Nucleon tomography from a calibrated diquark spectator model

Jakub Terlecki; Krzysztof Cichy; Shohini Bhattacharya

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Nucleon tomography from a calibrated diquark spectator model

Jakub Terlecki, Shohini Bhattacharya, Krzysztof Cichy

High Energy Physics - Phenomenology · Submitted 25 Oct 2026 · 87 references

Pith's verdict: New analytic expression for axial-vector diquark contribution to E^q is the main result; tomography claims are weakened by model's known unreliability at large t.

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CitationJakub Terlecki, Krzysztof Cichy, Shohini Bhattacharya. "Nucleon tomography from a calibrated diquark spectator model." arXiv:2610.34602 (2026). https://pith.science/paper/6ITRFZWD

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Editor's note

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This paper delivers a clean analytic result: the axial-vector diquark contribution to the helicity-flip GPD E^q (Eqs. 2.25–2.26). That formula is genuinely new and fills a gap in the spectator-model literature. The numerical calibration is careful, with two complementary fits that are internally consistent, and the uncertainty propagation is reasonable. The authors are honest about the model's limitations—they state upfront that form-factor predictions for t > 1 GeV² deteriorate because gluons are missing (Sec. 3.3), and they flag the quasi-hard cutoff as an ad hoc prescription (Appendix A). The tomography plots look qualitatively sensible.

That said, the soft spot the stress test hits is real and not merely a missed subtlety. The impact-parameter distributions are obtained by Fourier transforming the GPDs over all |Δ_⊥| up to ~2β, which for β ~ 0.7–0.8 GeV extends to t ~ 2.5 GeV². That is precisely the region where the model is known to be unreliable, and the GFFs already deviate from lattice data at moderate t (Fig. 15). The cutoff’s role in producing the oscillations that appear as negative densities at large x is acknowledged, but the authors do not test whether the shape of the tomography is robust under truncation of the large-|Δ_⊥| tail. Since the t-dependence is entirely an untested prediction—the calibration used only t=0 form factors—the claim of a “consistent tomographic picture” is softer than the text suggests.

The paper is a solid phenomenological study within its chosen framework, and the central analytic result deserves a place in the literature. But the tomography conclusions should be hedged more explicitly: they are model-dependent, driven by a regulator that is physically unmotivated, and are not validated against independent data at finite t. A serious referee would ask for a stability check or a clearer separation of what is constrained versus what is assumed.

Referee report

Significance

The analytic result for the axial-vector-diquark contribution to E^q (Eqs. 2.25–2.26 and 3.10) is a new addition to the spectator-model literature and may be useful for constructing GPD parametrizations. The two complementary calibration procedures demonstrate that including neutron data reduces parameter uncertainties and constrains the flavor decomposition. The PDFs obtained from the calibrated model are in reasonable agreement with JAM 2022 data, lending some support to the model. However, the central claim of providing a tomographic description of valence quarks is weakened because the Fourier transform extends to |Δ_⊥| values (∼1.6 GeV, t∼2.5 GeV^2) where the model is acknowledged to be unreliable due to the absence of gluon degrees of freedom. Without systematic checks on the robustness of the impact-parameter distributions to truncation of the large-t region, the tomographic results remain qualitative. The paper is suitable for publication after major revisions addressing the reliability of the tomography predictions.

Major objections (4)

