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Extraction of Pion Unpolarized Quark and Gluon Generalized Parton Distributions using Deep Neural-Networks

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a physics-informed neural network can extract the pion's quark and gluon GPDs from form-factor data while preserving PDF forward limits, with valence-quark results matching lattice QCD.

desk verdict First NN-based pion GPD extraction; the quark sector is a solid, useful result, but the gluon rescaling and internal inconsistencies in the forward-limit and Mellin-moment claims make the paper's framing run ahead of its equations. read the letter →

arxiv 2608.07085 v1 pith:VTHMAQQI submitted 2026-08-07 hep-ph

Satyajit Puhan , Shubham Sharma , Narinder Kumar This is my paper · ORCID
classification hep-ph
keywords piongeneralizedpartondistributionsdeepneuralnetworksphysics-informedelectromagneticformfactorgravitationallatticeQCDhadrontomography
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

This paper claims that a physics-informed neural network can extract the pion's unpolarized quark and gluon generalized parton distributions (GPDs) at zero skewness from the known pion PDFs, experimental pion electromagnetic form-factor data, and lattice-QCD determinations of the gluon gravitational form factor. The network is constructed so that each GPD reduces exactly to its input PDF in the forward limit, remains non-negative, and obeys the form-factor sum rules. The extracted valence-quark GPDs agree with independent lattice-QCD calculations over most of the explored kinematics, and the corresponding charge radius, about 0.67 fm, is close to the world average. The gluon sector matches the lattice gravitational form factor only after an overall rescaling of the gluon PDF input, which the paper traces to limited small-x gluon constraints. If the claim holds, the framework offers a flexible, largely model-independent route to pion tomography that extends to the nucleon.

What carries the argument

The load-bearing object is the multiplicative physics-informed parameterization $H^q(x,t)=q_v^{\pi}(x)\,e^{c|t|}\,\mathrm{NN}(x,t)$ and $H^g(x,t)=xg(x)\,e^{c_g|t|}\,\mathrm{NN}(x,t)$, which confines the network to learning the residual correction beyond the forward PDF and the Regge-inspired exponential $t$ dependence. Positivity is built in through exponential output activations, with the gluon output written $\exp[\tanh(\mathrm{NN}(x,t))|t|/(1+|t|)]$ so that $H^g(x,0)=xg(x)$ holds exactly. The loss function is a $\chi^2$ sum over pion EMFF data and squared EMFF data, plus a charge-normalization penalty and a regularization term for the quark sector; the gluon sector is trained on the lattice $A_g(t)$ data with its own regularization. All results are quoted at $\mu^2=4\,\mathrm{GeV}^2$, and the PDF replica ensemble is propagated through the network to produce $1\sigma$ GPD uncertainty bands.

What would settle it

Directly computing the $x$-dependent pion gluon GPD on the lattice over the same $|t|$ range used here would settle the claim: if the rescaled $H^g(x,t)$ disagrees with the lattice result at small $x$, the constant-rescaling assumption fails. Alternatively, a future measurement of the pion gluon PDF that yields $\int_0^1 dx\,xg(x)\approx0.55$ to $0.60$ at $\mu^2=4\,\mathrm{GeV}^2$ would confirm the normalization correction, while a value near the input PDFs' 0.26 to 0.37 would indicate the rescaling is absorbing a lattice-systematic effect rather than a genuine gluon deficit.

Watch

Extended reading notes

Core claim

The central claim is that a single leading-twist pion GPD in each sector can be determined nonparametrically through the multiplicative ansatz $H^q(x,t)=q_v^{\pi}(x)\,e^{c|t|}\,\mathrm{NN}(x,t)$ and $H^g(x,t)=xg(x)\,e^{c_g|t|}\,\mathrm{NN}(x,t)$, where the input PDFs fix the forward limits $H^q(x,0)=q_v^{\pi}(x)$ and $H^g(x,0)=xg(x)$, the exponential provides the momentum-transfer profile, and the network learns the residual $x$ and $t$ dependence. The network parameters are fixed by minimizing $\chi^2$ losses built on the sum rules $F_\pi(t)=\sum_q e_q\int_{-1}^{1}dx\,H^q(x,t)$ and $A_g(t)=\int_0^1 dx\,x\,H^g(x,t)$, with $\chi^2/N\approx1.4$ for 176 quark-sector data points and $\chi^2/N\approx0.84$ for 50 gluon-sector lattice points. The extracted quark GPDs are in good agreement with the renormalization-group-resummed lattice calculation except at large $x$ and large $|t|$, and the fit returns a pion charge radius of 0.668 and 0.667 fm for the two PDF inputs, close to the world-average 0.659 fm. Gluon results require rescaling the input gluon PDF by a factor of about 1.5 or 2.1 to reproduce the lattice $A_g(t)$; after that rescaling, the two PDF inputs yield nearly identical gluon GPDs.

