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REVIEW 3 major objections 5 minor 70 references

Hadronisation of in-medium $c\bar c$ pairs to the exotic $X(3872)$

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read A single Fermi Golden Rule overlap region in parameter space describes all LHC charm hadron ratios and implies the X(3872) is mostly a loosely bound molecule.

desk verdict A useful but overclaimed overlap formalism: the single common region is a real consistency result, but the in-medium sizes and the X(3872) compact fraction are fitted inputs, so the paper cannot 'imply' LEU dominance as presented. read the letter →

arxiv 2606.23261 v2 pith:CAS75JEE submitted 2026-06-22 hep-ph hep-ex

Henrique Legoinha , Bernardo Picão , Pedro Bicudo This is my paper · ORCID
classification hep-phhep-ex PACS 25.75.-q12.38.Mh14.40.Pq
keywords X(3872)quark-gluonplasmahadronisationFermiGoldenRulecharmoniumLowEnergyUniversalityheavy-ioncollisionswave-functionoverlap
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 measured production ratios of J/ψ, ψ(2S) and the exotic X(3872) in proton-proton and lead-lead collisions at the LHC are controlled by the spatial size of charm-anticharm pairs in the quark-gluon plasma. Using the Fermi Golden Rule, it models each ratio as a density-of-states factor times the squared overlap between a vacuum hadron wave function and a Gaussian-approximated in-medium wave function. A scan of the allowed pair sizes yields a single common region that reproduces all ratios across centrality classes. The authors conclude that the X(3872) must have a dominant loosely bound (Low Energy Universality) molecular component, with a smaller compact piece, and that in-medium pair sizes are about 0.85 fm in p-p collisions and 0.55 fm for the compact component in Pb-Pb collisions.

What carries the argument

The central object is Eq. (1), the Fermi Golden Rule ratio: the production ratio of two hadrons equals the ratio of their densities of states times the squared inner product between the in-medium modified c-cbar wave function (with the perturbation Hamiltonian applied) and the vacuum hadron wave function. In-medium wave functions are Gaussian ansätze whose widths are capped by measured source sizes (HBT radii), and the X(3872) wave function is a weighted sum of a compact Gaussian and an exponential Low Energy Universality tail with a ~10% compact admixture. This overlap machinery converts measured hadron ratios into constraints on pair sizes and the compact/molecular balance.

What would settle it

A concrete falsifier: measure the X(3872)-to-ψ(2S) ratio in Pb-Pb collisions with high precision as a function of centrality; the model predicts a single smooth common region with a specific centrality evolution, so a strong centrality dependence or a value far from the predicted ratio would rule out the overlap-dominated picture. Alternatively, compute the in-medium c-cbar wave function from finite-temperature lattice QCD; if its shape deviates substantially from a single Gaussian (e.g., showing long non-Gaussian tails or a distinctly different width), the Gaussian ansatz and the extracted 0.

Watch

Extended reading notes

Core claim

The central discovery is that a single family of in-medium c-cbar spatial sizes, combined with vacuum wave functions for J/ψ, ψ(2S) and a two-component X(3872), reproduces the full set of LHC charm hadron ratios through Fermi Golden Rule overlaps. The common overlap region fixes the p-p pair size near 0.85 fm and the compact Pb-Pb component near 0.55 fm. The X(3872) requires a dominant Low Energy Universality tail (about 90% of its wave function), not a purely compact state; including only the molecular component spoils the overlap, so a small compact piece is needed. The authors take this as evidence that the quark-gluon plasma acts as a spatial filter that resolves hadronic inner structure

Load-bearing premise

The load-bearing premise is that hadron ratios are exactly the density-of-states ratio times the squared overlap between a Gaussian in-medium c-cbar wave function (unmodified by the perturbation H') and vacuum hadron wave functions; if recombination, bound-state dynamics, or a significant H' modification of the wave function actually control hadronisation, the extracted pair sizes and the X(3872) conclusion do not follow.