[Section 3.4.2 (Tomography) and Section 3.3] The Fourier transform of zero-skewness GPDs to impact-parameter space integrates over |Δ_⊥| up to 2β, which is about 1.6 GeV for the calibrated β≈0.7–0.8 GeV, corresponding to t≈2.5 GeV^2. The authors state in Section 3.3 (page 24) that "form-factor predictions for t>1 GeV^2 deteriorate due to the absence of gluons." The calibration only uses t=0 data, so the t-dependence is an untested prediction. The tomography therefore relies on the least reliable part of the model. The negative values in the cross-sections are attributed to cutoff artifacts, but this does not demonstrate that the shape of the distributions is robust. The authors should quantify the sensitivity of the impact-parameter distributions to truncating the Fourier transform at a value of |Δ_⊥| where the model is considered reliable (e.g., |Δ_⊥| < 1 GeV) or provide a systematic uncertainty band from the large-|Δ_⊥| region. The authors acknowledge this: “Form factors values for−tlarger than1GeV 2 are not examined here, because of the simplified nature of the model, the main limitation being the absence of gluons.”
[Section 3.4.2] The claim to provide a "consistent tomographic description" is not supported by quantitative comparisons. The only benchmark in the paper is the PDF comparison to JAM 2022. No comparison is made to existing lattice-QCD tomography results (e.g., Ref. [61] cited by the authors) or to other GPD extractions that yield impact-parameter distributions. Although the authors note qualitative agreement with expectations (localization at large x, flavor-dependent distortion), quantitative validation is absent. The authors should either add a comparative discussion or explicitly note that the tomography is a model prediction that currently lacks independent verification.
[Appendix A (Cutoff discussion)] The quasi-hard cutoff (Eq. 1.5) is introduced to avoid unphysical growth of axial-vector GPDs at large |Δ_⊥|. However, the physical motivation for this specific form is limited: it is not derived from a vertex form factor or a natural scale. The authors examine alternative soft cutoffs but dismiss them because they are not radially symmetric in the GPD (rather than in the product of vertex functions). However, the sensitivity of the final tomography results to the choice of cutoff prescription is not studied. The authors should test at least one other cutoff (e.g., a Gaussian fall-off with a Δ_⊥-dependent width that preserves radial symmetry) to show that the qualitative tomography features are not an artifact of the quasi-hard cutoff.
[Section 3.2 (Calibration)] The calibration uses gravitational form factors A(0) and J(0) from lattice QCD (Hackett et al.) with an assigned 10% uncertainty. The reference values are described as "rounded" (page 19). The paper does not test whether the fitted parameters change appreciably if the lattice central values are used without rounding, or if the uncertainty is taken from the original lattice paper (which may be smaller or asymmetric). The robustness of the parameter extraction to this choice should be discussed.

Minor objections (5)

[General] Several typos: in the abstract, "decomposition constantλ" should be defined explicitly. In Eq. (2.33), the magnetic radius definition should use G_M(0) in the derivative, but the equation as written is ambiguous.
[Figure 19] The color scheme in Fig. 19 is difficult to distinguish in grayscale. Adding line styles (dashed for proton+neutron) would improve accessibility.
[Section 3.4.1] In the PDF comparisons (Fig. 17), the uncertainties from the JAM 2022 analysis are not shown. Including them would allow the reader to assess the level of agreement more rigorously.
[Section 2.2] The definition of δ after Eq. (2.7) is used only in the quasi-GPD appendix (B); it should be either defined closer to its use or omitted from Section 2.
[Appendix B] The convergence check of QGPDs to GPDs is performed with an uncalibrated parameter set (g=1, hard cutoff). The authors should note that this is a consistency check and does not validate the numerical results of the calibrated model.

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The paper

The paper claims

A diquark spectator model calibrated to nucleon data provides analytic GPDs and valence-quark tomography.

The reading

This paper aims to establish that a spectator model with both scalar and axial-vector diquark configurations can describe the twist-2 vector generalized parton distributions (GPDs) H and E of the nucleon at zero skewness, and thereby provide a tomographic picture of valence quarks. The central new result is an explicit analytic expression for the axial-vector diquark contribution to the helicity-flip GPD E^q. By introducing a momentum-transfer-dependent quasi-hard cutoff to regulate transverse-momentum integrals, and calibrating the model parameters to proton (and simultaneously proton plus neutron) electromagnetic and gravitational form factors, the authors obtain flavor-separated GPDs. From these, they compute valence-quark PDFs and impact-parameter distributions for unpolarized and transversely polarized nucleons, finding expected spatial localization and flavor-dependent distortions. The calibration procedures yield compatible parameter ranges, with the neutron-inclusive fit reducing uncertainties.