Load-bearing premise

The gluon-sector results stand on a single, $x$- and $t$-independent rescaling factor (1.46 for one PDF analysis and 2.07 for the other) that forces the input gluon PDF's second moment to match lattice $A_g(0)$; if the factor-of-two mismatch is concentrated at small $x$, the extracted gluon GPD has the wrong $x$-dependence.

Editorial extensions

If this is right

  • A future measurement of the pion electromagnetic form factor at larger $|t|$ than the fitted range would directly test the trained network's extrapolation, because $F_\pi(t)$ is an integral of the extracted quark GPD.
  • The same physics-informed ansatz can be applied to the nucleon, where two independent unpolarized GPDs enter and the constraints are richer; the paper identifies this as the natural next step.
  • The extraction quantifies pion tomography: Fourier transforming the zero-skewness GPDs yields transverse impact-parameter densities in which valence quarks localize at large $x$ and gluons dominate at small $x$.
  • The gluon-sector result implies that present pion gluon PDFs carry too little momentum at the reference scale unless rescaled, so improved small-$x$ pion gluon data would directly narrow the extracted gluon GPD uncertainty.

Reading between the lines

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

  • If the paper's constant-rescaling picture is correct, the same network trained without rescaling should fail the lattice $A_g(t)$ constraint by a factor that is independent of $t$; a useful diagnostic is to plot the ratio $A_g^{\mathrm{LQCD}}(t)/A_g^{\mathrm{model}}(t)$ and check whether the residual is flat in $t$.
  • A stronger version of the paper's normalization argument predicts that direct lattice calculations of the $x$-dependent gluon GPD, once available, will reproduce the rescaled $H^g(x,t)$ at small $x$; if they instead follow the raw input gluon PDF scaled only at $A_g(0)$, the rescaling is absorbing a low-$x$ deficit the paper does not model.
  • The near identity of the quark-sector results from the two independent PDF inputs suggests the electromagnetic form-factor data, rather than the valence PDF choice, controls the extracted $H^q(x,t)$; replacing the valence input with a third phenomenological PDF set would provide a cheap cross-check of that claim.
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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

4 major / 5 minor

Summary. The manuscript presents a deep neural-network (DNN) framework for extracting the pion's unpolarized quark and gluon generalized parton distributions (GPDs) at zero skewness. The GPDs are parameterized as input pion PDFs multiplied by an exponential Regge-inspired momentum-transfer factor and a trainable neural-network correction. The quark-sector network is fitted to pion electromagnetic form factor (EMFF) data, squared EMFF data, and lattice-QCD EMFF points, while the gluon-sector network is fitted to lattice-QCD determinations of the gluon gravitational form factor A_g(t). The authors use the JAM21 and xFitter PDF ensembles to propagate uncertainties and compare their extracted quark GPDs with lattice-QCD calculations. The paper claims that the framework preserves theoretical constraints such as the forward limit H(x,0)=PDF and the charge normalization F_pi(0)=1, and that the resulting quark GPDs agree with lattice QCD.

Significance. If the framework were fully consistent, this would be a useful contribution: it demonstrates a flexible, nonparametric way to map pion PDFs into GPDs, reports a genuinely good EMFF fit (chi^2/N about 1.4), and uses the full PDF replica ensemble to quantify uncertainties. The comparison of the extracted quark GPDs with lattice QCD is a meaningful independent check. However, several load-bearing issues currently prevent the central claims from being accepted: the quark forward limit is not enforced by the stated output layer, the gluon Mellin moment is used inconsistently between the formalism and the fit, the gluon normalization agreement is imposed by an ad hoc rescaling, and the text and loss function contradict each other on the normalization constraint. These issues affect the interpretation of the main results and require substantial revision.

