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REVIEW 3 major objections 4 minor 45 references

Study on the structure of the $Z_{c}(3900)$ state

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper argues that the Zc(3900) resonance is a coupled-channel mixture of a D Dbar* hadronic molecule and a dominantly diquark-antidiquark tetraquark, reproducing its measured mass and width in a Bethe-Salpeter calculation.

desk verdict A workmanlike B-S extension to Zc(3900) whose 'diquark dominance' conclusion is not actually established by the |g| comparison the paper makes. read the letter →

arxiv 2412.11144 v2 pith:JRPJV7KM submitted 2024-12-15 hep-ph hep-ex

Yan Ma , De-Shun Zhang , Cheng-Qun Pang , Zhi-Feng Sun This is my paper · ORCID
classification hep-phhep-ex
keywords Zc(3900)exotichadronhadronicmoleculediquark-antidiquarkBethe-Salpeterequationeffectivefieldtheorycoupledchannelshiddenlocalsymmetry
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

The paper argues that the observed $Z_c(3900)$ resonance is not purely a $D\bar D^*$ hadronic molecule nor purely a diquark-antidiquark tetraquark, but a mixture of both. Treating $D\bar D^*/D^*\bar D$ and $S_{cq}\bar A_{cq}/A_{cq}\bar S_{cq}$ as coupled channels in an effective field theory, the authors solve an on-shell Bethe-Salpeter equation and find a resonance pole whose mass and width match the experimental $Z_c(3900)$. The effective couplings indicate the diquark-antidiquark component is dominant. If right, this resolves a long-standing debate about the internal structure of this exotic hadron by showing both configurations coexist.

What carries the argument

The central object is the coupled-channel effective potential matrix $V$ entering the on-shell factorized Bethe-Salpeter equation $T=(I-VG)^{-1}V$, with the two-particle loop function $G$ regularized by a three-momentum cutoff and continued to the second Riemann sheet via $G^{II}_{ii}=G^{I}_{ii}+i\nu/(8\pi s)$. The mixing between the molecular and diquark-antidiquark configurations is encoded in the off-diagonal potentials $V_{12}$, which come from the meson-diquark-diquark Lagrangians with constants $e_8$ and $e_9$. The ratio of the resulting pole couplings $g_{S_{cq}\bar A_{cq}}/g_{D\bar D^*}\approx 1.8$ is the quantitative statement that the tetraquark component is dominant.

What would settle it

A direct test would be to compute the $Z_c(3900)$ pole with all $2^5$ sign choices for $e_3,e_5,e_6,h_2,h_5$: if no sign assignment keeps the pole in the $3888-3901$ MeV mass window with the observed width, the central conclusion fails. Alternatively, an experimental measurement of the $D\bar D^*$ coupling strength or a lattice calculation of the $I=1$, $J^{PC}=1^{+-}$ scattering amplitude near threshold would independently confirm or exclude the predicted dominance of the $S_{cq}\bar A_{cq}$ component.

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Extended reading notes

Core claim

Starting from hidden local gauge symmetry, the authors construct the interaction Lagrangians for charmed mesons with light mesons, charmed mesons with charmed and light diquarks, and charmed diquarks with light mesons, with all low-energy constants fixed from earlier quark-pair-creation model studies. From these they derive the effective potentials for the four coupled channels and solve the Bethe-Salpeter equation with a three-momentum cutoff. For cutoffs between 1400 and 1500 MeV, the pole on the second Riemann sheet sits at $3901-3888$ MeV with width $52-34$ MeV, in agreement with the measured $Z_c(3900)$ mass $3893.1\pm 2.2\pm 3.0$ MeV and width $44.4\pm 5.2\pm 14.0$ MeV and with the pole parameters of Ref. [27]. The extracted couplings $|g_{D\bar D^*}|≈ 8.4-8.6$ GeV and $|g_{S_{cq}\bar A_{cq}}|≈ 14.9-15.7$ GeV lead to the conclusion that the diquark-antidiquark channel dominates.

