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One-loop matching for leading-twist generalised transverse-momentum-dependent distributions

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Pith's one-line read All leading-twist GTMDs are matched onto GPDs at one loop, with T-even/T-odd mixing in the ERBL region.

desk verdict Complete one-loop GTMD-to-GPD matching with a new T-even/T-odd mixing claim; tables look right, but the regulator-independence of the imaginary part needs an explicit statement. read the letter →

arxiv 2502.07576 v2 pith:SCZV7X27 submitted 2025-02-11 hep-ph

Valerio Bertone , Miguel G. Echevarria , Óscar del Rio , Simone Rodini This is my paper · ORCID
classification hep-ph
keywords generalisedtransverse-momentum-dependentdistributionspartonone-loopmatchingoperatorproductexpansionT-oddeffectsERBLregionQCDfactorizationprotontomography
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

Generalised transverse-momentum-dependent distributions (GTMDs) are the most complete quark/gluon structure functions of the proton, depending on longitudinal momentum fraction, skewness, transverse momentum, and momentum transfer. This paper supplies the missing one-loop, $O(\alpha_s)$, matching coefficients that express every leading-twist GTMD of the proton in terms of generalised parton distributions (GPDs) at small transverse separation, the analogue for GTMDs of the standard TMD-to-PDF matching used in data analyses. It establishes that polarisations mix: an unpolarised quark GTMD receives contributions from transversely polarised gluon GPDs, and other channels mix similarly. It also finds that in the ERBL region ($\xi > x$) the gluon-induced matching functions and the evolution kernel pick up imaginary parts proportional to the Wilson-line direction, so time-reversal even (T-even) and time-reversal odd (T-odd) GTMD components rotate into each other. A sympathetic reader cares because this makes a complete, perturbatively controlled reconstruction of GTMDs feasible for the first time, which is needed before GTMDs can be extracted from or compared with data on exclusive processes.

What carries the argument

The central object is the one-loop matching function $C^{Y/\Gamma}_{i/j}$ of Eq. (14), which factorises the small-$|b|$ GTMD correlator as a convolution of a universal coefficient with a GPD correlator. The argument is carried by the newly computed residual functions $r^{Y/\Gamma,[1]}_{i/j}(y,\kappa)$ in Table 1, together with the decomposition of Eq. (21) that isolates the principal-value singularity at $y = 1/\kappa$ and the imaginary piece $-is\pi/\kappa \, \Sigma \, \delta(y - 1/\kappa)$ coming from the staple Wilson-line direction. The evolution side is carried by the Sudakov factor $R_i$ of Eq. (63), whose complex phase $\exp(i\phi)$ with $\phi = (\pi/2)\,\theta(\kappa-1)\,K_i(b,\mu)$ supplies the rotation that mixes the T-even and T-odd parts of a GTMD.

What would settle it

Recompute the one-loop GTMD matching functions with a different rapidity regulator (for instance off-the-light-cone Wilson lines) and check that the imaginary term $-is\pi/\kappa \, \Sigma \, \delta(y - 1/\kappa)$ in Eq. (21), and hence the T-even/T-odd rotation in Eq. (71), is reproduced with exactly the same phase; if the phase changes or disappears with the regulator, the claimed mixing is an artifact. Experimentally, a measurement of exclusive or double-Drell-Yan cross sections in ERBL kinematics that shows no dependence on the sign of the Wilson-line direction (e.g. no $\sin(2\theta_{k\Delta})$ modulation in the $S_{1,1b}$-type distribution) would violate the predicted pattern.

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Extended reading notes

Core claim

The paper computes the complete set of one-loop matching functions $C^{Y/\Gamma,[1]}_{i/j}$ that connect $b$-space GTMD correlators to collinear GPD correlators for all leading-twist quark and gluon polarisations. The new results are the residual functions $R^{Y/\Gamma,[1]}_{i/j}$ listed in Table 1 (and their singlet/non-singlet combinations in Tables 2-5), which supplement the known logarithmic and splitting-function terms in Eq. (14). Two structural discoveries follow. First, GTMD and GPD polarisations mix under matching: for example, unpolarised quark GTMDs receive contributions from transversely/linearly polarised gluon GPDs. Second, for gluon GPD channels in the ERBL region ($\kappa > 1$), the matching functions develop an imaginary part proportional to the Wilson-line direction $s$, of the form $-is\pi/\kappa \, \Sigma^{Y/\Gamma,+,[1]}_{i/j} \, \delta(y - 1/\kappa)$ in the singlet convolution; combined with an analogous imaginary phase in the Sudakov evolution of Eq. (63), this produces a rotation matrix in Eq. (71) that mixes T-even and T-odd components of each GTMD. Numerically the T-odd component is small but non-negligible in the ERBL region, and evolution, not matching, is the dominant source.

Load-bearing premise

The whole construction rests on a single set of universal one-loop coefficients — extracted from simple quark and gluon states — controlling how every polarisation of a proton's GTMD is built from GPDs at small transverse separation.

