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arxiv: 2604.11116 · v1 · submitted 2026-04-13 · ✦ hep-ph

Recognition: unknown

Extraction of Pion Unpolarized Quark Generalized Parton Distribution from Charge Form Factors

Satyajit Puhan , Shubham Sharma , Narinder Kumar , Harleen Dahiya
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:55 UTC · model grok-4.3

classification ✦ hep-ph
keywords piongeneralized parton distributionselectromagnetic form factorparton distribution functionszero skewnessimpact parameter spaceQCD meson structure
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The pith

A fit to pion form factor and PDF data extracts the pion's unpolarized quark GPDs at zero skewness.

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

The paper constructs a data-driven extraction of the unpolarized quark generalized parton distributions for the pion in the zero-skewness limit. It combines experimental measurements of the pion electromagnetic form factor with known parton distribution functions inside a model that adds a profile function to capture transverse spatial structure. This produces a consistent picture of how charge is distributed in the pion and how partons are arranged in impact-parameter space. The resulting distributions supply input for planned measurements at electron-ion colliders and existing fixed-target programs.

Core claim

By adopting a flexible parameterization of the pion electromagnetic form factor that is constrained by data and embedding that form factor inside a GPD model built from collinear PDFs plus a transverse profile function, the authors obtain explicit expressions for the unpolarized up- and down-quark GPDs of the pion at ξ = 0.

What carries the argument

The GPD framework that starts from collinear PDFs and adds a profile function encoding transverse dynamics, with the profile fixed by matching to the parameterized form factor.

If this is right

  • The extracted GPDs furnish impact-parameter-space densities that describe the spatial distribution of quarks inside the pion.
  • The distributions supply phenomenological input for the Sullivan process in deep-inelastic scattering at a future electron-ion collider.
  • They can be used to predict observables in exclusive π+ electroproduction at the 12 GeV Jefferson Lab program.
  • The same GPDs enter calculations for pion-induced exclusive reactions at COMPASS and proposed AMBER experiments.

Where Pith is reading between the lines

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

  • The same embedding technique could be applied to other light mesons once sufficiently precise form-factor and PDF data become available.
  • Extending the fit to include polarized observables would test whether the same profile function works for both unpolarized and polarized GPDs.
  • If the extracted GPDs are inserted into dispersion relations, they would generate predictions for the real part of Compton form factors that can be checked in future data.

Load-bearing premise

The profile function that encodes transverse dynamics can be chosen so that the resulting GPDs remain unbiased when the form factor is varied within its experimental uncertainties.

What would settle it

A direct lattice-QCD calculation of the same GPD moments at ξ = 0 that lies outside the uncertainty band obtained from the global fit would contradict the extraction.

Figures

Figures reproduced from arXiv: 2604.11116 by Harleen Dahiya, Narinder Kumar, Satyajit Puhan, Shubham Sharma.

Figure 2
Figure 2. Figure 2: FIG. 2: Pion PDFs for the valence quark and gluon: (a) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Unpolarized quark generalized parton distribution of the pion, shown as [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the extracted pion charge radius with avail [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Two-dimensional transverse tomography of the pion in the ( [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Unpolarized spin densities as functions of [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
read the original abstract

Based on a global fit to experimental measurements of the pion electromagnetic form factor and parton distribution functions (PDFs), we report a data-driven determination of the unpolarized quark generalized parton distributions (GPDs) for the case of pion in the zero-skewness limit ($\xi = 0$). The form factor is parameterized using a flexible functional form constrained by data and embedded into a GPD framework constructed from collinear PDFs and a profile function encoding transverse dynamics. This approach provides a unified description of the pion's electromagnetic structure and its spatial parton distributions. We present the extracted pion GPDs and their impact-parameter-space interpretations, offering new insights into the internal structure of the lightest QCD bound state and providing essential input for future electron-ion collider studies via the Sullivan process, as well as for the exclusive $\pi^+$ electroproduction at the 12~GeV Jefferson Lab program, pion-induced exclusive measurements at COMPASS, proposed pion-beam experiments at AMBER, and phenomenological and lattice investigations of the structure of the meson.

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 reports a data-driven extraction of the unpolarized quark GPDs of the pion at zero skewness (ξ=0) obtained via a global fit to pion electromagnetic form-factor data and collinear PDFs. A flexible parameterization of the form factor is embedded into a GPD model built from the PDFs plus a profile function that encodes t-dependence and transverse dynamics; the resulting GPDs and their impact-parameter representations are presented as input for EIC, JLab, COMPASS, and AMBER studies.

Significance. If the profile-function assumptions prove robust, the work supplies a unified, data-constrained framework for pion GPDs that could serve as useful phenomenological input for exclusive processes. The explicit linkage of form-factor and PDF constraints is a positive feature that may reduce inconsistencies between separate extractions. However, the significance is tempered by the modeling dependence inherent in the fixed profile function and the partial circularity between the fitted form factor and the GPD definition.