Editorial extensions

If this is right

  • If correct, the X(3872) is mostly a D0-D*0 molecule, with only about 10% compact (tetraquark-like) probability.
  • The in-medium c-cbar pair size in p-p collisions is pinned near 0.85 fm, while the compact Pb-Pb component is about 0.55 fm, giving quantitative targets for QGP transport models.
  • Only the ψ(2S)/J/ψ ratio receives a nontrivial density-of-states factor (0.293); the other ratios are pure wave-function-overlap effects, meaning the observed suppression is geometric rather than thermal in origin.
  • The common overlap region is stable across centrality bins, suggesting hadronisation ratios are robustly controlled by wave-function geometry once centrality is accounted for.
  • The same machinery can be applied to Upsilon states to extract bottom pair sizes and test whether the Υ(10753) has a loose component.

Reading between the lines

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

  • The overlap logic suggests a natural experimental analogue: measuring charm hadron ratios in smaller systems (p-Pb or high-multiplicity p-p) would test whether the p-p pair size of 0.85 fm remains universal or shifts with system size.
  • If the perturbation Hamiltonian H' were to modify the in-medium wave function appreciably, the extracted Gaussian widths would change; comparing with full quantum-evolution or lattice QCD computations of in-medium c-cbar wave functions would provide a direct test.
  • The X(3872) result implies that other near-threshold exotic candidates with large scattering lengths, such as the doubly charm tetraquark Tcc, could have their internal composition inferred from production ratios in heavy-ion collisions, offering a new diagnostic for compositeness.
  • Because the Gaussian ansatz caps the loose component at the HBT radius, the model implicitly assumes hadronisation happens at freeze-out; if hadronisation occurs earlier while the system is still expanding, the effective pair sizes could differ, which is a testable timescale ambiguity.
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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 proposes that the measured ratios of J/psi, psi(2S), and X(3872) production in pp and Pb-Pb collisions at the LHC can be described by Fermi Golden Rule ratios of overlaps between vacuum hadron wave functions and Gaussian-ansatz in-medium c-cbar wave functions. The in-medium widths (sigma for pp; sigma' and alpha for Pb-Pb) are floated within ranges, and the authors report a common overlapping region in the (sigma, sigma') plane that reproduces the measured R_AA, double ratio, and X(3872) data. They conclude that the X(3872) has a dominant loose (LEU) component and that the surviving compact in-medium fraction increases in peripheral collisions.

Significance. The paper usefully compiles a broad set of LHC charmonium and X(3872) data into a single framework and attempts a unified description of quarkonium suppression and exotic hadron production. The use of HBT radii to fix the loose-component width is an interesting phenomenological input, and the centrality dependence of the resulting alpha parameter is a plausible qualitative observation. However, the central compositeness claim is not an independent extraction: the X(3872) mixing fraction beta is a fixed input, not a fitted or marginalized parameter, and the reported in-medium rms sizes are the fitted Gaussian parameters themselves. The analysis also lacks a statistical measure of the 'common overlapping region' and propagates no uncertainties. These issues are fixable, but they currently weaken the claims substantially.