What carries the argument

The central object is a diquark spectator model containing both scalar and axial-vector diquark configurations, together with a momentum-transfer-dependent quasi-hard cutoff on transverse-momentum integrals (Λ = $\sqrt$($β^{2}$ - ⃗Δ_⊥^2/4)). The analytic expression for the axial-vector diquark contribution to E^q (Eqs. 2.25–2.26, 3.10) carries the argument, as it allows a complete flavor-decomposed calculation at zero skewness. The model's free parameters (quark mass m_q, scalar diquark mass m_s, axial-vector diquark mass m_a, decomposition constant λ, and transverse-momentum scale β) are calibrated to electromagnetic moments and radii plus gravitational form factors, either for the proton alone or for proton and neutron together.

Extended reading notes

Core claim

The central claim is that the axial-vector diquark contribution to the helicity-flip GPD E^q can be expressed in closed analytic form (Eqs. 2.25–2.26, with the integrated result Eq. 3.10). Combined with the known scalar diquark contribution and a novel quasi-hard cutoff that depends on the momentum transfer (Eq. 1.5), the model produces zero-skewness GPDs $H^{{u,d}}$(x,0,t) and $E^{{u,d}}$(x,0,t) once its parameters are fixed by matching proton (and neutron) charge radii, magnetic moments, and gravitational form factors. The paper demonstrates that this model, despite its simplicity, yields valence-quark PDFs consistent with phenomenological extractions, and impact-parameter distributions that localize large-x quarks near the transverse center of momentum, with a sideways shift for a transversely polarized nucleon controlled by E^q.

Load-bearing premise

The model assumes that all non-perturbative dynamics of the two spectator quarks can be represented by a single effective diquark with a point-like coupling to the active quark, and that the entire t-dependence arises from a quasi-hard cutoff and quark/diquark masses, neglecting explicit gluon degrees of freedom.

Editorial extensions

If this is right

The analytic expressions for E^q from the axial-vector diquark enable direct computation of impact-parameter distributions for transversely polarized nucleons, providing a phenomenological baseline for comparisons with lattice QCD and experimental extractions.
The calibrated model yields flavor-separated GPDs that can be used as input for phenomenological analyses of deeply virtual Compton scattering and meson production at zero skewness.
The demonstrated consistency between proton-only and proton+neutron calibrations supports the use of a common scalar-plus-axial-vector spectator structure for both nucleons, simplifying future model extensions.
The momentum-transfer-dependent cutoff suppresses unphysical growth of axial-vector GPDs at large momentum transfer, offering a regulator that respects symmetries and can be adopted in other spectator-model calculations.

Reading between the lines

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

The analytic axial-vector E^q expression fills a gap in spectator-model literature; it could be extended to nonzero skewness to study the ERBL region and to derive sum rules for orbital angular momentum.
The model's reliance on a point-like quark-diquark vertex and the absence of explicit gluon degrees of freedom likely limits its accuracy at momentum transfers above 1 GeV^2, as the authors note; a natural next step would be to incorporate a more realistic vertex form factor or gluon exchange.
The calibrated parameter sets provide a controlled starting point for matching the model to lattice QCD results at a common scale, after DGLAP evolution is applied.

