Nonforward Parton Distributions

A.V. Radyushkin

REVIEW

Nonforward Parton Distributions

A.V. Radyushkin

High Energy Physics - Phenomenology · Submitted 2 Apr 1997 · 69 references

Pith's verdict: The founding paper of the double-distribution formalism for GPDs; the derivations hold, and the one acknowledged open question is stated openly rather than papered over.

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arXiv IDhep-ph/9704207 (version 7)

Pith IDDYGXOOIK, Pith's permanent record of this paper and its review

CitationA.V. Radyushkin. "Nonforward Parton Distributions." arXiv:hep-ph/9704207 (1997). https://pith.science/paper/DYGXOOIK

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Editor's note

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Short version: this is the paper that built the double-distribution machinery for generalized parton distributions, and it holds up. The classifier called it a review because the abstract opens with 'we describe' [abstract:s1]. That label is wrong. The body derives new objects: the all-order α-representation proof of the support 0 ≤ {x,y,x+y} ≤ 1, the nonforward kernels W_ζ(X,Z) and the double-distribution kernels R(x,y;ξ,η), and their DGLAP (ζ=0) and Brodsky-Lepage (ζ=1) limits. The two headline functions F(x,y;t) and F_ζ(X;t) came from the author's earlier letters, and he says so; what is new here is the all-order spectral argument and the kernels, not the definitions. That is a fair accounting of the contribution and I would not hold the earlier letters against it. The paper is honest about its own lineage, which is more than most do. What it does well: the scalar toy model in Sec. II is worked out in closed form (Eqs. 2.8, 2.20, 2.27, 2.45) and can be checked by hand, which makes the whole construction readable. The reduction formulas (6.17) and (6.18) are a real cross-check on the kernels, not decoration. Sec. IX, comparing notations with Ji and with Collins-Frankfurt-Strikman, is the kind of section I wish more papers in this area had written. Soft spots, in proportion. The one that matters is the nonperturbative survival of F_ζ(0)=0. The perturbative proof is to all orders and the evolution preserves the vanishing (Sec. VI A), but the extrapolation to the real hadron is not proved, and the author says so plainly in Sec. VIII B. That is load-bearing for factorizing the real part of the DVCS amplitude, not for the double-distribution framework itself, and the stress-test note has this right. Second, smaller: in Sec. V B the transition-region kernel W^gQ_ζ for X ≤ ζ ≤ Z is asserted as 'our explicit calculation confirms' without the calculation shown. Minor and checkable, but it is the one place where the reader has to take something on faith. The Balitsky-Braun kernels are cited from published work and used as external input, which is fine. Who should read it: anyone working on DVCS or hard exclusive electroproduction, and anyone who wants the cleanest available derivation of the support properties. It is long and notation-heavy, so it is a reference rather than a weekend read. Recommendation: send it to a serious referee. I would cite it, and I would put it on a reading group list for the toy-model section alone.

Referee report

Significance

This is the foundational reference for the double-distribution formulation of generalized parton distributions, and it remains the standard place to learn why the ζ-independent and ζ-dependent descriptions are equivalent and how each behaves under evolution. The paper's strengths are concrete and checkable. The one-loop scalar model yields closed forms (Eqs. (2.8), (2.20), (2.27), (2.45)) that can be verified by hand, and I did verify the reduction formula (2.21) and the two branches of Eq. (2.25) against Eq. (2.20). The evolution kernels are not merely asserted: the reduction formulas (6.17) and (6.18) are nontrivial cross-checks, and I confirmed that the ζ → 0 limits of Eqs. (5.26)–(5.29) reproduce the DGLAP kernels P^{qq}, P^{qg}, P^{gq}, P^{gg} exactly, including the correct factor of X/Z required by the convention F^g_{ζ=0}(X) = X f_g(X). The 1/ζ² in L^{Qg}_ζ of Eq. (5.16) is likewise consistent with W^{Qg} = −∂_X M^{Qg}. There are no free parameters and no fits; the only external input is the published light-ray kernel of Refs. [33,34]. The paper also states the F_ζ(0) = 0 requirement as an explicit, falsifiable condition for factorization rather than assuming it silently, in

