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REVIEW 3 major objections 5 minor 23 references

Composite nature of exotic states from data analysis

T0 review · 3 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read CDD-pole fits to X(3872), Zb, and Tcc spectra quantify their hadronic-molecule content

desk verdict A conference proceedings that usefully compacts the author's own CDD-pole compositeness analyses, but the 'deeper insights' claim outruns what the production model can actually support. read the letter →

arxiv 2509.01289 v1 pith:436FY4AX submitted 2025-09-01 hep-ph hep-exnucl-th

Xian-Wei Kang This is my paper · ORCID
classification hep-phhep-exnucl-th
keywords compositenessCDDpoleexotichadronshadronicmoleculesX(3872)Zb(10610)Zb(10650)Tcc
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 proposes that a two-hadron scattering amplitude with exactly one Castillejo-Dalitz-Dyson (CDD) pole—an extra zero of the amplitude that unitarity and analyticity do not fix—plus the standard two-point loop function can describe the measured spectra of near-threshold exotic hadrons, even where the effective-range expansion fails. From the fitted amplitude one reads the compositeness, the weight of the two-hadron (molecule) component in the state's wave function. Applied to X(3872), Zb(10610), Zb(10650), and Tcc, the method reports that Tcc has a two-hadron weight of 0.23 (+0.40/−0.09) and therefore a large elementary (tetraquark-like) component, that the two Zb states have weights between 0.4 and 1, and that X(3872) data alone leave its weight anywhere from 0 to 1. This matters because it turns ordinary line-shape data into a structural statement about what these states are made of, without committing to a specific quark model.

What carries the argument

The load-bearing object is the CDD pole—an extra zero of the scattering amplitude whose position and residue are not fixed by unitarity and analyticity. The amplitude is t(s) = [γ²/(s−M_CDD²)+G(s)]⁻¹, with G the two-point loop function; non-relativistically it becomes t(E) = 8π m_th [λ/(E−M_CDD)+β−ik]⁻¹. The quantity fitted to data is |d(E)|², with d(E) = [1+(E−M_CDD)(β−ik)/λ]⁻¹, which removes the zero at the CDD pole. Compositeness is X = |γ² dG(s_R)/ds_R| at the resonance pole. Thus the CDD-pole location sets the molecule-versus-elementary mix: far from threshold gives molecule-dominated, near threshold gives elementary-dominated.

What would settle it

Fit the same Tcc spectrum after adding a second subtraction constant or a second channel, and check whether the pole residue (hence X) moves outside the quoted errors; if it does, the single-pole |d(E)|² ansatz is incomplete. Separately, measure the near-threshold effective range r for Tcc: Eq. (7) predicts r from the fitted CDD-pole position, so a precise measured r that disagrees with that prediction would falsify the amplitude.

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Extended reading notes

Core claim

The central claim is that the position of the CDD pole, not only the proximity of a pole to a two-hadron threshold, controls whether a near-threshold state is a hadronic molecule or an elementary object. For Tcc, the fit yields poles on both the physical and unphysical Riemann sheets and, invoking Morgan's rule, the paper concludes that Tcc has a large elementary (tetraquark-like) component; the compositeness is quoted as 0.23 (+0.40/−0.09). For Zb(10610) and Zb(10650), moving the CDD pole from threshold to farther away changes the compositeness from 0.39/0.36 up to 1. For X(3872), bound, virtual, and higher-order virtual-pole scenarios all fit the data, leaving the compositeness anywhere be

Load-bearing premise

The extraction assumes that the measured signal shape is produced entirely by the two-hadron rescattering function d(E) of Eq. (8), which contains one CDD pole and one subtraction constant; if direct production of a compact state or additional channels shape the spectrum, the quoted compositeness values are not physical probabilities.

Editorial extensions

If this is right

  • Tcc's two-hadron component is 0.23 central, so the state is mostly not a D-D* molecule; the dominant part is elementary/tetraquark-like.
  • Zb(10610) and Zb(10650) are molecule-dominated but not pinned: compositeness ranges from 0.39/0.36 to 1 as the CDD pole moves.
  • X(3872)'s line shape is compatible with compositeness anywhere from 0 to 1, including bound, virtual, and higher-order virtual poles; current data do not decide.
  • When the CDD pole sits near threshold, the effective range becomes huge, so effective-range expansion fails just where these states live; the CDD-pole parameterization stays valid.
  • The fitting procedure turns any measured two-body line shape into a pole residue and a compositeness value, so improved spectra will directly sharpen these numbers.