Original abstract

Generalized parton distributions (GPDs) provide a unified description of the longitudinal-momentum and transverse spatial structure of the nucleon. We calculate the twist-2 vector GPDs $H^{q}$ and $E^{q}$ in a spectator model containing both scalar and axial-vector diquark configurations. A central new result is the explicit analytic expression for the axial-vector-diquark contribution to the helicity-flip GPD $E^{q}$. To regulate the transverse-momentum integrations while retaining a physically sensible dependence on the momentum transfer, we introduce a momentum-transfer-dependent quasi-hard cutoff. The model parameters are determined through two complementary calibrations: one using proton observables only and another simultaneously including proton and neutron observables. Electromagnetic moments and radii, together with selected constraints on gravitational form factors, determine the quark and diquark masses, the relative scalar and axial-vector contributions, and the transverse-momentum scale. The two calibrations yield compatible parameter ranges, while the inclusion of neutron observables reduces the parameter uncertainties and more strongly constrains the flavor structure. From the resulting zero-skewness GPDs $H^{u,d}(x,0,t)$ and $E^{u,d}(x,0,t)$, we calculate the valence-quark PDFs and the impact-parameter distributions of unpolarized up and down quarks in both unpolarized and transversely polarized nucleons. The transverse distributions exhibit the expected localization of large-$x$ quarks near the transverse center of momentum and a flavor-dependent distortion when the nucleon is transversely polarized. Our results provide a unified phenomenological description of the longitudinal-momentum and transverse-position structure of valence quarks in the nucleon.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-27: 1 request, 7,828 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.09 · compares with prior results 0.15 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Generalized parton distributions (GPDs) provide a unified description of the longitudinal-momentum and transverse spatial structure of the nucleon. We calculate the twist-2 vector GPDs $H^{q}$ and $E^{q}$ in a spectator model containing both scalar and axial-vector diquark configurations. A central new result is the explicit analytic expression for the axial-vector-diquark contribution to the helicity-flip GPD $E^{q}$. To regulate the transverse-momentum integrations while retaining a physically sensible dependence on the momentum transfer, we introduce a momentum-transfer-dependent quasi-hard cutoff. The model parameters are determined through two complementary calibrations: one using proton observables only and another simultaneously including proton and neutron observables. Electromagnetic moments and radii, together with selected constraints on gravitational form factors, determine the quark and diquark masses, the relative scalar and axial-vector contributions, and the transverse-momentum scale. The two calibrations yield compatible parameter ranges, while the inclusion of neutron observables reduces the parameter uncertainties and more strongly constrains the flavor structure. From the resulting zero-skewness GPDs $H^{u,d}(x,0,t)$ and $E^{u,d}(x,0,t)$, we calculate the valence-quark PDFs and the impact-parameter distributions of unpolarized up and down quarks in both unpolarized and transversely polarized nucleons. The transverse distributions exhibit the expected localization of large-$x$ quarks near the transverse center of momentum and a flavor-dependent distortion when the nucleon is transversely polarized. Our results provide a unified phenomenological description of the longitudinal-momentum and transverse-position structure of valence quarks in the nucleon.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Generalized parton distributions (GPDs) provide a unified description of the longitudinal-momentum and transverse spatial structure of the nucleon.	background 1.00	0.04	0.05	asserted 1.00
s1	We calculate the twist-2 vector GPDs $H^{q}$ and $E^{q}$ in a spectator model containing both scalar and axial-vector diquark configurations.	contribution 0.61	0.96	0.04	asserted 1.00
s2	A central new result is the explicit analytic expression for the axial-vector-diquark contribution to the helicity-flip GPD $E^{q}$.	result 0.81	0.95	0.48	asserted 1.00
s3	To regulate the transverse-momentum integrations while retaining a physically sensible dependence on the momentum transfer, we introduce a momentum-transfer-dependent quasi-hard cutoff.	method 0.99	0.95	0.02	asserted 1.00
s4	The model parameters are determined through two complementary calibrations: one using proton observables only and another simultaneously including proton and neutron observables.	method 0.98	0.93	0.03	asserted 1.00
s5	Electromagnetic moments and radii, together with selected constraints on gravitational form factors, determine the quark and diquark masses, the relative scalar and axial-vector contributions, and the transverse-momentum scale.	method 0.99	0.80	0.05	asserted 1.00
s6	The two calibrations yield compatible parameter ranges, while the inclusion of neutron observables reduces the parameter uncertainties and more strongly constrains the flavor structure.	result 0.91	0.95	0.11	asserted 1.00
s7	From the resulting zero-skewness GPDs $H^{u,d}(x,0,t)$ and $E^{u,d}(x,0,t)$, we calculate the valence-quark PDFs and the impact-parameter distributions of unpolarized up and down quarks in both unpolarized and transversely polarized nucleons.	contribution 0.44	0.97	0.11	asserted 1.00
s8	The transverse distributions exhibit the expected localization of large-$x$ quarks near the transverse center of momentum and a flavor-dependent distortion when the nucleon is transversely polarized.	result 0.80	0.93	0.15	asserted 0.99
s9	Our results provide a unified phenomenological description of the longitudinal-momentum and transverse-position structure of valence quarks in the nucleon.	interpretation 0.85	0.92	0.81	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from simulation or computation 0.58.