Major objections (2)

[§VIII B and §IX] The paper is explicit that the vanishing of F_ζ(X;t) at X = 0 is proved only diagram by diagram in perturbation theory, and that its nonperturbative survival is a hope rather than a theorem ('a cautious statement might be that if in perturbation theory some function does not vanish at a particular end-point, it is unlikely that it will vanish nonperturbatively'). I credit the authors for stating this openly, and I do not treat it as a new finding. What remains unresolved, and what I would ask the authors to add, is the consequence side of the argument. The manuscript should say plainly what goes wrong if F_ζ(0) ≠ 0: the P∫_0^1 F_ζ(X)/(X−ζ) dX in Eq. (7.4) and the P∫ F_ζ(X)/X dX implicit in Eq. (7.2) acquire an unintegrable end-point singularity, so the leading-twist real part of the DVCS amplitude is not expressible through F_ζ alone and the handbag factorization of Sec. VII A fails for the real part while surviving for the imaginary part (7.3). Stated that way, the assumption becomes a testable statement about the X → 0 behaviour of the DVCS real part rather than a technicality. It would also strengthen the section to point forward to the paper's own strongest evidence, namely the Sec. VI A result that F_ζ(X;μ → ∞) ∝ X(1 − X/ζ) and the Sec. IX observation that even the δ(x) asymptotic of F(x,y;μ → ∞) yields a regular F_ζ confined to 0 ≤ X ≤ ζ; these show that the property is stable under evolution, which is more than a perturbative accident.
[§VII B, Eq. (7.14)] The gluonic matrix element is parametrized with the weighting F^g_ζ(X)/[(X − iε)(X − ζ + iε)], and this weighting is the step that produces the imaginary part of the longitudinal amplitude in Eq. (7.18), Im T_LL ∝ F^g_ζ(ζ)/ζ. The text motivates the double denominator by the light-cone-gauge representation (7.13), A_μ(z;q') = q'^ρ ∫_0^∞ G_{μρ}(z + σq') e^{−εσ} dσ, but the intermediate step is not shown: the reader is asked to accept that the two σ-integrals attached to the two vector potentials generate exactly 1/(X − iε) and 1/(X − ζ + iε) with unit residues and no additional X- or ζ-dependent numerator. Since Eq. (7.18) is a physical prediction and since the whole comparison with Ref. [9] in the paragraph following Eq. (7.18) hinges on this normalization, please insert the explicit derivation, or at minimum state the convention under which the residues are fixed. This is the one place in the paper where a load-bearing formula is asserted rather than derived.

Minor objections (7)