Reading between the lines

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

  • A natural test the paper does not report is to fit Tcc again with a second channel or a second subtraction constant; if the extracted residue changes beyond the quoted errors, the 0.23 value is parameterization-dependent rather than intrinsic.
  • If CDD-pole distance from threshold is the controlling variable, then other near-threshold exotics—such as charmed pentaquark candidates—could be classified by the same line-shape analysis instead of by quark-model prejudice.
  • The Tcc result implies that a sharp peak sitting exactly at a threshold is not by itself evidence for a molecule; a compact state with a near-threshold CDD pole can produce a similar line shape, so threshold proximity alone should not drive molecule claims.
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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. This proceedings paper introduces a CDD-pole-modified two-body scattering amplitude (Eq. (1), with non-relativistic form Eq. (4)) and uses it to extract compositeness values for the exotic candidates X(3872), Zb(10610), Zb(10650), and Tcc. The signal shape is taken as |d(E)|^2, where d(E) is defined in Eq. (8), with an additive background in the spectral fits. The paper reports that X(3872) can be bound or virtual with compositeness from 0 to 1, that Zb states have compositeness 0.4--1, and that Tcc has compositeness 0.23^{+0.40}_{-0.09}, which is interpreted as indicating a large elementary/tetraquark component because both physical and unphysical Riemann sheet poles are found. The quantitative support for these claims is largely delegated to cited papers.

Significance. The CDD-pole parameterization itself is a standard and internally consistent extension of the effective-range expansion, and the paper honestly acknowledges the wide ranges obtained for X(3872) and Zb. If the extracted compositeness values were robust, they would give useful constraints on the nature of these states. However, the central numerical claims are not verifiable from this manuscript: the fits, pole searches, and uncertainties are in Refs. [14,15,17,19,21], and this paper does not provide the fitted parameters, residues, pole positions, or error budgets. In addition, the extraction rests on the production-model assumption that the observed spectrum is, up to an additive background, exactly |d(E)|^2, with no coherent direct-production amplitude. The paper's own closing caveat in Sec. 4 that a more rigorous coupled-channel study is ongoing signals that the present numbers are provisional. The abstract's claim of 'deeper insights' therefore overstates the robustness of the reported compositeness values.

major comments (3)
  1. [Sec. 4, Eq. (8)] The central quantitative claim is obtained by setting the signal shape equal to |d(E)|^2, where d(E) is a single-channel final-state interaction with one CDD pole and one subtraction constant. This equates the observable spectrum with the two-hadron FSI alone, plus an additive background. A coherent direct-production amplitude for a compact/tetraquark component, or an additional coupled-channel production mechanism, would interfere with this term and change the fitted pole residue gamma^2, and hence X = |gamma^2 dG/ds|. The additive background in Fig. 2 does not cure this. This is not a testable assumption as presented; the paper would need, at minimum, a comparison with an alternative production parameterization or an explicit statement of the production model's experimental validity. This point is load-bearing because the Tcc conclusion about a large elementary component depends direct
  2. [Sec. 4] The abstract promises compositeness values and deeper insights, but the support is not in this paper. For X(3872) the text says only that 'the compositeness could range from 0 to 1'; for Zb the manuscript states that M_CDD can vary over a wide range and gives no fit uncertainties; for Tcc only the central value and asymmetric uncertainty of X are given. There is no table of fitted parameters, pole positions, residues, chi^2 values, or correlation matrices. Figures 1 and 2 show curves and data, but a reader cannot check whether the claimed compositeness values follow from the displayed fits. The provenance of the numbers in Refs. [14,19,21] is fine, but for a paper whose stated purpose is to report these values, the omission makes the central claims unverifiable from the manuscript itself.
  3. [Sec. 4, Tcc paragraph] The text calls the value 0.23^{+0.40}_{-0.09} a 'predicted' compositeness, but it is an output of the fit to the same data from which M_CDD and the residue are determined (Eqs. (1)--(8)). It is therefore a fit result, not a prediction in the sense of a parameter fixed independently of the fitted spectrum. The subsequent inference 'both poles are found in the physical and unphysical Riemann sheets, which indicates that Tcc has large portion of elementary degree of freedom' rests on the Morgan rule and on the production-model assumption in the previous comment. The manuscript itself notes 'A detailed and more rigorous study in the coupled channel is ongoing', which is an explicit admission that the present single-channel FSI-only extraction is provisional. This caveat should be reflected in the abstract and conclusions.
minor comments (5)
  1. [Eq. (3)] The formula for kappa_± is garbled by typesetting; as printed it reads 'kappa_± = ...' with the square-root argument split across lines in an ambiguous way. Please rewrite the equation cleanly.
  2. [Fig. 2] The horizontal axis label appears as an encoded string ('/s32/s116/...') rather than readable text, and the axis label 'Events(/500keV)' is not standard notation. Please fix the figure text.
  3. [Sec. 4] The phrase 'meadited' should be 'mediated'; other typos include 'compositenss', 'quantatified', and 'interpretated'. A careful proofread is needed.
  4. [Sec. 4, Zb paragraph] The two scenarios for Zb are described with M_CDD values and X values, but the figure does not show which curve corresponds to which parameter set beyond the solid/dashed distinction. Please give the exact parameters or a small table in the caption or text.
  5. [Sec. 4, X(3872) paragraph] The statement that X(3872) can be a bound and/or virtual state, and possibly a higher-order virtual-state pole in the limit of vanishing D*0 width, is an important caveat, but it is stated without references to the specific pole positions or compositeness values. A brief numerical summary would make the 'range 0 to 1' claim concrete.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the compositeness values are derived outputs of fits to experimental spectra, not inputs or renamed fit parameters.