Load-bearing premise

The whole calculation stands on the low-energy constants $e_i$ and $h_j$ taken from Ref. [37], whose signs the authors say cannot be fixed for $e_3,e_5,e_6,h_2,h_5$; if those signs or their normalization are wrong, the effective potentials change and the predicted pole may no longer match the $Z_c(3900)$.

Editorial extensions

If this is right

  • The $Z_c(3900)$'s measured mass and width can be reproduced without assuming it is a pure molecule or a pure compact tetraquark; both configurations are required.
  • The diquark-antidiquark component dominates, so searches for its decay patterns should see signatures of correlated $cq$ and $\bar c\bar q$ substructure.
  • Because the same framework with the same coupling constants already described $Z_{cs}(4000)$, $Z_{cs}(4220)$, $Z_b(10610)$, $Z_b(10650)$, and $X(4500)$ as mixtures, the claim extends a single mixing mechanism across several exotic states.
  • The pole position depends mildly on the cutoff: changing $q_{\rm max}$ from 1400 to 1500 MeV shifts the mass by 13 MeV and the width by 18 MeV, quantifying the regularization uncertainty.

Reading between the lines

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

  • If the $Z_c(3900)$ is indeed a coupled-channel object with dominant diquark content, the same two-channel mixing matrix should also predict the line shape in $e^+e^-\to \pi D\bar D^*$; a precise measurement of the $D\bar D^*$ invariant mass distribution could discriminate this from a pure tetraquark or pure molecular pole.
  • The undetermined signs of $e_3,e_5,e_6,h_2,h_5$ leave a finite set of alternative sign choices; scanning those $2^5$ combinations would show how robust the pole is and could identify sign patterns that also reproduce other exotics.
  • Lattice QCD with $J^{PC}=1^{+-}$ and $I=1$ channels could compute the two-channel scattering phase shifts; matching them to the effective potentials would provide an ab initio check of the dominant diquark component.
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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 / 4 minor

Summary. The paper constructs a coupled-channel effective field theory, using hidden local symmetry, for the Zc(3900) with IG(JPC)=1+(1+-). It derives effective potentials in the channels D Dbar*/D* Dbar and S_cq A_cq/A_cq S_cq, solves the on-shell factorized Bethe-Salpeter equation with a three-momentum cutoff, and reports a pole at 3901-i26 to 3888-i17 MeV for qmax=1400-1500 MeV. On the basis of the residue moduli in Table III, the authors conclude that Zc(3900) is a mixture of the two configurations with the diquark-antidiquark component dominant.

Significance. If fully established, the result would support a coupled-channel interpretation of Zc(3900) in which both molecular and compact diquark-antidiquark configurations contribute, with the latter numerically dominant. The manuscript is carefully constructed: it provides explicit Lagrangians, flavor/color wave functions, effective potentials, and a cutoff-regularized BS equation, giving a concrete and reproducible framework. The central structural claim, however, rests on a comparison of residue couplings that does not by itself determine channel composition, and the quantitative agreement with the experimental mass/width is not accompanied by an uncertainty estimate from the model parameters.