[§II E, Eqs. (2.47) and (2.48)] As typeset in the version I read, Eq. (2.48) behaves as T√(1+T²) ln(T + √(1+T²)) → 0 as t → 0, which contradicts Eq. (2.23) and the X = ζ value of Eq. (2.27), both of which give the finite constant g²/(16π²m²). Carrying out the y-integral in Eq. (2.46) with the model distribution (2.45) I obtain ln(T + √(1+T²))/(T√(1+T²)), which does tend to unity; the prefactor in Eq. (2.48) appears to be inverted. The same integral suggests that the X ≥ ζ branch of Eq. (2.47) is missing a factor X̄ = 1 − X, since the t → 0 limit should reproduce (1 − X)/(1 − ζ) from Eq. (2.27). Please check both prefactors against the explicit integral.
[§VII A, Eqs. (7.2) and (7.4)] Eq. (7.2) contains two denominators, 1/(X − ζ + iε) and 1/(X − iε), but Eq. (7.4) for the real part retains only the principal value of the first. The principal value P(1/X) of the second term is nonzero and, with F_ζ(X) vanishing linearly at the origin, gives a finite contribution. Please clarify whether Eq. (7.4) is meant to be the full real part or only the s-channel piece; if the latter, say so explicitly, since the two readings differ by a term of the same order as the one retained.
[§II C and Abstract] The ζ-independence of F(x,y;t) is stated without qualification in Sec. II C and in the abstract, but for a massive target the physically accessible t-range is ζ-dependent through the kinematic bound (4.5), −t ≳ ζ²M²/ζ̄, which the paper itself quotes in Sec. IV A. A short caveat at the point of first use would prevent the reader from over-reading the ζ-independence as holding at fixed physical t.
[§IX, Eq. (9.9)] The rebuttal of the Collins-Frankfurt-Strikman normalization f_g(x₁,x₂;t) = F̃^g_ζ(X;t)/[X(X−ζ)] rests on the observation that P^{gQ}_ζ(X,Z) of Eq. (5.28) does not vanish at X = ζ. That is a statement about the evolution kernel, not about F^g_ζ itself, so the conclusion that f_g must be singular at x₂ = 0 is conditional on F^g_ζ(ζ) ≠ 0. The argument is plausible and probably right, but it should be phrased as conditional, or supported by the explicit one-loop model of Sec. II where F^g_ζ(ζ) can be computed.
[§III B, Fig. 6b] The power-counting estimate t_H(Q) ≲ Q^{−1} for the configuration of Fig. 6b is quoted without showing which external lines carry which twist. A single sentence listing the external lines of H and their twists would let the reader reproduce the 4 − N − Σt_i arithmetic of Eq. (3.7) for this case as easily as for the leading case.
[§VI A, Eq. (6.12)] The text correctly warns that the Gegenbauer inversion is only legitimate for ζ = 1 and that Eq. (6.12) must be read as a distributional identity, but this warning comes several sentences after the equation. A parenthetical in the equation itself, or a forward reference, would keep a casual reader from using (6.12) as a functional representation.
[Notation and figures] The spin-flip distributions are denoted K^a_ζ(X;t) here while the comparison section maps them onto Ji's E^a; a compact conversion table (this paper, Ji, Collins-Frankfurt-Strikman) would be more useful than the prose of Sec. IX. The momentum-flow labels in Figs. 2 and 3 are difficult to read at reproduced size and would benefit from redrawing. There are also several typographical errors in the figure captions and body ('electoproduction' in the captions of Figs. 9 and 10, 'distibution' in Sec. VII B, 'excercise' in Sec. VI A).

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The paper

The paper claims

The paper proposes that a single skewness-independent double distribution can generate the nonforward distributions needed for hard exclusive scattering.

The reading

The paper sets out to describe hard exclusive processes such as deeply virtual Compton scattering and exclusive meson electroproduction, where the long-distance information sits in a matrix element $langle p'|O(0,z)|prangle$ rather than in the usual forward parton density. Its proposal is to use two linked functions: a double distribution $F(x,y;t)$ that records the fractions of the initial momentum $p$ and the momentum transfer $r$ carried by the active parton, and a nonforward distribution $F_zeta(X;t)$ that records only the total fraction $X$ for a given skewness $zeta$. The paper argues that the double distribution has clean spectral support, $0le{x,y,x+y}le1$, and that both objects obey evolution equations whose kernels reduce to the usual DGLAP kernels at $zeta=0$ and to the Brodsky-Lepage kernels at $zeta=1$. If this holds, a single $zeta$-independent object could organize a family of skewed distributions and make factorization for hard exclusive amplitudes look much like the ordinary parton picture.

What carries the argument

The load-bearing device is the $alpha$-representation of Feynman diagrams, used to read off the allowed momentum fractions from the denominator structure before spin numerators or gauge complications are added. In that representation the double distribution is defined by delta functions fixing $x$ and $y$ in terms of the $alpha$-parameters, which makes the support $0le{x,y,x+y}le1$ manifest diagram by diagram. The bridge to evolution is the light-ray operator equation with kernels $B(u,v)$; inserting it into nonforward matrix elements and Fourier transforming with respect to $(pz)$ yields the convolution kernels $W_zeta(X,Z)$ for $F_zeta(X;t)$, with the limits $zeta=0$ and $zeta=1$ recovering DGLAP and Brodsky-Lepage evolution.

