formula mu = sqrt(M^2 - (2 M_c)^2) used to choose the QCD spectral density energy scale for every current. It is inherited from a previous fit by the same author [41] and sets the overall scale of all predicted masses. A different M_c shifts the whole spectrum.
  • Per-current continuum threshold sqrt(s0) = e.g., 4.40 GeV for [uc]S[dc]S; values in Table 3
    Chosen as M_Z + 0.58/0.59 GeV, using the experimental mass of the state being assigned, and then adjusted by trial and error to satisfy pole dominance and OPE convergence criteria. Each of the 20 channels has its own threshold; these are inputs from experiment, not outputs of the calculation.
  • Borel parameter windows T^2 = values in Table 3, e.g., 2.7-3.1 GeV^2 for [uc]S[dc]S
    Selected by trial and error so that the extracted mass is stable (Borel platform), pole contributions are 40-60%, and |D(10)| is small. This is a standard QCD sum rule practice, but it introduces additional freedom in matching the central values.
assumptions (4)
  • domain assumption Quark-hadron duality: the Borel-transformed QCD side equals the hadronic side below the continuum threshold s0.
    Standard QCD sum rule postulate. Invoked implicitly in Eq. (14) where the hadron spectral integral is matched to the QCD spectral density integral.
  • domain assumption The local diquark-antidiquark currents J(x) have non-zero overlap with the physical XYZ resonances, so isolating the ground-state pole in Eq. (11) is valid.
    The entire assignment Table 5 depends on the assumption that each chosen four-quark current couples to the observed state; if a current has negligible overlap, the extracted mass belongs to a different or spurious state.
  • ad hoc to paper Energy scale formula mu = sqrt(M^2 - (2 M_c)^2) with universal M_c = 1.82 GeV determines the optimal scale and weakens scale dependence.
    This is a phenomenological postulate specific to the author's program (Refs. [19,41]). It is not derived from QCD and is used to fix mu for every channel; the paper itself says a hand-chosen mu that violates this formula can reproduce Zc(4200) and Zc(4100), so the formula carries a major part of the predictive weight.
  • domain assumption The Lucha-Melikhov-Sazdjian cancellation of factorizable diagrams at O(alpha_s^k), k <= 1 is wrong; tetraquark states contribute starting at O(alpha_s^0).
    The paper's feasibility discussion depends on the author's refutation in Ref. [37] and arXiv:1910.09981, which are not reproduced here. The conclusion is asserted with a qualitative argument about Landau singularities and confinement.

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Pith. "Pith review of Analysis of the hidden-charm tetraquark mass spectrum with the QCD sum rules." pith.science (2026). https://pith.science/paper/7ZHQNIKD

@misc{pith2026190807914,
  author       = {Pith},
  title        = {Pith review of: Analysis of the hidden-charm tetraquark mass spectrum with the QCD sum rules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZHQNIKD}},
  note         = {Machine review of arXiv:1908.07914}
}
abstract

In this article, we take the pseudoscalar, scalar, axialvector, vector, tensor (anti)diquark operators as the basic constituents, and construct the scalar, axialvector and tensor tetraquark currents to study the mass spectrum of the ground state hidden-charm tetraquark states with the QCD sum rules in a comprehensive way. We revisit the assignments of the $X$, $Y$, $Z$ states, such as the $X(3860)$, $X(3872)$, $X(3915)$, $X(3940)$, $X(4160)$, $Z_c(3900)$, $Z_c(4020)$, $Z_c(4050)$, $Z_c(4055)$, $Z_c(4100)$, $Z_c(4200)$, $Z_c(4250)$, $Z_c(4430)$, $Z_c(4600)$, etc in the scenario of tetraquark states in a consistent way based on the QCD sum rules. Furthermore, we discuss the feasibility of applying the QCD sum rules to study the tetraquark states and tetraquark molecular states (more precisely, the color-singlet-color-singlet type tetraquark states), which begin to receive contributions at the order $\mathcal{O}(\alpha_s^0)$, not at the order $\mathcal{O}(\alpha_s^2)$.

Figures

Figures reproduced from arXiv: 1908.07914 by the authors.

Figure 1
Figure 1. The masses of the [uc]S[dc]A − [uc]A[dc]S(I) and [uc]S[dc]A + [uc]A[dc]S(II) axialvector tetraquark states with variations of the Borel parameters T 2 . X(1+−) → ηJ/ψ , ηψ′ , ηhc , ωηc , (DD¯ ∗ ) 0 , (D∗D¯) 0 , (D∗D¯ ∗ ) 0 , X(0++) → ηηc , ηχc1 , ωJ/ψ , ωψ′ , (DD¯) 0 , (D∗D¯ ∗ ) 0 , X(1++) → ηχc1 , ωJ/ψ , ωψ′ , (DD¯ ∗ ) 0 , (D∗D¯) 0 , (D∗D¯ ∗ ) 0 , X(2++) → ηηc , ηχc1 , ωJ/ψ , ωψ′ , (DD¯) 0 , (D∗D¯ ∗ ) 0 , (25) with… view at source ↗

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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. Full citation record

  1. Investigating triply heavy tetraquark states through QCD sum rules

    hep-ph 2024-12 conditional novelty 5.0 of 10

    QCD sum rules with condensates up to dimension 9 predict triply heavy tetraquark masses of 5.4 to 6.2 GeV for charm systems and 14.9 to 15.7 GeV for bottom systems.

  2. Radiative decays of $X(3872)$ within $D{\bar D}^*$ molecular framework

    hep-ph 2026-07 conditional novelty 4.0 of 10

    Using nonrelativistic effective field theory, the X(3872) is treated as a D*D molecule to predict radiative decay widths to D D gamma, finding a strong neutral-over-charged hierarchy and quantifying D D rescattering effects.

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