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sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
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levels run

P abstract

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** A schematic transition from a physical model of the proton, in which, in addition to the valence quarks, the internal dynamics of the particle involves a gluon-quark sea, to a DSM, in which the two valence quarks are combined into a single effective diquark D, and the dynamics of the gluon-quark sea is accounted for within the quark-diquark pair itself. H(E) up can be expressed as the sum of the scalar diquark GPDs H(E) S and the axialvector diquark GPDs H(E) A, and the down quark GPDs H(E) down are the same as the pure axial-vector diquark GPDs, Hup = λHS + (2 − λ)HA, Hdown = HA,

**Figure 2.** A schematic view of a hadron polarized along the x-axis and moving in the z-direction, depicted as a sphere rotating about the x-axis. It can be seen that, as a result of the rotation about the x-axis, for one half of the sphere (from the center upwards), the rotational momentum is greater than the total momentum of the particle, whereas for the other half of the sphere, the momentum is correspondingly smaller. Similar interpretation can be found in [10]. polarized transversely to its direction of motion, the density of partons will be distorted by terms coming from the Fourier…

**Figure 3.** The Feynman diagram for the leading-twist cut vertex GPD. The vertical blue zigzag line represents the cut vertex γ +, which effectively replaces the interaction between quarks and photons. The thick solid line represents the incoming (outgoing) hadron with momentum p(p ′ ), the solid thin line represents the quark interacting with the photon, and the dashed line is the spectator diquark. Gordon identities1 and performing the contour integral over k −, and changing integral coordinates of ⃗k⊥ to polar ones, one obtain: H(x, ξ, t) =    0 −1 ≤ x ≤ −ξ, g 2 (x + ξ)(1 +…

**Figure 4.** Normalized scalar PDF (left panel) and normalized axial PDF (right panel) for x = 0, as a function of mq/M and ms/a/M for M = 1 GeV, β = 1 GeV. of the zero-skewness GPDs. The limit x → 0 + determines whether the model can reproduce the steep rise of the valence-quark PDFs, whereas the limit x → 1 is essential for a consistent tomographic interpretation, as it requires the t dependence of the GPDs to vanish (see Eq. (3.13) and Eq. (3.14)). Later in this section, we therefore examine the effect of the model parameters on the PDFs and on the zero-skewness GPDs. This is particularly…

**Figure 5.** PDF as a function of x for β = 1 GeV and M = 0.939 GeV. Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. trends anticipated from Eq. (3.12): decreasing mq or increasing ms/a enhances the PDFs at small x. Obviously, the normalization condition implies that the small-x enhancement is accompanied by PDF suppression at intermediate x. The curves corresponding to different masses cross at x ≈ 0.2 and for lighter diquarks, the maximum of the PDF moves…

**Figure 6.** Zero momentum transfer E GPD (forward limit) as a function of x for β = 1 GeV and M = 0.939 GeV. Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. H GPDs. Moreover, because of the negative value of the forward limit axial E GPD, one needs to be careful when transferring interpretations from the scalar and axial E GPDs to the up quark E GPD. For the purposes of tomography, it is essential that the following equation is satisfied: lim x→1 d dtH(x,…

**Figure 7.** H GPD as a function of x for β = 1 GeV, M = 0.939 GeV, |∆⃗ ⊥| = 0.5 GeV (solid line), |∆⃗ ⊥| = 1.0 GeV (dashed line), |∆⃗ ⊥| = 1.5 GeV (dotted line). Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. 0 0.5 1 1.5 2 2.5 3 3.5 mq=0.50 mq=0.40 mq=0.30 mq=0.20 mq=0.10 mq=0.01 Scalar E GPD -2.5 -2 -1.5 -1 -0.5 0 0 0.2 0.4 0.6 0.8 1 mq=0.50 mq=0.40 mq=0.30 mq=0.20 mq=0.10 mq=0.01 Axial x 0 2 4 6 8 10 12 m=0.70 m=0.85 m=1.00 m=1.20 m=1.50 m=2.00 Scalar E…