Extended reading notes

Core claim

The central claim is that nonforward matrix elements of quark and gluon light-cone operators can be parametrized by a $zeta$-independent double distribution $F(x,y;t)$ whose support is restricted to $0le xle1$, $0le yle1$, and $x+yle1$, and that integrating over $y$ produces the $zeta$-dependent nonforward distribution $F_zeta(X;t)$ used in hard exclusive amplitudes. The same construction supplies evolution: the kernels $W_zeta(X,Z)$ for $F_zeta$ are obtained from the known light-ray kernels $B(u,v)$, and they interpolate between DGLAP evolution in the region $X>zeta$ and distribution-amplitude (Brodsky-Lepage) evolution in the region $X<zeta$. The paper also argues that $F_zeta(X;t)$ vanishes at $X=0$ for any finite double distribution, which is what keeps the $1/X$ endpoint in the hard amplitude integrable and allows factorization of the real part of the deeply virtual Compton amplitude.

Load-bearing premise

The spectral constraints, especially the vanishing of $F_zeta(X;t)$ at $X=0$, are established diagram by diagram in perturbation theory through the $alpha$-representation, and the paper explicitly treats their survival to the nonperturbative distribution as a cautious expectation rather than a proof.

Editorial extensions

If this is right

Deeply virtual Compton scattering can be analyzed with the same parton-language convolution used for inclusive processes, but with $F_zeta(X;t)$ replacing the ordinary density.
A single $zeta$-independent double distribution can generate the whole family of skewed distributions needed at different momentum asymmetries.
Evolution connects the two standard limits: ordinary DGLAP evolution for $X>zeta$ and Brodsky-Lepage evolution for $X<zeta$.
The vanishing of $F_zeta$ at $X=0$ removes the unintegrable endpoint singularity that would otherwise spoil factorization of the real part.
The gluon distribution can be defined through a gauge-invariant light-cone operator and used in hard exclusive meson electroproduction, with the same $1/(X-zeta+i epsilon)$ structure producing the imaginary part.

Reading between the lines

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

If the endpoint vanishing survives nonperturbatively, the real part of DVCS becomes a clean probe of the $X<zeta$ distribution-amplitude region, where the skewed function behaves like a meson distribution amplitude rather than a parton density.
The double distribution suggests a practical reduction: fit one $zeta$-independent function and predict the skewed distributions at all $zeta$, then test the prediction against DVCS and exclusive meson data at several kinematics.
Because the same kernels interpolate between DGLAP and Brodsky-Lepage limits, the construction offers a bridge between inclusive parton densities and exclusive distribution amplitudes; a single fit could in principle connect two normally separate QCD industries.
A lattice calculation of the first few $X$-moments of $F_zeta$ at nonzero $zeta$ would test whether the $alpha$-representation support constraints are genuine nonperturbative properties or only perturbative bookkeeping.

Original abstract

Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements <p'|O|p> of quark and gluon light-cone operators. We describe two types of nonperturbative functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F_\zeta (X;t), discuss their spectral properties, evolution equations which they satisfy, basic uses and general aspects of factorization for hard exclusive processes.

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arXiv listing abstract v7. Results and the main claim are underlined; show every role.

Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements <p'|O|p> of quark and gluon light-cone operators. We describe two types of nonperturbative functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F_\zeta (X;t), discuss their spectral properties, evolution equations which they satisfy, basic uses and general aspects of factorization for hard exclusive processes.

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	Applications of perturbative QCD to deeply virtual Compton scattering and hard exclusive electroproduction processes require a generalization of usual parton distributions for the case when long-distance information is accumulated in nonforward matrix elements <p'\|O\|p> of quark and gluon light-cone operators.	background 1.00	0.06	0.11	asserted 1.00
s1	We describe two types of nonperturbative functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions F_\zeta (X;t), discuss their spectral properties, evolution equations which they satisfy, basic uses and general aspects of factorization for hard exclusive processes.	contribution 1.00	0.84	0.32	asserted 0.98

What kind of paper the classifier reports this is: review 0.51 · evidence from proof or derivation 0.51.

