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arxiv: 2512.24855 · v2 · submitted 2025-12-31 · ✦ hep-ph · nucl-th

Recognition: 2 theorem links

· Lean Theorem

QCD Wehrl and entanglement entropies in a gluon spectator model at small-x

Gabriel Rabelo-Soares , Reinaldo Francener , Gabriel S. Ramos , Giorgio Torrieri
Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:07 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords Wehrl entropyentanglement entropygluon spectator modelHusimi distributionWigner distributionsmall-x QCDsaturation scaleAdS/QCD
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The pith

In a small-x gluon spectator model, the Wehrl entropy from the Husimi distribution decomposes into an entanglement entropy term plus a transverse residual term, with numerical values matching CMS proton data.

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

The paper develops a phase-space description of gluon entropy in the proton using a light-front spectator model inspired by soft-wall AdS/QCD. By smearing the Wigner distribution with a Gaussian of width set by the saturation scale to form the Husimi distribution, it shows that the Wehrl entropy naturally splits into an entanglement entropy component and a term capturing transverse momentum degrees of freedom. This goes beyond earlier longitudinal-only entanglement measures tied to multiplicity. Calculations yield proton entanglement entropy values that are compared directly to CMS experimental results, while the full Wehrl entropy is tracked across different virtualities.

Core claim

From a normalized Husimi distribution one can decompose the Wehrl entropy into an entanglement entropy term and a residual term associated with transverse degrees of freedom. In the gluon light-front spectator model with AdS/QCD wave functions constrained by NNPDF, the Husimi is formed by Gaussian smearing of the Wigner distribution using the GBW saturation scale. Numerical results for the proton entanglement entropy agree with CMS data, and the Wehrl entropy is computed as a function of virtuality.

What carries the argument

The decomposition of the normalized Husimi distribution's Wehrl entropy into an entanglement entropy term and a transverse residual term.

If this is right

  • The entanglement entropy extracted this way quantifies the quantum correlations in the proton's gluon field at small x.
  • Changes in virtuality directly affect the Wehrl entropy through the transverse smearing width.
  • This model allows consistent computation of both PDFs and phase-space distributions from the same wave functions.
  • Comparison to CMS data validates the link between multiplicity and entanglement entropy in this framework.

Where Pith is reading between the lines

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

  • If the saturation scale in the GBW model is varied, it would alter the Husimi width and thus the split between entanglement and transverse terms.
  • Extending the spectator model to include sea quarks or antiquarks could test whether the decomposition holds for the full parton content.
  • This phase-space entropy might relate to other information measures like von Neumann entropy in QCD simulations.

Load-bearing premise

The underlying wave functions come from the soft-wall AdS/QCD-inspired spectator model with parameters fixed by NNPDF fits, and the Husimi smearing width is set exactly by the GBW saturation scale.

What would settle it

A direct comparison showing that the model's predicted entanglement entropy deviates from CMS hadronic multiplicity data by more than the model's uncertainties at small x would falsify the proposed decomposition.

Figures

Figures reproduced from arXiv: 2512.24855 by Gabriel Rabelo-Soares, Gabriel S. Ramos, Giorgio Torrieri, Reinaldo Francener.

Figure 1
Figure 1. Figure 1: FIG. 1: Unpolarized gluon PDF [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The panels displays the Kharzeev-Levin entanglement entr [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: This figure shows the plots for the first moment of the Wigne [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Plots of the Wehrl entropy as a function of the longitudinal [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Recent studies have shown that hadronic multiplicity in deep inelastic scattering can be associated with entanglement entropy. However, such definitions are intrinsically longitudinal and do not capture the full phase-space structure of the proton. In this work, we investigate the proton Wehrl entropy constructed from the gluon Husimi distribution, which provides a positive phase-space description within the present definitions and model calculations. Within this framework, we employ a gluon light-front spectator model based on soft-wall AdS/QCD-inspired wave functions, with free parameters constrained by global NNPDF fits, allowing us to compute both parton distribution functions and Wigner distributions. The Husimi distribution is obtained via Gaussian smearing of the Wigner distribution with width given by the saturation scale in the GBW model. We show that from a normalized Husimi distribution one can decompose the Wehrl entropy into an entanglement entropy term and a residual term associated with transverse degrees of freedom. Numerical results for the proton entanglement entropy are shown and compared with CMS data, while the Wehrl entropy is presented for different values of the virtuality.

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

3 major / 1 minor

Summary. The paper claims that in a gluon spectator model at small-x using soft-wall AdS/QCD-inspired wave functions constrained by NNPDF fits, the Wehrl entropy can be computed from the Husimi distribution obtained by Gaussian smearing the Wigner distribution with the GBW saturation scale. This allows a decomposition of the Wehrl entropy into an entanglement entropy term and a residual term for transverse degrees of freedom. Numerical results for the proton entanglement entropy are presented and compared with CMS data, and the Wehrl entropy is shown for different virtualities.