**Figure 8.** E GPD as a function of x for β = 1 GeV, M = 0.939 GeV, |∆⃗ ⊥| = 0.5 GeV (solid line), |∆⃗ ⊥| = 1.0 GeV (dashed line), |∆⃗ ⊥| = 1.5 GeV (dotted line). Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. is also satisfied. As for the H GPDs, this equality is proven analytically for both scalar and axial-vector diquarks. – 18 –

**Figure 9.** Proton magnetic moment, electromagnetic radii, and GFFs reference values with uncertainties (black) and model results (blue) relative to the reference value. For a fit including only quantities describing the proton. for the neutron magnetic moment is about 10% lower than the reference value. The worst result is for the mean-square charge radius of the neutron, which is almost two times smaller than the reference value. However, this value is very small, so the percentage difference does not translate directly into a difference in absolute value; even so, the reference value lies…

**Figure 10.** Neutron magnetic moment, electromagnetic radii, and GFFs reference values with uncertainties (black) and model results (blue) relative to the reference value. For a fit including only quantities describing the proton. observables are included. We collect the results of the fit in Tab. 2. Following the same

**Figure 11.** Proton magnetic moment, electromagnetic radii and GFFs reference values with uncertainties (black) and model results (orange) relative to the reference value. For a fit including only quantities describing simultaneously the proton and the neutron

**Figure 12.** Neutron magnetic moment, electromagnetic radii and GFFs reference values with uncertainties (black) and model results (orange) relative to the reference value. For a fit including only quantities simultaneously describing the proton and the neutron. to note that the parameter values obtained from the proton-only and proton+neutron fits agree with each other, as shown in

**Figure 13.** Comparison of obtained model parameters for the fit including only the proton (p) and the fit simultaneously including the proton and the neutron (p+n). The results are shown as a ratio to the central value obtained for the fit including only the proton. which leaves room for correlated compensation among them. In the remainder of this paper, we will present both the results for a fit that includes only the proton (represented in blue) and those for a fit that also takes the neutron into account (represented in orange). One might also ask whether a neutron-only fit would be…

**Figure 14.** EMFFs as a function of −t obtained for the proton and the neutron, calculated from results for a fit including only the proton (blue), and a fit that included both the proton and the neutron (orange), compared to experimental data (black) [86]. both nucleons, show greater discrepancies from the reference data [32] (in this case, lattice data rather than experimental data) than those for EMFFs. In addition, all features related to the characteristics of the charts and their relationships with each other for both kinds of fit for EMFFs apply to GFFs. It can be seen here that both…

**Figure 15.** GFFs as a function of −t obtained for the proton and the neutron, calculated from results for a fit including only the proton (blue) and a fit that includes both the proton and the neutron (orange), compared to lattice data (black) [32]. lattice lattice

**Figure 16.** GFFs as a function of −t obtained for up and down quarks in the proton, calculated from results for a fit including only the proton (blue), and a fit that included both the proton and the neutron (orange), compared to lattice data (black) [32]. fit that considers only the proton. 3.4 PDFs, GPDs, and tomography results In this section, we present the main results of this paper: the PDFs, the GPDs, and the proton tomography for both up and down quarks. It should be emphasized that the model was tuned with respect to the FF-related quantities at t = 0, so all the results in the…

**Figure 17.** The PDF and the absolute value of the E GPD in the forward limit as a function of x, obtained for up and down quarks in the proton, calculated from results for the fit including only the proton (blue), and fit that includes both the proton and the neutron (orange), compared to JAM 2022 for the PDF (black) [87]. The absolute value for the E GPD only affects the down quark plot, because the values for the up quark E GPD are positive over the whole x range, in contrast to the down quark GPD, which is negative over the entire range. We present results for the down quark forward limit E…