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arXiv listing abstract v7

not examined

full text
figures and tables
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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.** FIG. 1. Scalar model analogs of

**Figure 2.** FIG. 2. Longitudinal momentum flow for two components of the a

**Figure 4.** FIG. 4. Handbag diagram for deeply virtual Compton scatteri

**Figure 5.** FIG. 5. Four-point amplitude corresponding to the deeply vi

**Figure 6.** FIG. 6. Some regimes responsible for powerlike contributio

**Figure 9.** FIG. 9. Hard exclusive meson electoproduction process:

**Figure 10.** a as T α g (p, p′ , q′ ) = Z hq ′ , M|ψ¯(0) γ αS c (−z1)τ a γ µS c (z1 − z2)τ b γ νψ(z2)|0i hp ′ |A a µ (z1)A b ν (z2)|pi d 4 z1d 4 z2 , (7.12) where τ a , τ b are the SU(3) color matrices. The first matrix element here can be expressed through the meson distribution amplitude ϕ(τ) while the second one is related to the asymmetric gluon distribution. Other 3 lowestorder diagrams can be written in a similar way. Applying formally the power counting (see Eq. (3.7) and discussion preceding it), we may conclude that each gluonic contribution has an extra Q2 f

**Figure 11.** FIG. 11. Double-flow regime for the scalar analog of a meson fo

Showing 8 of the paper's 11 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9704207 by the authors.

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Forward citations

Cited by 60 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Kinematic power corrections to DVCS to twist-six accuracy
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Complete kinematic power corrections up to twist-6 are derived for nucleon DVCS, and the series converges best when organized in powers of 1/(Q²+t).
Generalized parton distributions of the nucleon in the chiral quark soliton model: Sum over all quark states at nonzero skewness and momentum transfer
hep-ph 2026-09 conditional novelty 7.0 of 10

First sum-over-Dirac-continuum computation of the subleading unpolarized GPD combinations H^{u-d} and E^{u+d} at nonzero skewness and momentum transfer in the chiral quark soliton model.
Exclusive Leptonium Electroproduction
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First complete calculation of exclusive vector-leptonium electroproduction finds Bethe-Heitler dominance, predicts dimuonium discovery potential at EIC, and computes NLO QCD corrections that significantly modify the d...
Matching collinear factorization with color-glass condensate for inclusive and exclusive deep inelastic scattering
hep-ph 2026-05 accept novelty 7.0 of 10

Collinear factorization amplitudes exactly reproduce the large-Q² expansion of CGC amplitudes for inclusive DIS, DVCS, and DVMP at the amplitude level.
Exclusive photoproduction of a di-meson pair with large invariant mass
hep-ph 2026-05 unverdicted novelty 7.0 of 10

Computes LO amplitudes for di-meson photoproduction channels sensitive to quark GPDs and reports high cross sections suitable for GPD studies at JLab energies.

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A flexible two-component model of pion-nucleon transition distribution amplitudes is fitted to CLAS data and used to predict cross-sections and three leading-twist spin asymmetries for backward pion electroproduction.
Resummation for Lattice QCD Calculation of Generalized Parton Distributions at Nonzero Skewness
hep-ph 2025-01 conditional novelty 7.0 of 10

A threshold factorization and resummation scheme for quasi-GPD matching in LaMET is derived and shown to be self-consistent on a GPD model.
Gluon Unpolarized, Polarized, and Transversity GPDs from Lattice QCD: Lorentz-Covariant Parametrization (Part I)
hep-lat 2024-12 conditional novelty 7.0 of 10