major comments (4)
  1. [Sec. II / Appendix Eq. (23)] The forward-limit constraint is not enforced by the stated quark output layer. Equation (3) defines H_q(x,t)=q_v(x)e^{c|t|}NN(x,t), and Eq. (23) gives NN_q(x,t)=exp(NN(x,t)) for the valence-quark sector. Consequently, H_q(x,0)=q_v(x)exp(NN(x,0)), which equals q_v(x) only if the network output happens to vanish at t=0. No such pointwise constraint is listed in the loss function, and the claim in Sec. IV that "NN(x,0)=1" and that the forward limit is "satisfied exactly by construction" is therefore incorrect. This also undermines the interpretation of the -t=0 lattice comparison in Fig. 10, since the quark GPD at t=0 is not guaranteed to reduce to the input PDF. The authors must either modify the quark output layer to enforce NN_q(x,0)=1, add an explicit penalty, or remove the exactness claim and re-interpret the lattice comparison.
  2. [Sec. II Eq. (6) / Sec. III Eq. (19)] The gluon gravitational form factor is defined inconsistently. Equation (6) states A_g(t)=∫ dx x H_g(x,t), which is the second Mellin moment, while Eq. (19), used in the fit, states A_g(t)=∫ dx H_g(x,t), which is the first Mellin moment. These are different quantities. Since Eq. (19) is the formula actually implemented in the loss function, the quantity constrained by the lattice A_g(t) data is the first moment of H_g, not the second moment entering the sum rule of Eq. (6). The reported values A_g(0)=0.37 (JAM21) and 0.26 (xFitter) are consistent with the PDF gluon momentum fraction ∫ xg(x)dx, not with the second Mellin moment of Eq. (6). The authors should specify which sum rule is being fitted and correct either Eq. (6) or Eq. (19) and all subsequent interpretation.
  3. [Sec. IV, gluon rescaling paragraph] The introduction of rescaling factors 1.46 (JAM21) and 2.07 (xFitter) makes the agreement between the rescaled gluon GPD normalization and the lattice A_g(0) a construction, not a validation. A single x-independent multiplicative constant cannot undo a mismatch that the authors themselves attribute to the poorly constrained small-x gluon PDF; the x-dependence of the extracted gluon GPD remains inherited from the input PDF, and the fit only constrains the t-dependence of a single moment integral. The gluon extraction should be presented as a model-dependent estimate with a dedicated systematic uncertainty associated with the rescaling, rather than as an independent determination of the gluon GPD.
  4. [Sec. III Eq. (15) / Sec. IV] The text and the loss function contradict each other on the charge-normalization constraint. The loss function in Eq. (11) explicitly includes chi^2_norm = [(F_pi(0)-1)/0.01]^2, and Fig. 6 shows a separate chi^2_F(0) loss component, but Sec. IV states that "No additional normalization penalty is required for the valence-quark sector" because the forward limit is satisfied by construction. These statements cannot both be true. The authors should state clearly whether F_pi(0)=1 is imposed by a penalty, by the network architecture, or by both, and adjust the text and the loss accordingly.
minor comments (5)
  1. [Eq. (6) and Eq. (19)] Both equations use the symbol A_g(t) for different Mellin moments; please use distinct notation (e.g., A_g^{(2)}(t) and A_g^{(1)}(t)) or state explicitly which moment is being considered in each context.
  2. [Fig. 11 caption] The legend entries "LQCD" and "Lattice QCD" appear to refer to the same dataset; unify the naming to avoid confusion.
  3. [Eq. (22) and text] The parameter T_max is used in the cosine-annealing schedule and also identified as the total number of training epochs; if these are intended to be the same, please say so explicitly, otherwise use separate symbols.
  4. [Sec. III, text after Eq. (16)] The regularization term L_reg is defined in Eq. (16) for the quark sector, but the paragraph discussing gluon regularization mentions "additional regularization terms" without giving their explicit form; provide the gluon L_reg expression or remove the statement.
  5. [Sec. III, output layer sentence] The sentence "The exact form of the output layer have been discussed in Appendix VI" contains a subject-verb agreement error and should refer to the appendix without a section number that duplicates the main text numbering.

Circularity Check

2 steps flagged · score 5.0 of 10

Partial circularity: gluon normalization is imposed by a fitted rescaling factor, and the quark t=0 lattice 'agreement' is built into the forward-limit ansatz; the nonzero-t quark extraction retains independent content.

  1. fitted input called prediction [Section IV (gluon GFF discussion, around Fig. 5 and Fig. 9); Section III, Eqs. (17)-(19)]
    "This discrepancy in the overall normalization motivates the introduction of a rescaling factor in the gluon-sector fit, which has been incorporated into all gluon GPDs results presented in this work."