major comments (3)
  1. [Sec. VI; Table III; Eq. (75)] The conclusion that the S_cq A_cq component is dominant is based entirely on |g_S_cq A_cq| approximately 15 GeV versus |g_D Dbar*| approximately 8.5 GeV. In the on-shell Bethe-Salpeter formalism, Eq. (75) defines couplings g_i through the residue of the T-matrix; these are not channel probabilities. A quantitative statement about the composition of the state requires a normalized wavefunction or a compositeness coefficient that also involves, for example, dG_i/ds and the channel normalizations. Because the D Dbar* and S_cq A_cq channels have very different thresholds and phase space, the larger residue for the heavier channel does not establish dominance. This point is load-bearing for the abstract and Sec. VI claim, and the authors should either compute the composition properly or weaken the claim.
  2. [Sec. V; Eqs. (49)-(68); Table II] The parameter dependence of the pole is not assessed. The potentials used in the coupled-channel calculation depend on e2, e8, e9, h1, and h4, whereas Table II gives an uncertainty only for e2 and no uncertainty for e8/h1/h4; the text's caveat about the signs of e3, e5, e6, h2, and h5 is irrelevant because these constants do not appear in Eqs. (49)-(68). The quoted 13 MeV spread of the real part of the pole comes only from the arbitrary cutoff interval qmax=1400-1500 MeV. The authors should propagate the LEC uncertainties and vary e8/e9/h1/h4 within reasonable ranges to show that the pole remains compatible with the Zc(3900) mass and width.
  3. [Sec. IV; Eqs. (71)-(74)] The manuscript states that the resonance appears on 'the second Riemann sheet' but defines four different sheet combinations for the two-channel G-matrix: GI=(G_I_11,G_I_22), GII=(G_II_11,G_I_22), GIII=(G_II_11,G_II_22), GIV=(G_I_11,G_II_22). For each cutoff in Table III, the authors should specify which sheet combination contains the pole and verify that the pole can be continuously traced as qmax changes; otherwise it is unclear whether the same physical state is being followed. This is a technical but important reproducibility issue.
minor comments (4)
  1. [Sec. VI] The word 'modules' is used where 'moduli' is standard in English; this appears also in Sec. V and Table III.
  2. [Sec. II] The sentence in the Introduction contains the typo 'charned diquark-charned diquark-light meson'; it should read 'charmed diquark-charmed diquark'.
  3. [Sec. V] The determination of e8 and e9 is described only as 'using the phase in Ref. [35]'; since these are the dominant constants in the inter-channel potential V12, a more explicit explanation or a reference to the relevant equations of Ref. [35] would help.
  4. [Sec. IV] The paper does not provide a derivation or reference for the on-shell factorization of the Bethe-Salpeter equation beyond Eq. (71); adding a brief justification would make the approximation transparent.

Circularity Check

2 steps flagged · score 5.0 of 10

Same-author LEC input plus cutoff scan calibrated to the target weaken, but do not fully erase, the independent BSE content; score 5.

  1. self citation load bearing [Sec. V, Lagrangian constants, Table II (after Eqs. (76))]
    "In the Lagrangians L2 and L3, there are two sets of coupling constants ei (i = 1, 2,··· , 9) and h j ( j = 1, 2,··· , 5), which are still unknown. In Ref. [37], these constants were determined by naively using the quark-pair-creation model with the diquark masses taken from Ref. [43]. Their values are listed in TABLE II. ... For e7, e8 and e9, we get the values of them using the phase in Ref. [35] explaining the Zcs states well."

    The entire numerical output (pole position, width, and especially the residue moduli used for the diquark-dominance claim) is computed with low-energy constants imported from Refs. [35,37], which are by the same authors. These constants are not rederived or independently checked here, and the paper itself states that the signs of e3, e5, e6, h2, and h5 cannot be fixed. The central conclusion that Zc(3900) can be explained therefore leans on an unverified same-group citation chain for its quantitative content; if those constants or phase choices are wrong, the pole and the dominance conclusion are not obtained.

  2. fitted input called prediction [Sec. V, Numerical results, Table III]
    "If the cutoff is chosen from qmax = 1400− 1500 MeV, the obtained pole appears on the second Riemann sheet, of which the corresponding mass varies from 3901 − 3888 MeV , and the width from 52− 34 MeV . These results are in good agreement with the mass and width of Zc(3900) given in Refs. [23, 27], which indicates that the Zc(3900) can be explained as the mixture of D ¯D∗ and S cq ¯Acq components."