A Lorentz-covariant tensor basis and projection set is constructed that isolates all eight leading-twist gluon GPDs from lattice QCD matrix elements for spin-0 and spin-1/2 hadrons.
The two-loop coefficient functions for double deeply virtual Compton scattering
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The paper derives the two-loop coefficient functions for the operator product expansion of two electromagnetic currents in general kinematics, the central ingredient for next-to-next-to-leading-order DDVCS predictions.
Gluon gravitational form factors of the nucleon and the pion from lattice QCD
hep-lat 2018-10 conditional novelty 7.0 of 10

Lattice QCD on a single 32^3 x 96 ensemble at m_pi ~ 450 MeV yields the first determination of the gluon D-term form factor of the nucleon and the gluon gravitational form factors of the pion.
Elements of a theory for multiparton interactions in QCD
hep-ph 2011-11 conditional novelty 7.0 of 10

Multiple hard parton scattering is not power suppressed once final-state transverse momenta are measured, and its cross section factorizes into transverse-momentum-dependent two-parton distributions whose color, spin,...
Computability of GPDs near $x=\pm\xi$ in Lattice QCD
hep-lat 2026-09 accept novelty 6.0 of 10

GPDs near x=+-xi are shown to be calculable in lattice QCD via LaMET, with the soft-quark contribution power-suppressed, and reanalysis of lattice data confirms continuity at x=+-xi.
Twist-3 gluon GPDs for the nucleon in a spectator model
hep-ph 2026-09 conditional novelty 6.0 of 10

The complete set of twist-3 gluon GPDs for the nucleon at nonzero skewness is computed for the first time in a spectator model with dipolar form factors and a spectral function.
Kaon gravitational form factors and mechanical structure in a three-flavor NJL model
hep-ph 2026-08 accept novelty 6.0 of 10

The kaon's gravitational form factors are computed in a symmetry-preserving NJL model, showing that explicit SU(3) breaking makes the kaon's mechanical structure more compact than the pion's.
Extraction of DVCS amplitudes off the nucleon
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A global neural-network fit of proton and neutron DVCS data yields Compton form factors, including first helicity-flip amplitudes, with comparisons to TCS and lattice QCD.
Multipole structure of the $N \to \Delta$ Transition Generalized Parton Distributions
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The nucleon unpolarized generalized form factors and Mellin moments up to fourth order
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Gradient flow for parton distribution functions: first application to the pion
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Generalized parton distributions of the kaon and pion within the nonlocal chiral quark model
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Gluon Tomography from Deeply Virtual Compton Scattering at Small-x
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Kernel methods for evolution of generalized parton distributions
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hep-ph 2026-08 unverdicted novelty 3.0 of 10

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hep-ph 2026-09 unverdicted novelty 2.0 of 10

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hep-ph 2025-12 unverdicted novelty 2.0 of 10

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hep-ph 2020-01 accept novelty 2.0 of 10

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Azimuthal and Single Spin Asymmetries in Hard Scattering Processes
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Generalized Parton Distributions
hep-ph 2003-07 accept novelty 2.0 of 10

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How later papers used it

206 later papers cite it

Evidence: substantial. 200 of the citing papers share no author with it, and 17 of those carry a confident label from Pith's citation classifier.

Pith read 15 of the 295 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

7 later papers by other authors are listed below: 7 used its method or tool.

Uses its method or tool 7 of 18 papers with this label

Used the Goloskokov-Kroll model, itself based on this paper's standard double distribution model of GPDs, for its GPD input.

Complete 1-loop study of exclusive $ J/\psi $ and $ \Upsilon $ photoproduction with full GPD evolution (2024) · 0.83 · paraphrased: the licence does not allow quoting
Analyzed the process in terms of generalized parton distributions from this paper.

$p\bar{p} \to \Lambda_c \bar{\Lambda}_c$ within a Handbag Picture -- Cross Section and Spin Observables (2009) · 0.81 · paraphrased: the licence does not allow quoting
Worked within the alpha-representation, following this paper's lines of thought on deeply virtual Compton scattering, to analyze the four-particle amplitude in the phi^3_E model.

