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arxiv: 2508.05171 · v3 · submitted 2025-08-07 · ✦ hep-ph

An analysis of nuclear parton distribution function based on relative entropy

Shu-Man Hu , Ao-Sheng Xiong , Ji Xu , Fu-Sheng Yu , Ji-Xin Yu This is my paper

Pith reviewed 2026-05-19 00:59 UTC · model grok-4.3

classification ✦ hep-ph
keywords nuclear parton distribution functionsrelative entropyKullback-Leibler divergenceEMC effectquark distributionsgluon nPDFsglobal fitsEPPS21
0
0 comments X p. Extension

The pith

Minimum relative entropy determines the shape of nuclear quark distributions in the intermediate-x region and matches global fits.

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

The paper applies relative entropy, or Kullback-Leibler divergence, to quantify how nuclear parton distributions differ from those in free nucleons. Imposing a minimum relative entropy hypothesis together with constraints fixes the form of the quark structure function at intermediate momentum fractions. This derived form reproduces the behavior found in the latest global analyses of data and ties directly to the EMC effect. The same procedure applied to gluon distributions shows that the central values from the EPPS21 fit sit closer to the hypothesized minimum than those from nNNPDF3.0. The method supplies an independent way to constrain distributions in regions with limited experimental or theoretical input.

Core claim

By introducing certain constraints and the minimum relative entropy hypothesis, the shape of the quark structure function in the intermediate-x region is determined and agrees with the latest global fits; the same procedure applied to gluon nPDFs shows that the central values of EPPS21 align more closely with the hypothesis than those of nNNPDF3.0. This agreement suggests that the relative entropy-based methodology may provide novel insight into the structure of nucleons, particularly in cases where experimental data and theoretical QCD constraints are limited.

What carries the argument

The minimum relative entropy hypothesis, which selects the nuclear parton distribution minimizing the Kullback-Leibler divergence from the free-nucleon distribution subject to chosen constraints.

If this is right

  • Quark nuclear modifications at intermediate x are fixed by entropy minimization rather than additional free parameters.
  • The EMC effect acquires a possible information-theoretic origin through the minimum-entropy condition.
  • Gluon nPDF central values should be chosen to lie closer to the minimum relative entropy solution in future global analyses.
  • The approach supplies an extra constraint usable in data-sparse kinematic regions for both quarks and gluons.

Where Pith is reading between the lines

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

  • The same minimum-entropy selection could be tested on other nuclear modifications such as shadowing at small x.
  • Electron-ion collider data on nuclear structure functions could provide a direct experimental test of the predicted intermediate-x shapes.
  • The principle offers one possible criterion for choosing among discrepant global-fit results when data alone cannot decide.

Load-bearing premise

The minimum relative entropy hypothesis is a valid physical principle for selecting the shape of nuclear parton distributions when experimental data and QCD constraints are limited.

What would settle it

A new global fit or direct measurement that produces a quark structure function in the intermediate-x region whose shape deviates substantially from the one obtained by minimizing relative entropy under the same constraints.

Figures

Figures reproduced from arXiv: 2508.05171 by Ao-Sheng Xiong, Fu-Sheng Yu, Ji-Xin Yu, Ji Xu, Shu-Man Hu.

Figure 1
Figure 1. Figure 1: FIG. 1: The KL divergence as a function of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The obtained structure function ¯p [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The obtained structure function ¯p [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The KL divergence as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The obtained structure function ¯p [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

In this work, we propose a method to quantify the difference between nuclear parton distribution functions in different nuclei and parton distribution functions in free nucleons using the relative entropy (also known as Kullback-Leibler divergence), a measure widely employed in quantum information theory. By introducing certain constraints and the ``minimum relative entropy" hypothesis, we can determine the shape of the structure function in the intermediate-$x$ region, which is intimately connected with the renowned EMC effect. For quark structure functions, our results align with the latest global fits to experimental data. This agreement suggests that the relative entropy-based methodology may provide novel insight into the structure of nucleons, particularly in cases where experimental data and theoretical QCD constraints are limited, such as those pertinent to gluon nPDFs. Therefore, we applied this methodology to gluon nPDFs, analyzing the results from two commonly used global fitting groups, EPPS21 and nNNPDF3.0. Our analysis suggests that the central values of EPPS21 align more closely with the ``minimum relative entropy" hypothesis. This finding underscores the utility of the proposed method and provides a valuable reference for future global fitting of nPDFs.

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 / 2 minor

Summary. The manuscript proposes a method to quantify differences between nuclear parton distribution functions (nPDFs) and free-nucleon PDFs using relative entropy (Kullback-Leibler divergence). By imposing the minimum relative entropy hypothesis together with constraints such as normalization and momentum sum rules, the authors determine the shape of the quark structure function in the intermediate-x region and report that the results align with global fits EPPS21 and nNNPDF3.0. The same procedure is applied to gluon nPDFs, with the conclusion that the central values of EPPS21 align more closely with the hypothesis than those of nNNPDF3.0.