**Figure 18.** Zero transverse momentum transfer (|∆⃗ ⊥| = 0) H and E GPDs with ξ = 0.2 as a function of x obtained for up and down quarks in a proton, calculated from results for the fit including only the proton (blue) and the fit that includes both the proton and the neutron (orange). The boundary of the ERBL region is indicated by a vertical dashed line. A natural extension of the calculations involving the forward limit GPDs is the calculation of nonzero momentum transfer GPDs in this model for our calibrated parameters. We show results for both kinds of fits, for ξ = 0.2 and |∆⃗ ⊥| = 0 in

**Figure 19.** Zero skewness GPDs as function of x and √ −t. For the fit including the proton (blue) and both the proton and the neutron (orange). The results are presented only for the central value without uncertainty bands. Note that the vertical axis uses a logarithmic scale. GPDs, and to see whether this model can fill the gaps in lattice calculations. We tested the model for several different values of ξ. For any value of ξ, the scalar and axial GPDs always exhibit a transition from the ERBL to the DGLAP region, in which the function increases in the ERBL region and decreases in the DGLAP…

**Figure 20.** Proton GPDs in the impact parameter space for both the up and the down quark. Results for the central value obtained for the fit taking only the proton into consideration are shown in blue; results for the fit that take both the proton and the neutron into account are shown in orange. The proton is polarized along the x axis. Because the results go to zero more slowly for the proton-only fit, we expect them to be narrower in tomography after the Fourier transform than the results for the proton+neutron fit. For the latter, we expect a slower decay as b⊥ increases. For the same…

**Figure 21.** Nucleon tomography for unpolarized and transversely polarized proton for x = 0.1. The left column shows the cross-section for bx = 0, of the heat map shown in the right column. We present both the result for the fit including only protons (blue) and both nucleons (orange). We show the central result on the heatmap. -6 -4 -2 0 2 4 6 q(x=0.3, b) by [GeV ] -1 bx [GeV ] -1 b y [GeV ] -1 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 0.1 0.075 0.05 0.025 0 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 proton proton+neutron (a) x = 0.3, unpolarized up quark in proton

**Figure 22.** Nucleon tomography for unpolarized and transversely polarized proton for x = 0.3. The left column shows the cross-section for bx = 0, of the heatmap shown in the right column. We present both the result for the fit including only the proton (blue) and both nucleons (orange). We show the central result on the heatmap. – 34 –

**Figure 23.** g+ ⃗k⊥ + ∆⃗ ⊥/2 · g− ⃗k⊥ − ∆⃗ ⊥/2 value for form factor given by Eq. (A.1), for different values of ∆⃗ ⊥, and Λ = 2.5 [GeV]. All values in plots are given in GeV. functional forms exhibiting significant variation in values within the range from ⃗k⊥±∆⃗ ⊥/2 = 0 to the specified cutoff value, where the function’s value should fade away. Instead, we are looking for form factors such that the resulting value is an almost constant function, up to a certain limit beyond which it begins to decay rapidly. We tested several forms of exponential functions g±( ⃗k⊥ ± ∆⃗ ⊥/2) such that…

**Figure 24.** g+ ⃗k⊥ + ∆⃗ ⊥/2 · g− ⃗k⊥ − ∆⃗ ⊥/2 value for form factor given by Eq. (A.2), for different values of ∆⃗ ⊥, Λ = 2.5 [GeV] and α = 0.05. All values in plots are given in GeV. -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k 2 k1 Δ⊥=(0,0) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k 2 k1 Δ⊥=(2.5,0) -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 k 2 k1 0 0.2 0.4 0.6 0.8 1 Δ⊥=(2.5,2.5)

**Figure 25.** Soft cutoff value given by Eq. (A.2), for different values of ∆⃗ ⊥, Λ = 2.5 √ 2 [GeV] and α = 0.05. All values on plots are given in GeV. behavior where, for larger values of |∆⃗ ⊥|, axial-vector diquark GPDs, after an initial drop, begin to rise to infinity as |∆⃗ ⊥| increases. For this reason, we propose a hard cutoff with integration limits dependent on the value of |∆⃗ ⊥| as an alternative to the standard hard cutoff. A cutoff defined in this way is referred to as a quasi-hard cutoff. In the GPD structure, ⃗k⊥ ± ∆⃗ ⊥/2 naturally appears as the transverse momentum of the hadron…