    The rescaling factor (1.46 for JAM21, 2.07 for xFitter) is a free normalization adjusted so that the model reproduces the lattice A_g(t) data that the gluon network is trained on (Eq. 18). Since it is applied to all reported gluon GPDs, the normalization of the final gluon GPDs—and hence A_g(0)—is forced to match the lattice input by construction. Any 'agreement' in gluon normalization is therefore a fitted input, not an independent output of the PDF+NN extraction; only the residual x- and t-shape beyond the global factor is data-sensitive, and that shape is largely inherited from the input xg(x).

  2. self definitional [Section II, Eqs. (3)-(4); Section IV (Fig. 10 discussion); Appendix, Eq. (23)]
    "This parameterization satisfies the forward-limit constraint. In the limit t→0, the quark and gluon GPDs reduce to their corresponding collinear pion PDFs ... In the forward limit (−t=0), the phenomenological and lattice-QCD results exhibit excellent agreement over the entire x range."

    The equality H_q(x,0)=q_v(x) is inserted into the ansatz H_q(x,t)=q_v(x) e^{ct} NN(x,t), so the quoted 'excellent agreement' at t=0 is a restatement of the input PDF, not a test of the GPD extraction. Moreover, the stated quark output layer NN_q(x,t)=exp(NN(x,t)) (Eq. 23) does not force NN(x,0)=1, while the loss includes a χ²_norm penalty (Eq. 15) that only imposes Fπ(0)=1. Thus the t=0 comparison is either definitionally determined by the input PDF or a normalization fitted to data—not an independent GPD prediction.

full rationale

The extraction is not globally circular. The valence-quark GPD at t>0 is genuinely constrained by experimental EMFF data and then compared with independent lattice-QCD results [48], and the PDF-replica uncertainty propagation is a real calculation. However, two validation steps are partly by construction. First, the gluon-sector normalization is not predicted: a global rescaling factor is fitted to the same lattice A_g(t) data used in the loss and then applied to all reported gluon GPDs, so the normalization agreement is forced. Second, the t=0 'excellent agreement' between quark GPDs and lattice is a built-in consequence of the forward-limit ansatz, not an independent test. The paper compounds this by claiming the forward limit is satisfied exactly 'by construction' while the stated quark output layer does not enforce NN(x,0)=1, and by including a χ²_norm penalty (Eq. 15) that contradicts the claim that no normalization penalty is required. These are correctness flaws as well as circularity indicators. The self-citations ([6], [47], [65]) are not load-bearing for the main derivation. The inconsistent definitions of A_g(t) in Eq. (6) (with an explicit x factor) and Eq. (19) (without) are a separate consistency error rather than a circular reduction, but they further weaken the gluon-sector claims. Overall, the central nonzero-t quark extraction retains independent content, so the circularity is partial rather than total.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central extraction rests on standard GPD sum rules, on the choice of PDFs and lattice data as inputs, and on three modeling assumptions: the multiplicative ansatz with exponential t dependence, the zero-skewness DGLAP-only kinematics, and the x-independent gluon rescaling. The forward-limit constraint is also assumed rather than guaranteed by the stated activation. The NN weights and the Regge slopes c, c_g, plus the gluon rescaling factor are fitted to data.

free parameters (5)
  • Quark Regge slope c = 0.681 (JAM21), 0.655 (xFitter)
    Trainable coefficient in e^{c|t|}; fitted to EMFF data; sets |t|-dependence of quark GPD.
  • Gluon Regge slope c_g = 0.377 (JAM21), 0.383 (xFitter)
    Trainable coefficient in e^{c_g|t|}; fitted to lattice A_g(t).
  • Gluon rescaling factor = 1.46 (JAM21), 2.07 (xFitter)
    Multiplicative normalization applied to all gluon GPDs after the fit to force agreement with lattice A_g(0); fitted to lattice normalization.
  • NN weights and biases = not reported (trained)
    Thousands of parameters optimized to minimize χ²; effectively free parameters with regularization.
  • Regularization strength λ and L2 weight decay = not reported
    Hand-chosen hyperparameters; influence smoothness and the resulting GPD shape.
assumptions (7)
  • standard math Pion GPD sum rules: first Mellin moment of quark GPD gives EMFF (Eq. 5); ∫ dx H_g gives A_g(t) (Eq. 19).
    Standard QCD sum rules, assumed as input.
  • domain assumption Zero-skewness, DGLAP-region parametrization is sufficient to describe EMFF over fitted t range.
    Neglects ERBL region and skewness; standard for this type of extraction but an assumption.
  • ad hoc to paper The factorization H_q=q_v e^{c|t|} NN and H_g=xg e^{c_g|t|} NN is flexible enough to represent the true GPDs.
    The profile form is an ansatz; the NN only corrects residuals around the chosen functional form.
  • domain assumption Input PDFs from JAM21 and xFitter at µ²=4 GeV² are reliable.
    The extraction inherits all PDF uncertainties and systematics.
  • domain assumption Lattice A_g(t) data are accurate and comparable to the continuum model.
    The fit trusts lattice systematics, including renormalization and excited-state contamination.
  • ad hoc to paper The gluon normalization mismatch can be corrected by an x-independent constant rescaling.
    No derivation; the paper applies fitted factors 1.46 and 2.07.
  • ad hoc to paper The forward limit H(x,0)=PDF holds exactly.
    The paper asserts NN(x,0)=1 by construction, but the quark output activation exp(NN(x,t)) does not enforce it; the trained network is assumed to satisfy it.