Duality between different mechanisms of QCD factorization in \gamma^*\gamma collisions (2008) · 0.96 · paraphrased: the licence does not allow quoting
Employed the alpha-representation, citing this paper, to study the four-particle amplitude in detail.

Duality and Factorization Theorem in QCD (2008) · 0.98 · paraphrased: the licence does not allow quoting
Used a toy GPD model based on this paper's double-distributions ansatz that allows a simple analytic representation.

The two-loop coefficient functions for double deeply virtual Compton scattering (2024) · 0.93 · paraphrased: the licence does not allow quoting
Employed the alpha-representation, citing this paper, to study the four-particle amplitude in detail.

Photon-photon collision: ambiguity and duality in QCD factorization theorem (2009) · 0.96 · paraphrased: the licence does not allow quoting
Found this paper's double distribution parametrization of nucleon matrix elements of light-ray vector- and axial-vector operators most convenient at intermediate stages of the calculation.

Kinematic power corrections to DVCS to twist-six accuracy (2025) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

182 arXiv papers cite this in 266 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 2 papers

We discuss two types of functions parametrizing such matrix elements: double distributions F(x,y;t) and nonforward distribution functions \cal F_\zeta (X;t) and also their relation to usual parton densities f(x).

Deeply Virtual Compton Scattering: Facing Nonforward Distributions · 0.85
I describe two types of nonperturbative functions parametrizing such matrix elements: double distributions and skewed parton distributions.

Generalized Parton Distributions · 0.81

Papers citing it · 85 sentences without a confident label

Builds on it 2 papers

In Sec. [ref] we extend the relation between FFs and GPDs [ref] found in [ref] to compute the gluon distribution functions.

Gluon matter distribution in the proton and pion from extended holographic light-front QCD · Introduction intro · 0.96
The scale dependence of nucleon-to-meson TDAs is governed by the evolution equations which turn to be an extension of the evolution equations for baryon DAs [ref] and GPDs [ref] .

Transition distribution amplitudes and hard exclusive reactions with baryon number transfer · QCD evolution equations for TDAs · 0.87

Uses its method or tool 20 papers

In this work we consider a procedure similar to the one discussed in [ref] for the soft-wall case, i.e. we perform a matching of the nucleon electromagnetic form factors considering two main ideas: we use sum rules, derived in QCD [ref] , which contain the GPDs for the valence quarks, and we consider specific integral representations obtained in the AdS/QCD hard-wall model [ref] .

Generalized parton distributions in an AdS/QCD hard-wall model · Introduction · 0.80
In this paper, using reparametrization procedure described in [ref] , we establish the link between the dual parametrization of GPDs and the famous Radyushkin double distribution Ansatz [ref] employed in numerous phenomenological applications.

Dual parametrization of GPDs versus the double distribution Ansatz · Introduction · 0.88
In Fig. [ref], we present the same helicity-dependent proton GPDs as in Fig. [ref], but evolved to a scale $Q^2 = 4$GeV$^2$ using leading-order kernels [ref] .

Quark spin and orbital angular momentum from proton GPDs · Helicity-dependent proton GPDs · 1.00

Cites it as background 138 papers

Both processes rely on the collinear factorization (CF) scheme [ref] .

Backward-angle Exclusive pi0 Production above the Resonance Region · GPDs and Skewed Parton Distributions (SPDs) · 0.84
The following statements hold for both for the skewness parameter $ζ$ of Radyushkin [ref] and for the parameter $ξ$ of Ji [ref] , given their relation $ξ= ζ/ (2 - ζ)$.

Extracting Mellin moments of double parton distributions from lattice data · Dependence on the skewness zeta · 0.85
However, because there is a transfer of four-momentum between initial and final-state hadrons, $p$ and $p'$, the description of this process in the framework of collinear factorization [ref] does not involve usual PDFs, but the so-called generalized parton distributions (GPDs) [ref] .

Exclusive meets inclusive at small Bjorken-$x_B$: how to relate exclusive measurements to PDFs based on evolution equations · Introduction · 0.80

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.

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