major comments (3)
  1. [Abstract / method] Abstract and method section: the GPD H^q(x,0,t) is constructed so that its x-integral exactly recovers the parameterized electromagnetic form factor F_π(t). Because the flexible parameterization of F_π(t) is itself fitted to data, the extracted t-dependence of the GPD is partly defined by the same fitted quantity it is supposed to predict, introducing a circularity that weakens the claim of an independent, data-driven determination.
  2. [Method] Method section (parameter count and fit procedure): only two free parameters are reported, evidently associated with the form-factor parameterization. The profile function that supplies the t-dependence is not stated to be varied or profiled in the global fit. If its functional form and parameters are held fixed, this modeling choice is load-bearing for the x- and t-dependence of H(x,0,t) and for all impact-parameter interpretations; the manuscript must demonstrate that the extracted GPDs remain stable under reasonable variations of the profile.
  3. [Results] Results section: no quantitative assessment of error propagation from the form-factor fit into the GPD uncertainties, nor validation against independent observables (e.g., lattice moments or other exclusive channels), is provided. Without these, the reliability of the reported GPDs and their spatial distributions cannot be judged.
minor comments (2)
  1. [Abstract] The abstract should explicitly state the functional form chosen for the profile function and the number of parameters it contributes (or confirm that it is fixed).
  2. [Introduction / method] Notation for the GPD (H vs. H^q) and the precise definition of the zero-skewness limit should be introduced consistently in the first section where the model is defined.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment point by point below, indicating the revisions we plan to make to strengthen the presentation and analysis.

read point-by-point responses

  1. Referee: [Abstract / method] Abstract and method section: the GPD H^q(x,0,t) is constructed so that its x-integral exactly recovers the parameterized electromagnetic form factor F_π(t). Because the flexible parameterization of F_π(t) is itself fitted to data, the extracted t-dependence of the GPD is partly defined by the same fitted quantity it is supposed to predict, introducing a circularity that weakens the claim of an independent, data-driven determination.

    Authors: We agree that the t-dependence of H^q(x,0,t) is fixed by construction through the sum rule to match the parameterized F_π(t). This is an inherent feature of the GPD framework we employ, chosen to guarantee consistency between the electromagnetic form factor and the parton distributions. The data-driven character of the work arises from the global fit of the flexible form-factor parameterization to experimental data combined with the use of measured collinear PDFs for the x-dependence. We will revise the abstract and method section to state this linkage explicitly and to remove any phrasing that could suggest an independent prediction of the t-dependence. The resulting unified description remains a useful phenomenological input for future studies. revision: partial

  2. Referee: [Method] Method section (parameter count and fit procedure): only two free parameters are reported, evidently associated with the form-factor parameterization. The profile function that supplies the t-dependence is not stated to be varied or profiled in the global fit. If its functional form and parameters are held fixed, this modeling choice is load-bearing for the x- and t-dependence of H(x,0,t) and for all impact-parameter interpretations; the manuscript must demonstrate that the extracted GPDs remain stable under reasonable variations of the profile.

    Authors: The profile function is taken from standard forms employed in meson GPD phenomenology. Its parameters are held fixed at values that reproduce known transverse momentum properties of the pion. We recognize that demonstrating stability is important. In the revised version we will add a sensitivity analysis in which the profile parameters are varied within physically reasonable ranges, and we will show the resulting variations in the GPDs and impact-parameter distributions. This will quantify the modeling uncertainty associated with the profile choice. revision: yes

  3. Referee: [Results] Results section: no quantitative assessment of error propagation from the form-factor fit into the GPD uncertainties, nor validation against independent observables (e.g., lattice moments or other exclusive channels), is provided. Without these, the reliability of the reported GPDs and their spatial distributions cannot be judged.

    Authors: We will include error bands on the extracted GPDs and impact-parameter representations obtained by propagating the uncertainties of the form-factor fit. We will also add a comparison of the lowest moments of the GPDs with existing lattice QCD results for the pion. While direct experimental validation against other exclusive channels is limited by the current scarcity of data, we will discuss consistency with phenomenological expectations from related meson studies. revision: yes

Circularity Check

1 steps flagged

GPD at ξ=0 constructed to integrate exactly to fitted form-factor parameterization

specific steps
  1. fitted input called prediction [Abstract]
    "The form factor is parameterized using a flexible functional form constrained by data and embedded into a GPD framework constructed from collinear PDFs and a profile function encoding transverse dynamics."

    The GPD H(x,0,t) is defined such that its x-integral equals the electromagnetic form factor F(t). By parameterizing and fitting F(t) to data and then embedding that parameterization into the GPD via the profile, the extracted GPD's t-dependence is identical to the fitted input by construction; the 'data-driven determination' therefore reduces to re-expressing the form-factor fit in GPD language rather than providing an independent extraction.

full rationale

The paper's central extraction embeds a flexible, data-fitted parameterization of the pion electromagnetic form factor directly into a GPD model whose defining property is that ∫dx H(x,0,t) recovers F(t). Because the t-dependence of the extracted GPD is supplied by this same fitted F(t) via the profile function, the reported GPDs are not an independent determination but are forced by construction to reproduce the input fit. The collinear PDFs supply the x-dependence at t=0, but the transverse (t) structure is inherited from the form-factor fit rather than constrained separately. This matches the 'fitted input called prediction' pattern and produces partial circularity (score 6) without requiring self-citation chains.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard GPD modeling assumptions and on parameters fitted to form-factor and PDF data; no new particles or forces are introduced.

free parameters (2)
  • parameters of the flexible functional form for the pion form factor
    The form factor is parameterized with a flexible functional form whose coefficients are adjusted to experimental data.
  • parameters of the profile function
    The profile function that encodes transverse dynamics is part of the GPD construction and must be specified or fitted.
axioms (2)
  • domain assumption Unpolarized quark GPDs at zero skewness can be constructed from collinear PDFs plus a profile function encoding transverse dynamics
    This is the standard phenomenological framework invoked in the abstract.
  • domain assumption The zero-skewness limit (ξ = 0) is sufficient for the intended applications
    Explicitly stated in the abstract as the regime of the extraction.

pith-pipeline@v0.9.0 · 5493 in / 1534 out tokens · 41315 ms · 2026-05-10T15:55:24.241083+00:00 · methodology

discussion (0)

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Reference graph

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