**Figure 26.** Zero transverse momentum transfer H GPD as a function of x for β = 1 GeV, M = 0.939 GeV and ξ = 0.2. Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. In the calculations for this paper, we used a function describing the E GPD for an axialvector diquark, a result that was previously not available in the literature. In addition, we also calculated the quasi-GPD (QGPD) for this diquark. Following [58], we calculated QGPDs for both Γ = γ 0 and Γ =…

**Figure 27.** Zero transverse momentum transfer E GPD as a function of x for β = 1 GeV, M = 0.939 GeV and ξ = 0.2. Left panel: results for different values of mq (ms = ma = 1 GeV). Right panel: results for different values of ms = ma = m (mq = 0.35 GeV). The mass values in the legend are given in GeV. model, we refer to [58] and [79]. For Γ = γ 0 , we get: 2 δ N a E(0) = −2 m2 q m2 a M2 + 2M mq m2 a M2 k 0 δP3 − 1 − t 4 1 + k 0 δP3 − 4k 0 mq δP3 − 4M2x k 0 δP3 + 1 + 2 k 0 δP3 M2 m2 a 2x(P 3 ) 2 + x M2 − t 4 − 2 M2 m2

**Figure 28.** E QGPD for axial-vector diquark as a function of x for different values of ξ and P 3 . Left panel: results for γ 0 . Right panel: results for γ 3 . The standard axial-vector diquark E GPD is shown for comparison. The limits of the ERBL region are indicated by vertical dashed lines. – 47 –

**Figure 29.** Nucleon tomography for unpolarized and transversely polarized proton for x = 0.5. The left column shows the cross-section for bx = 0, of the heat map shown in the right column. We present both the result for the fit including only protons (blue) and both nucleons (orange). We show the central result on the heatmap. -6 -4 -2 0 2 4 6 q(x=0.7, b) by [GeV ] -1 bx [GeV ] -1 b y [GeV ] -1 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 0.1 0.075 0.05 0.025 0 6 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 proton proton+neutron (a) x = 0.7, unpolarized up quark in proton. –

**Figure 30.** Nucleon tomography for unpolarized and transversely polarized proton for x = 0.7. The left column shows the cross-section for bx = 0, of the heat map shown in the right column. We present both the result for the fit including only the proton (blue) and both nucleons (orange). We show the central result on the heatmap. 0.02 -6 -4 -2 0 2 4 6 q(x=0.9, b) by [GeV ] -1 bx [GeV ] -1 b y [GeV ] -1 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 0.1 0.075 0.005 0.0025 0 6 0.015 0.01 0.005 0 proton proton+neutron (a) x = 0.9, unpolarized up quark in proton. – 51 –

**Figure 31.** Nucleon tomography for unpolarized and transversely polarized proton for x = 0.9. The left column shows the cross-section for bx = 0, of the heatmap shown in the right column. We present both the result for the fit including only protons (blue) and both nucleons (orange). We show the central result on the heatmap. – 52 –

Figures reproduced from arXiv:2610.34602 by the authors.

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [61]

K. Cichy, M. Constantinou, P. Sznajder and J. Wagner,Nucleon tomography and total angular momentum of valence quarks from synergy between lattice QCD and elastic scattering data,Phys. Rev. D110(2024) 114025 [2409.17955]

View on Pith Pith review arXiv 2024

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D. Müller, D. Robaschik, B. Geyer, F.M. Dittes and J. Hořejši,Wave functions, evolution equations and evolution kernels from light ray operators of QCD,Fortsch. Phys.42(1994) 101 [hep-ph/9812448]

View on Pith Pith review arXiv 1994

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View on Pith Pith review arXiv 1997

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View on Pith Pith review arXiv 1997

[5] [4]

A.V. Radyushkin,Scaling limit of deeply virtual Compton scattering,Phys. Lett. B380 (1996) 417 [hep-ph/9604317]

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