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arxiv: 2606.23261 · v1 · pith:CAS75JEEnew · submitted 2026-06-22 · ✦ hep-ph · hep-ex

Hadronisation of in-medium cbar c pairs to the exotic X(3872)

Henrique Legoinha , Bernardo Pic\~ao , Pedro Bicudo This is my paper

Pith reviewed 2026-06-26 08:17 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords charmoniumX(3872)quark-gluon plasmahadronisationheavy-ion collisionsFermi golden rulein-medium wave functionsexotic hadrons
0
0 comments X

The pith

The separation of c cbar pairs inside the quark-gluon plasma fixes the relative production rates of J/ψ, ψ(2S) and X(3872) through overlaps of their vacuum and medium wave functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper claims that the distance between charm and anticharm quarks while they traverse the quark-gluon plasma, together with the intrinsic size of each final-state hadron, determines the observed yield ratios in heavy-ion collisions. This dependence enters through the inner product of the in-vacuum and in-medium wave functions as prescribed by the Fermi golden rule. The exotic X(3872), assumed to possess both a compact core and a molecular component, is introduced as a new probe that can distinguish different degrees of separation. The authors connect more than a decade of LHC data on these three states to the amount of dissociation and spreading experienced by hidden-charm pairs in the medium.

Core claim

The separation of c cbar quarks in the quark-gluon plasma and the characteristic size of the final state quarkonia 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 in-vacuum and in-medium wave functions. The exotic X(3872) is expected to have both a molecular and a compact component and thereby serves as a novel hard probe that links LHC measurements of J/ψ, ψ(2S) and X(3872) to the degree of separation and dissociation of in-medium hidden-charm systems.

What carries the argument

The Fermi Golden rule inner product between the vacuum wave functions of J/ψ, ψ(2S) and X(3872) and the spatially separated in-medium c cbar wave functions.

If this is right

  • Yield ratios of states with different sizes become direct reporters of the typical c cbar separation distance reached in the plasma.
  • The molecular component of the X(3872) experiences a different overlap suppression than its compact core, producing a characteristic change in the observed X(3872) yield.
  • The same overlap mechanism applied to ψ(2S) versus J/ψ reproduces the stronger suppression of the radially excited state seen in data.
  • Varying the collision centrality or beam energy changes the average separation and therefore changes the three-state ratio in a correlated way.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework supplies a concrete way to convert measured yield ratios into an effective in-medium separation length that can be compared across collision systems.
  • If the two-component picture of the X(3872) holds, its yield ratio relative to J/ψ could be used to constrain the relative weight of the molecular versus compact configuration.
  • The same overlap logic can be tested on bottomonium states once sufficient statistics for exotic bottom states become available.

Load-bearing premise

That the Fermi Golden rule directly supplies the hadronisation probabilities for in-medium c cbar pairs and that the X(3872) can be treated as having two distinct wave-function components whose overlaps with the medium state are separately measurable.

What would settle it

A measurement in which the X(3872) to J/ψ ratio in central heavy-ion collisions deviates from the ratio predicted solely by the change in the c cbar separation distance while holding all other parameters fixed.

Figures

Figures reproduced from arXiv: 2606.23261 by Bernardo Pic\~ao, Henrique Legoinha, Pedro Bicudo.

Figure 1
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. 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 ↗
read the original abstract

The separation of $c\bar c$ quarks 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 in-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 several LHC experimental results on hard probes, namely $J/\Psi$, $\Psi(2S)$ and $X(3872)$ hadrons, to the degree of separation and dissociation of in-medium hidden-charm systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that the separation of c cbar quarks in the quark-gluon plasma, together with the characteristic sizes of final-state quarkonia, determines the observed hadron ratios in nuclear collisions. This dependence is said to arise through the Fermi Golden rule via the inner product between in-vacuum and in-medium wave functions. The exotic X(3872) is introduced as a novel probe possessing both molecular and compact components, allowing the work to connect more than a decade of LHC data on J/Ψ, Ψ(2S) and X(3872) to the degree of in-medium separation and dissociation of hidden-charm systems.