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Pith. "Pith review of Extraction of Pion Unpolarized Quark and Gluon Generalized Parton Distributions using Deep Neural-Networks." pith.science (2026). https://pith.science/paper/VTHMAQQI

@misc{pith2026260807085,
  author       = {Pith},
  title        = {Pith review of: Extraction of Pion Unpolarized Quark and Gluon Generalized Parton Distributions using Deep Neural-Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VTHMAQQI}},
  note         = {Machine review of arXiv:2608.07085}
}
abstract

We present a deep neural-network (DNN) extraction of the pion unpolarized quark and gluon generalized parton distributions (GPDs) using the corresponding parton distribution functions (PDFs) from the JAM21 and xFitter analysis, together with experimental measurements of the pion electromagnetic form factor (EMFF) and lattice quantum chromodynamics (QCD) results. The GPDs are parameterized using a physics-informed neural-network (PINN) that incorporates the known PDF behavior, an exponential momentum-transfer dependence, and a trainable neural network (NN) component. The network parameters are determined by minimizing a $\chi^2$-based loss function. For the valence-quark GPDs, the loss function includes contributions from the EMFF, squared EMFF, charge-normalization constraints, and regularization terms. For the gluon GPDs, it incorporates constraints from the gluon gravitational form factors together with regularization. This framework enables a flexible, nonparametric extraction while preserving the essential theoretical and phenomenological constraints. By employing the full ensemble of available PDF replicas, we quantify the uncertainties of the extracted GPDs over a broad kinematic range in the longitudinal momentum fraction and momentum transfer, with the uncertainty bands corresponding to the $1\sigma$ confidence interval. The extracted valence-quark GPDs are found to be in good agreement with available lattice-QCD calculations. Our study demonstrates that DNN-based methods provide a flexible and robust framework for extracting pion GPDs and probing the multidimensional internal structure of the pion, offering a promising avenue for future investigations of hadron tomography.

Figures

Figures reproduced from arXiv: 2608.07085 by the authors.

Figure 1
Figure 1. FIG. 1: Valence-quark PDFs of the pion obtained using the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Gluon PDFs of the pion obtained using the central [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Schematic illustration of the NN framework used for the extraction of valence quark GPDs. Valence-quark PDFs, pion [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the NN fits with the experimental [ [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: NN fit to the gluon GFF [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Evolution of the individual loss components and the total loss function as functions of the training epoch up to 10 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Training history of the total loss function and [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Comparison of the extracted quark GPDs obtained using the JAM21 and xFitter pion PDF inputs at the scale [PITH_FULL_IMAGE:figures/full_fig_p006_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Gluon GPD extracted using the JAM21 and xFitter pion PDF parameterizations as inputs at the scale [PITH_FULL_IMAGE:figures/full_fig_p006_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: The calculated quark GPDs from the NN framework are shown as functions of the longitudinal momentum fraction [PITH_FULL_IMAGE:figures/full_fig_p007_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The calculated quark GPDs from the NN framework are shown as functions of the momentum transfer [PITH_FULL_IMAGE:figures/full_fig_p008_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Impact-parameter dependent parton distribution [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Impact-parameter-dependent quark distribution, 2 [PITH_FULL_IMAGE:figures/full_fig_p010_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: Impact-parameter-dependent quark distribution, 2 [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: Two-dimensional transverse impact-parameter distributions of the gluon GPD, [PITH_FULL_IMAGE:figures/full_fig_p011_16.png]
Figure 17
Figure 17. Figure 17: FIG. 17: Two-dimensional transverse impact-parameter distributions of the gluon GPD, [PITH_FULL_IMAGE:figures/full_fig_p011_17.png]

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