    The cutoff qmax is a free regularization parameter that is not fixed by any observable before the comparison. It is scanned to 1400–1500 MeV precisely because that range makes the BSE pole sit on the experimental Zc(3900) mass and width. The statement that the state 'can be explained' is therefore partly a consequence of choosing the input parameter after seeing the target; the agreement is not parameter-free and the pole position is not a prediction in the usual sense.

full rationale

The paper does not exhibit a definitional equivalence: the coupled-channel BSE calculation, thresholds, and the sign choices do provide content beyond the input constants. However, the numerical content is dominated by low-energy constants taken from the same group's earlier work ([35,37]), with no independent derivation or stability scan, and the only genuinely free parameter, the cutoff, is selected so that the resulting pole matches the Zc(3900) mass and width. The diquark-dominance conclusion is additionally inferred solely from the residue moduli |g_SA| > |g_DD*|, which is not a channel-probability measure; that is a validity problem rather than a circularity problem. The paper also admits that the signs of five LECs cannot be fixed, although those particular constants do not enter the displayed potentials (Eqs. (49)-(68), which use e2, e8, e9, h1, and h4). Taken together, the central claim is partially calibrated rather than fully parameter-free, but it is not forced by construction, so the score is 5 rather than higher.

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

The results depend critically on the model couplings from Refs [35,37] and the arbitrary cutoff; these are inherited, not derived here.

free parameters (3)
  • Momentum cutoff qmax = 1400 to 1500 MeV
    Regulator for the loop function; the pole position varies from 3901 to 3888 MeV as qmax changes in this range, and the authors use this interval to obtain agreement with the measured Zc(3900) mass and width.
  • Coupling constants e_i and h_j = values in Table II
    These 14 constants are taken from the authors' previous work [37], where they were determined using the quark-pair-creation model. The signs of e3, e5, e6, h2, and h5 are not fixed, giving additional freedom in the effective potentials.
  • Diquark masses m_S_cq and m_A_cq = not given in this paper
    Taken from the quark model in Ref. [43]; they set the thresholds of the diquark channels and therefore affect the pole energy and width.
assumptions (5)
  • domain assumption The HLS effective theory with G_global x H_local symmetry provides the correct interaction vertices.
    Standard framework for chiral symmetry and vector mesons; invoked in Sec. II.
  • domain assumption Only color-antitriplet diquarks are relevant; sextet contributions are neglected.
    Based on the attractive nature of antitriplet quark-quark interaction; stated in Sec. II.
  • standard math The on-shell factorized Bethe-Salpeter equation with three-momentum cutoff is a valid approximation.
    Common approximation in hadron EFT; Sec. IV.
  • ad hoc to paper The quark-pair-creation model used in Ref. [37] reliably determines the low-energy constants for this system.
    The e_i and h_j are imported from the same group's previous calculation; their validity for the Zc(3900) is assumed without independent verification.
  • domain assumption Unitary gauge sigma=0 can be chosen without loss of physical content.
    Standard gauge fixing in HLS; used in Sec. V.

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Cite this review

Pith. "Pith review of Study on the structure of the $Z_{c}(3900)$ state." pith.science (2026). https://pith.science/paper/JRPJV7KM

@misc{pith2026241211144,
  author       = {Pith},
  title        = {Pith review of: Study on the structure of the $Z_c(3900)$ state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JRPJV7KM}},
  note         = {Machine review of arXiv:2412.11144}
}
abstract

In this work, we studied the $Z_{c}(3900)$ state within the framework of effective field theory. We firstly show the construction of the Lagrangian describing meson-meson-meson and meson-diquark-diquark interactions. By using the Feynman rule, we calculate the effective potentials corresponding to the coupled channels of $D\bar{D}^{*}/D^{*}\bar{D}$ and $S_{cq}\bar{A}_{cq}/A_{cq}\bar{S}_{cq}$ with $S_{cq}$ ($A_{cq}$) the scalar (axial vector) diquark composed of $c$ and $q$ quarks. After solving the Bethe-Salpeter equation of the on-shell parametrized form and compare our numerical results with the experimental mass and width of $Z_{c}(3900)$, we find that the $Z_{c}(3900)$ state can be explained as the mixture of $D\bar{D}^{*}/D^{*}\bar{D}$ and $S_{cq}\bar{A}_{cq}/A_{cq}\bar{S}_{cq}$ components.

Figures

Figures reproduced from arXiv: 2412.11144 by the authors.

Figure 1
Figure 1. FIG. 1: The diagrams needed to be considered in the present work. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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