major comments (3)
  1. [Vacuum Wave Functions, Eq. (6)] The X(3872) mixing parameter beta=10% and the compact width sigma_X (rms ~0.40 fm) are fixed inputs taken from external composition estimates. With beta fixed, the analysis cannot 'imply' a dominant LEU component; the only beta test is the limiting case beta=0, which is a single point and not a scan. Please scan beta over a physically plausible range (e.g., 0-30%) and sigma_X over hadronic scales, and show whether the common overlapping region persists only near beta=10%. As written, the conclusion 'our analysis implies the X(3872) indeed has a dominant LEU component' is circular.
  2. [Results / Fig. 2] The existence of a 'single common overlapping region' is established by visual intersection of individual-ratio bands. No goodness-of-fit, confidence level, or uncertainty propagation is provided for the extracted widths (sigma, sigma') or for alpha. The extracted rms values (0.85 fm, 0.55 fm) are the fitted Gaussian parameters themselves, so they are not independent predictions. Please provide a quantitative overlap criterion (e.g., chi-square contours or a p-value for the common region) and propagate the experimental uncertainties through the fit.
  3. [In-medium c-cbar Wave Function, Eq. (1)] The Fermi Golden Rule ratio in Eq. (1) is asserted without derivation, and the perturbation Hamiltonian H' is never specified. The statement 'we do not expect H' to significantly modify the WF' is in tension with the notation |~Psi> = H'|Psi>; if H' has no dynamical effect, the in-medium modification is entirely encoded in the Gaussian widths and alpha, making the model a parametrization rather than a Golden-Rule prediction. Please clarify the role of H' or justify why the overlap ansatz captures the relevant physics.
minor comments (5)
  1. [Density of States, Eq. (3)] The integral limits in Eq. (3) are rendered as \int_{1.6}^{1.6}; they should be \int_{-1.6}^{1.6}.
  2. [Introduction] Typo: 'femptoscopic' should be 'femtoscopic'.
  3. [In-medium c-cbar Wave Function] Duplicate word: 'consistent with with the formation of a large system'.
  4. [Fig. 2] The top axis of the inclusive panel is described in the caption but is not labeled on the plot itself; please label it (e.g., 'total Pb-Pb rms [fm]') for clarity.
  5. [References] Reference [12] lacks a publication year/volume; reference [73] formatting is inconsistent with the rest of the bibliography.

Circularity Check

1 steps flagged · score 6.0 of 10

X(3872) 'dominant LEU' conclusion is built into Eq. (6): β=10% is fixed before the fit, so the Golden-Rule analysis cannot imply it.

  1. self definitional [Eq. (6) in 'Vacuum Wave Functions'; 'Results' section]
    "For the exotic X(3872) hadron there is a prediction for the molecular part of the WF steaming from Low Energy Universality (LEU) [43]. ... We consider a mixing parameter β=10% and σX such that the compact WF features a √⟨r2⟩ ≈0.40 fm, respecting the landscape of charmonia states, including hidden-charm tetraquarks [45]. Still, due to its dominant loose nature, the X(3872) WF remains very spatially extended, with √⟨r2⟩ ≈6.44 fm. ... Our analysis implies the X(3872) indeed has a dominant LEU component."

    The 90% LEU/10% compact composition is inserted into Ψ_X by Eq. (6) before any overlap is computed; β=10% and σX are fixed inputs, not fitted or marginalized parameters. The only reported variation is β=0 (pure LEU), so the data can at most distinguish β=0 from β=10% and cannot select 'dominant LEU'. The conclusion 'Our analysis implies the X(3872) indeed has a dominant LEU component' restates the ansatz as a result rather than deriving it from the Fermi Golden Rule overlap analysis.

full rationale

The paper is a fit-and-consistency study, not a closed-form prediction: σ, σ′, and α are explicitly floated within physically motivated ranges, and the 'single common overlapping region' is a nontrivial intersection of several LHC observables under the model of Eq. (1). I therefore do not count the extracted in-medium sizes (~0.85 fm and ~0.55 fm) as circular: they are fitted parameters, and the paper does not disguise them as independent predictions. The density-of-states input and the vacuum J/ψ and ψ(2S) wave functions come from independent Schrödinger-potential/thermal inputs, and there is no load-bearing self-citation chain that forces the result. The central circular element is the X(3872) composition claim: Eq. (6) fixes β=10% and σX so that Ψ_X is 90% LEU by construction, and the later statement that the analysis 'implies' a dominant LEU component treats that input as an output. Because this concerns the paper's headline conclusion and is not merely a minor self-citation, the appropriate score is 6: partial circularity, with the rest of the consistency analysis retaining independent content.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The analysis rests on a phenomenological Golden Rule factorization and on Gaussian ansätze for in-medium wave functions; the main extracted quantities are fitted parameters, while vacuum wave functions, HBT radii, and the X(3872) mixing fraction are imported from prior work. No genuinely new physical entities are introduced.