full rationale

The paper's derivation chain is: choose a single-channel scattering amplitude with one CDD pole (Eq. 1), construct d(E) (Eq. 8), fit |d(E)|^2 to the measured invariant-mass spectra, locate the pole and residue in the fitted amplitude, and then evaluate X = |gamma^2 dG/ds_R|. The data are spectra; the compositeness is never used to define or constrain the amplitude. It is a nonlinear function of fitted pole parameters, and different parameter choices give different X values (e.g., the Z_b range 0.4-1). Calling the T_cc value 'predicted' is loose terminology, but it is not a fitted parameter renamed as a prediction. The cited self-references (Refs. 5, 14, 15, 17, 19, 21) point to parameter-free formal criteria or to independent fits of the same type using external data; they do not inject the target compositeness values as assumptions. The Morgan rule is an external statistical criterion. The closing caveat that a more rigorous coupled-channel study is ongoing is a model-limitation statement, not evidence of circularity: the production assumption that the observed spectrum equals |d(E)|^2 with background could be wrong, but that is a physical/model-selection risk, not a reduction of the derivation to its own input. No quoted equation or fitted parameter is shown to be equivalent to the compositeness result by construction.

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

All compositeness values are derived from fitted parameters (M_CDD, λ, α, and background terms) via X = |γ² dG/ds|. The Tcc value quoted as a 'prediction' is computed from the fitted pole position and residue, so by the paper's own equations it reduces to a function of the fit output. The central interpretive output is a re-expression of the data fit, not an independent constraint.

free parameters (4)
  • CDD pole mass M_CDD for each state = For Zb(10610): M_CDD = m_th (X=0.39) or M_CDD - m_th = 12.57 MeV (X=1); similar values for other states, not quoted in t
    Free parameter in Eq. (1) and (4); fitted to experimental mass spectra. The resulting X depends directly on its value.
  • CDD pole residue λ (or γ²) = not quoted per state
    Residue of the CDD pole, Eq. (5); fitted to data and determines the pole residue of t(s), hence X.
  • Subtraction constant α(μ²) = not quoted
    Renormalization constant in the loop function Eq. (2), absorbed into β in Eq. (5); fitted.
  • Background parameters in spectral fits = not specified in this paper
    The paper mentions 'background shape' in Sec. 4 but does not list the fitted background parameters for any state.
assumptions (4)
  • standard math The amplitude with one CDD pole, Eq. (1), follows from N/D method and satisfies analyticity and unitarity.
    Invoked in Sec. 3 with Ref. [9].
  • domain assumption For a resonance with M_R > m_th, compositeness X = |γ² dG/ds| at the unphysical-sheet pole is a valid probability weight.
    Adopted from Guo-Oller Ref. [4] and used for all four states in Sec. 4.
  • domain assumption The event distribution is proportional to |d(E)|² (Eq. (8)), i.e., the signal is entirely generated by the two-hadron final-state interaction.
    This is the modeling choice that connects the fitted amplitude to data; stated in Sec. 4.
  • domain assumption A CDD pole close to threshold indicates a large 'elementary' component in the wave function (Morgan rule).
    Used in Sec. 4 to interpret the Tcc result and in Sec. 3 bullet points.

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Cite this review

Pith. "Pith review of Composite nature of exotic states from data analysis." pith.science (2026). https://pith.science/paper/436FY4AX

@misc{pith2026250901289,
  author       = {Pith},
  title        = {Pith review of: Composite nature of exotic states from data analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/436FY4AX}},
  note         = {Machine review of arXiv:2509.01289}
}
abstract

We introduce two basic concepts: the compositeness for resonance and the Castillejo-Dalitz-Dyson (CDD) pole. Applying them to hadron states $X(3872)$, $Z_b(10610)$ and $Z_b(10650)$, and $T_{cc}$, we obtain their compositeness values and achieve deeper insights into their inner structure.

Figures

Figures reproduced from arXiv: 2509.01289 by the authors.

Figure 1
Figure 1. Mass spectra of 𝑍𝑏 (10610) and 𝑍𝑏 (10650). The left (right) one is for 𝐵𝐵¯∗ (𝐵 ∗𝐵¯∗ ) invariant mass spectrum and the relevant 𝑍𝑏 (10610) (𝑍𝑏 (10650)) resonance. Points with error bar are the experimental data from Belle Collaboration [20]. The vertical lines correspond to the 𝐵 (∗)𝐵¯∗ thresholds. In each subfigure, the solid line corresponds to 𝑀CDD = 𝑚th case, where the value of compositeness 𝑋 is smallest, 0.39 f… view at source ↗
Figure 2
Figure 2. Mass spectrum for the 𝐷 0𝐷 0𝜋 + decay channel. The data are from the LHCb collaboration [22]. The solid line represents our total result, with dotted line showing the background and dashed line corresponding to without the Gaussian resolution. 5. Conclusion and outlook We proposed a new parameterization with inclusion of the CDD pole to treat the nontrivial and nonperturbative two-body interaction. By fitting to the… view at source ↗

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