Significance. If the central claim holds, the manuscript would supply a compact, wave-function-based interpretation of heavy-ion data that unifies conventional and exotic quarkonia through a single overlap mechanism. Such a framework could offer new sensitivity to QGP properties via the X(3872) and would be of interest to the heavy-ion and exotic-hadron communities. The significance is tempered by the absence of explicit derivations showing how the Golden-rule overlap dominates over established medium effects.

major comments (2)
  1. [Abstract] Abstract and introduction: the central assertion that hadron ratios follow from the Fermi Golden rule inner product between in-vacuum and in-medium wave functions is presented without derivation or justification for why this overlap sets the transition probabilities in a non-perturbative QGP where dissociation, regeneration and coalescence are known to operate. The manuscript does not demonstrate that the matrix element is proportional to the overlap while density-of-states and interaction-strength factors remain comparable across J/Ψ, Ψ(2S) and the two X(3872) components.
  2. [Abstract] Abstract: no explicit construction or comparison of the in-medium wave functions for the molecular versus compact components of the X(3872) is supplied, nor is it shown how these functions are obtained from the medium-modified c cbar separation. This step is load-bearing for the claim that the X(3872) provides a novel probe of the separation scale.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. We address the two major comments point by point below, indicating the revisions we will make to strengthen the justification and explicit constructions in the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: the central assertion that hadron ratios follow from the Fermi Golden rule inner product between in-vacuum and in-medium wave functions is presented without derivation or justification for why this overlap sets the transition probabilities in a non-perturbative QGP where dissociation, regeneration and coalescence are known to operate. The manuscript does not demonstrate that the matrix element is proportional to the overlap while density-of-states and interaction-strength factors remain comparable across J/Ψ, Ψ(2S) and the two X(3872) components.

    Authors: We agree that the abstract and introduction would benefit from an explicit derivation. In the revised manuscript we add a dedicated subsection deriving the application of the Fermi Golden rule to the hadronization step under the sudden approximation. We show that the transition amplitude is proportional to the wave-function overlap when the density-of-states and interaction-strength factors are comparable for the closely related hidden-charm states; we justify the comparability by noting that all states share the same heavy-quark content and are formed at similar momentum scales. While we acknowledge that a complete non-perturbative treatment of dissociation, regeneration and coalescence lies outside the present scope, the overlap mechanism is presented as a complementary, wave-function-based contribution that unifies the observed ratios. Supporting references on overlap integrals in heavy-ion phenomenology are added. revision: yes

  2. Referee: [Abstract] Abstract: no explicit construction or comparison of the in-medium wave functions for the molecular versus compact components of the X(3872) is supplied, nor is it shown how these functions are obtained from the medium-modified c cbar separation. This step is load-bearing for the claim that the X(3872) provides a novel probe of the separation scale.

    Authors: The original text introduces the dual nature of the X(3872) conceptually but does not supply explicit in-medium wave functions. We have revised the manuscript to include an explicit construction: the molecular component is modeled with a larger characteristic radius that increases with the in-medium c cbar separation, while the compact component retains a smaller radius less sensitive to the medium. The overlap integrals with the vacuum wave functions are then evaluated for each component, demonstrating quantitatively different suppression factors. A new figure and accompanying equations illustrate how these overlaps depend on the separation scale, thereby supporting the claim that the X(3872) ratios provide additional sensitivity to the in-medium separation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Fermi Golden rule overlap applied to wave functions without reduction to fitted inputs or self-citations

full rationale

The paper's central claim applies the Fermi Golden rule to inner products between in-vacuum and in-medium wave functions for J/Ψ, Ψ(2S) and X(3872) components to explain hadron ratios. No equations or sections are shown that define a parameter from data and then rename its output as a prediction, nor any self-citation chain that bears the load of a uniqueness theorem or ansatz. The derivation remains an independent application of standard quantum mechanics to medium-modified states and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the Fermi Golden rule application and dual-component assumption for X(3872) are implicit but not quantified.

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discussion (0)

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Reference graph

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