free parameters (7)
  • sigma (pp in-medium pair width) = rms ≈ 0.85 fm
    Gaussian width of the in-medium c-cbar wave function in pp collisions; floated and fixed by the common-solution fit.
  • sigma' (Pb-Pb compact width) = rms ≈ 0.55 fm
    Width of the compact component of the Pb-Pb in-medium wave function; floated within hadronic scales and fixed by the fit.
  • alpha (Pb-Pb compact fraction) = 0.29–0.39 by centrality
    Relative balance between compact and dissociated components of the Pb-Pb in-medium wave function; scanned to produce the common overlap region.
  • zeta (Pb-Pb loose component width) = rms = R_HBT (3.7–6.0 fm)
    Width of the dissociated tail fixed to measured HBT source sizes; the Gaussian form is an ad hoc modelling choice.
  • beta (X(3872) compact mixing fraction) = 10%
    Fraction of compact component in the X(3872) wave function, taken from cited literature; the paper's conclusion depends on this input.
  • sigma_X (X(3872) compact Gaussian width) = rms ≈ 0.40 fm
    Chosen to match the charmonia/tetraquark size landscape; not fitted to the hadron-ratio data.
  • LEU scattering length a = 9.6 fm
    Scattering length from cited X(3872) analyses; shapes the loose LEU tail of the X(3872) wave function.
assumptions (7)
  • domain assumption Fermi Golden Rule factorization: hadron ratios are proportional to density-of-states ratio times squared overlap of vacuum and in-medium wave functions (Eq. 1).
    This is the central bridge between data and wave functions; it is asserted in the Introduction without derivation.
  • ad hoc to paper In-medium c-cbar wave functions are Gaussian ansätze, with a coherent compact+loose superposition in Pb-Pb (Eqs. 7-8).
    The Gaussian form is a modeling choice introduced to make the calculation tractable; no QCD derivation is provided.
  • domain assumption The perturbation Hamiltonian H' does not significantly modify the in-medium wave function.
    Stated explicitly in the 'In-medium c-cbar Wave Function' section; if false, the Gaussian ansatz is not a good representation.
  • domain assumption Statistical hadronization at T ≈ 158 MeV gives the density of states entering Eq. (2).
    Used to compute rho_a/rho_b; relies on the thermal-statistical model being valid in the chosen p_T/rapidity region.
  • domain assumption Vacuum J/ψ and ψ(2S) wave functions from the Schrödinger potential model Eq. (4).
    The potential-model WFs are taken from the literature; the potential parameters are not re-derived and uncertainties are not propagated.
  • domain assumption HBT radii characterize the maximum separation achieved by dissociated c-cbar pairs before hadronization.
    The loose component width is identified with R_HBT; this identification is physically plausible but not derived.
  • domain assumption X(3872) wave function is a coherent sum of a compact Gaussian and a LEU tail, with β = 10% fixed.
    The form and the mixing fraction are imported from cited literature; the conclusion about the compact component partly assumes it.

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Cite this review

Pith. "Pith review of Hadronisation of in-medium $c\bar c$ pairs to the exotic $X(3872)$." pith.science (2026). https://pith.science/paper/CAS75JEE

@misc{pith2026260623261,
  author       = {Pith},
  title        = {Pith review of: Hadronisation of in-medium $c\bar c$ pairs to the exotic $X(3872)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CAS75JEE}},
  note         = {Machine review of arXiv:2606.23261}
}
abstract

The separation of $c\bar c$ pairs in the quark-gluon plasma and the characteristic size of the final state quarkonia should lead to the observed hadron ratios in nuclear collisions. Such dependence manifests itself through the Fermi Golden rule, where hadron ratios are sensitive to the inner product between vacuum and in-medium wave functions. A novel hard probe is the exotic $X(3872)$, which is expected to have a molecular and a compact component. We bridge more than a decade of LHC experimental results on hard probes, namely regarding the $J/\Psi$, $\Psi(2S)$ and $X(3872)$ hadrons, to the degree of separation and dissociation of in-medium hidden-charm systems.

Figures

Figures reproduced from arXiv: 2606.23261 by the authors.

Figure 1
Figure 1. FIG. 1: In-medium hidden-charm modified WFs and in-vacuum states WFs. In this plot, the modified WF of the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The 2-D plots for several centrality bins as a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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