Significance. If the results hold, this work provides a novel phase-space formulation of entanglement entropy in the proton that includes transverse momentum information, extending beyond purely longitudinal definitions used in prior DIS studies. The consistent use of a single spectator model for both PDFs and Wigner distributions is a strength, as is the direct comparison to CMS data. However, the heavy reliance on external phenomenological inputs (NNPDF fits and GBW scale) means the entanglement entropy values are not independent predictions but derived quantities, which tempers the overall significance.

major comments (3)
  1. [Decomposition section] The decomposition S_Wehrl = S_entanglement + S_transverse follows directly once the Husimi distribution is normalized; the manuscript should demonstrate that this split is robust under variations in the smearing procedure rather than relying solely on the GBW scale choice.
  2. [Numerical results] No error bars or uncertainty estimates are provided for the entanglement entropy values despite the use of fitted parameters from NNPDF and GBW; this omission weakens the quantitative comparison to experimental data.
  3. [Model construction] The width of the Gaussian smearing for the Husimi distribution is taken from the GBW saturation scale without validation against the transverse momentum distributions inherent to the soft-wall AdS/QCD spectator model; any inconsistency here directly impacts the extracted entanglement entropy.
minor comments (1)
  1. [Abstract] The phrase 'within the present definitions and model calculations' in the abstract is unclear and should be elaborated upon in the introduction or methods section.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We provide point-by-point responses to the major comments below and describe the planned revisions to the manuscript.

read point-by-point responses

  1. Referee: The decomposition S_Wehrl = S_entanglement + S_transverse follows directly once the Husimi distribution is normalized; the manuscript should demonstrate that this split is robust under variations in the smearing procedure rather than relying solely on the GBW scale choice.

    Authors: We concur that illustrating the robustness of this decomposition is important. In the revised version, we will add numerical checks by varying the smearing width around the GBW saturation scale (e.g., factors of 0.5 to 2 times the nominal value) and show that the entanglement entropy component remains largely insensitive to these variations, while the transverse term absorbs the changes. This will be presented in an extended decomposition section. revision: yes

  2. Referee: No error bars or uncertainty estimates are provided for the entanglement entropy values despite the use of fitted parameters from NNPDF and GBW; this omission weakens the quantitative comparison to experimental data.

    Authors: This is a valid point. Although the model uses central values, we will include uncertainty estimates in the revised manuscript by considering the error bands from the NNPDF gluon distributions and variations in the GBW parameter Q_s^2 within its phenomenological range. These will be shown as shaded regions in the plots comparing to CMS data. revision: yes

  3. Referee: The width of the Gaussian smearing for the Husimi distribution is taken from the GBW saturation scale without validation against the transverse momentum distributions inherent to the soft-wall AdS/QCD spectator model; any inconsistency here directly impacts the extracted entanglement entropy.

    Authors: We appreciate this observation. The GBW scale is selected for its consistency with small-x DIS data, and the spectator model is tuned to NNPDF fits that include such data. To validate, we will add a comparison in the model construction section between the model's computed <k_T^2> from the Wigner function and the GBW scale, confirming they are of the same order in the relevant x and Q^2 range. Any minor inconsistencies will be discussed in terms of their effect on the entropy. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs gluon Wigner distributions from a soft-wall AdS/QCD spectator model whose parameters are fixed externally by NNPDF global fits, then forms the Husimi distribution by Gaussian smearing with the GBW saturation scale (also an external phenomenological input). The Wehrl entropy is computed directly from the normalized Husimi function via its standard definition, and the decomposition into an entanglement term plus transverse residual is presented as a mathematical splitting of that integral over phase space. This splitting does not reduce the final numerical values to the input fits by construction; it is a definitional partitioning whose numerical output still depends on the model wave functions. Comparison of the extracted entanglement entropy to CMS data is an external validation step, not an internal prediction forced by the same data used in the fit. No self-citation chains, uniqueness theorems, or ansatze smuggled via prior author work appear in the derivation. The calculation is therefore a standard phenomenological model evaluation whose central results remain independent of the target observables.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The calculation rests on the light-front spectator model with soft-wall AdS/QCD wave functions and the GBW saturation scale for smearing; both are taken from prior literature and global fits rather than derived inside the paper.

free parameters (2)
  • saturation scale in GBW model
    Sets the Gaussian smearing width for the Husimi distribution
  • parameters in soft-wall AdS/QCD wave functions
    Constrained by global NNPDF fits to parton distributions
axioms (2)
  • domain assumption Gluon light-front spectator model based on soft-wall AdS/QCD-inspired wave functions
    Provides the Wigner distribution from which the Husimi is obtained
  • domain assumption Gaussian smearing with GBW saturation scale yields a valid positive Husimi distribution
    Assumed to produce the phase-space description used for entropy

pith-pipeline@v0.9.0 · 5503 in / 1597 out tokens · 50046 ms · 2026-05-16T19:07:40.823984+00:00 · methodology

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Reference graph

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