Significance. If independently justified, the minimum-relative-entropy approach could supply a useful additional constraint for nPDFs in data-sparse regions such as gluons and could illuminate nuclear modifications associated with the EMC effect. The work introduces a novel information-theoretic perspective to the field and provides a concrete comparison between two widely used global-fit sets; these elements would constitute a modest but genuine contribution if the central hypothesis can be shown to be more than a restatement of the imposed constraints.

major comments (3)
  1. [Abstract and §2] Abstract and §2 (Methodology): The minimum-relative-entropy hypothesis is introduced as an additional physical principle without derivation from QCD dynamics, sum rules, or nuclear models. Because the subsequent agreement with global fits is offered as evidence that the hypothesis selects the correct intermediate-x shape, the lack of independent motivation is load-bearing for the central claim.
  2. [§4] §4 (Quark results): The statement that the derived quark structure functions 'align with' EPPS21 and nNNPDF3.0 is presented without quantitative measures (e.g., integrated difference, χ², or overlap integrals), error propagation, or a demonstration that the result is independent of the parametrization choices already present in those fits.
  3. [§5] §5 (Gluon nPDFs): The claim that EPPS21 central values align more closely than nNNPDF3.0 is evaluated by comparing both sets to the same minimum-relative-entropy construction; this comparison risks circularity because both global fits already incorporate experimental data and modeling assumptions that may overlap with the imposed constraints.
minor comments (2)
  1. [§2] Notation for the relative-entropy functional and the explicit form of the constraints should be collected in a single equation block for clarity.
  2. [Figures 2–4] Figure captions should state the precise x-range and Q² value used for each comparison to allow direct reproduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below, indicating revisions where appropriate while maintaining the integrity of our approach.

read point-by-point responses

  1. Referee: [Abstract and §2] The minimum-relative-entropy hypothesis is introduced as an additional physical principle without derivation from QCD dynamics, sum rules, or nuclear models. Because the subsequent agreement with global fits is offered as evidence that the hypothesis selects the correct intermediate-x shape, the lack of independent motivation is load-bearing for the central claim.

    Authors: We acknowledge that the minimum-relative-entropy hypothesis is introduced as a guiding principle rather than a direct consequence of QCD. It is motivated by the principle of minimum relative entropy (or maximum entropy) from information theory and statistical inference, which selects the distribution that incorporates the given constraints with minimal additional assumptions. We do not claim a first-principles QCD derivation, as the hypothesis is intended to provide insight in data-limited regimes. The agreement with global fits is offered as empirical support for its utility, not as the sole justification. In revision we will expand §2 with additional discussion of this motivation, including references to analogous information-theoretic methods in high-energy physics. revision: partial

  2. Referee: [§4] The statement that the derived quark structure functions 'align with' EPPS21 and nNNPDF3.0 is presented without quantitative measures (e.g., integrated difference, χ², or overlap integrals), error propagation, or a demonstration that the result is independent of the parametrization choices already present in those fits.

    Authors: We agree that quantitative measures would improve clarity and rigor. In the revised manuscript we will add explicit calculations of the integrated absolute difference and a normalized overlap integral between the minimum-relative-entropy quark shapes and the central values of both global fits. We will also discuss propagation of uncertainties from the published fit error bands. Our construction depends only on the relative-entropy minimization subject to normalization and momentum sum rules; these constraints are independent of the specific functional parametrizations chosen by the fitting groups. We will add text in §4 to make this independence explicit. revision: yes

  3. Referee: [§5] The claim that EPPS21 central values align more closely than nNNPDF3.0 is evaluated by comparing both sets to the same minimum-relative-entropy construction; this comparison risks circularity because both global fits already incorporate experimental data and modeling assumptions that may overlap with the imposed constraints.

    Authors: We respectfully disagree that the comparison is circular. The minimum-relative-entropy construction is obtained solely from the hypothesis plus the normalization and momentum sum rules, without reference to any nuclear-modification data. Both global fits are determined from experimental data, but neither explicitly enforces the minimum-relative-entropy condition. Shared modeling elements such as the sum rules are already common to all three approaches; the additional constraint therefore provides an independent test of consistency. We will revise §5 to state this distinction more clearly and to note the data-independent nature of our construction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies an explicit hypothesis to constraints without reducing to inputs by construction

full rationale

The paper introduces the minimum relative entropy hypothesis as an additional selection principle and combines it with explicit constraints (normalization, momentum sum rules, and EMC-related bounds) to derive a functional form for the intermediate-x shape of nuclear structure functions. This derived form is then compared against independent global fits for validation rather than being fitted to them; the gluon analysis similarly evaluates two pre-existing parametrizations (EPPS21 and nNNPDF3.0) against the same external criterion. No equation reduces the claimed shape or alignment result to a fitted parameter or self-citation by construction, and the central claim remains an application of the stated information-theoretic ansatz rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven minimum-relative-entropy hypothesis and on the assumption that the chosen constraints are sufficient and physically motivated; no free parameters are explicitly fitted in the abstract, but the hypothesis itself functions as an ad-hoc selection rule.

axioms (1)
  • ad hoc to paper The minimum relative entropy hypothesis selects the physically realized nuclear parton distributions when data are limited.
    Invoked in the abstract to determine the shape of the structure function in the intermediate-x region.

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discussion (0)

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    Relation between the paper passage and the cited Recognition theorem.

    By introducing certain constraints and the 'minimum relative entropy' hypothesis, we can determine the shape of the structure function in the intermediate-x region... We assume that, with the endpoints fixed, the actual structure function of the nucleus tends to minimize the KL divergence

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Reference graph

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