Editorial extensions

If this is right

  • With these coefficients, all leading-twist proton GTMDs can be reconstructed from GPDs at small $b$ (large $k_T$) at one-loop accuracy, in the same way TMDs are matched onto PDFs in global fits.
  • Polarisation mixing means any extraction of one GTMD flavour must account for GPDs of different polarisations; e.g. unpolarised quark GTMDs receive contributions through the U/T channel from linearly polarised gluon GPDs.
  • T-even and T-odd components of a GTMD mix both under $O(\alpha_s)$ matching onto gluon GPDs and under DGLAP-type evolution in the ERBL region, so T-odd contributions cannot be treated as purely non-perturbative there.
  • The matching functions are real in the quark-GPD channels and acquire an imaginary part only through gluon GPDs in the ERBL region; the imaginary part scales with the sign $s$ of the Wilson line, making the mixing process-dependent.
  • The kinematic point $x = \xi$ is out of reach of the transverse-momentum factorisation used here: the $\log|1-\kappa^2|$ divergence at $x = \xi$ is argued to persist to all orders because one parton carries zero longitudinal momentum, so double-Drell-Yan observables must avoid this region.

Reading between the lines

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

  • If the matching is process-universal as claimed, existing high-precision TMD fits (which already constrain the $b$-space non-perturbative input) could be adapted to seed GTMD models for the DGLAP region, though the ERBL region also needs the new imaginary matching terms; this is my inference, not a proposal in the paper.
  • The complex phase in the Sudakov factor implies that GTMDs reconstructed in $b$-space are genuinely complex-valued even when the input GPDs are real; an observable built from ERBL-kinematics GTMDs in forward or diffractive processes should therefore show a T-odd modulation with a characteristic $\log(\mu/\mu_b)$ growth, which would be a testable quantitative prediction of this one-loop picture.
  • A natural extension the paper does not attempt is to compute the two-loop matching functions to check whether the imaginary part and the T-even/T-odd rotation persist beyond $O(\alpha_s)$; the paper's claim that one-loop coefficients are the only relevant ones for now is a scope judgement, not a proof of convergence.
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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

0 major / 6 minor

Summary. The paper computes the complete set of one-loop matching coefficients that connect all leading-twist generalised transverse-momentum-dependent distributions (GTMDs) of the proton to generalised parton distributions (GPDs). The coefficients are extracted from partonic bilocal correlators with staple-like Wilson lines, using the δ-regulator for rapidity divergences and a b-space OPE at small transverse separation. The paper presents the residual functions in Tab. 1, the singlet/non-singlet combinations in Tabs. 2-5, and the forward limit in Tab. 6. It further derives imaginary parts in the matching functions and in the evolution kernel, leading to a rotation between T-even and T-odd GTMDs in the ERBL region, and illustrates the effect with numerical results based on the GK model and the PARTONS/APFEL++/NangaParbat codes.

Significance. If the results are correct, this is the first complete one-loop matching kernel set for all leading-twist GTMDs, and the paper fills a genuine gap between TMD and GPD phenomenology. The central novelty is the derivation of an imaginary part in the matching functions and in the evolution kernel that induces perturbative mixing of T-even and T-odd GTMDs in the ERBL region; this is a concrete, falsifiable prediction for future exclusive and double-Drell-Yan phenomenology. The paper makes good use of internal consistency checks: the forward limit reproduces known TMD matching results (Tab. 6), the sign of the Wilson-line-dependent imaginary part is carefully tracked in App. A, and the numerical implementation is based on publicly available codes, which is a substantial reproducibility strength. There is no circularity in the derivation: the coefficients are computed from partonic matrix elements rather than fitted to data.

minor comments (6)
  1. [Sec. 3, Eq. (25)] In the expression for R^{U/U,-,[1]}_2(y,κ), the factor 'CF' appears twice, reading '2y(1−κ)CF'; this is presumably a typo and the intended expression is CF·2y(1−κ)/(1−κ²y²). Please correct it.
  2. [App. A.2, Eqs. (87)-(94)] The imaginary part of the residual function is derived with the δ-regulator and light-cone gauge, and the text explicitly demonstrates regulator independence only for the UV-renormalisation phase in A.1, not for the residual function. In my reading the concern is not fatal, because the sign of the iδ term is fixed by the Wilson-line boundary condition in Eq. (87), so any regulator preserving the staple direction gives the same δ-function coefficient; nevertheless, adding a sentence in A.2 stating this explicitly would preempt an otherwise natural objection.
  3. [App. A.1, Eq. (77)] The identity in Eq. (77) relies on a specific branch of the logarithm; with the principal branch it is correct, but the branch convention should be stated explicitly for completeness.
  4. [Sec. 3, Tab. 1] The derivation of the entries in Tab. 1 is delegated to Refs. [30,56,57]. Since this table is the central result, including one explicit worked example for a new entry in an appendix or an ancillary file would improve the verifiability of the calculation.
  5. [Sec. 6 and Fig. 3] The statement that evolution 'proves' to be the dominant source of T-odd effects is stronger than what a single model-dependent numerical study can establish; I suggest softening this to 'indicates' or 'demonstrates in this implementation'.
  6. [Sec. 7] The claim that the reconstructed GTMDs 'should be regarded as realistic' is somewhat overstrong given the ad hoc scaling factor c=10^{-1} for linearly polarised gluon GPDs and the absence of direct data validation; please qualify this statement.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: matching coefficients are obtained from explicit partonic matrix-element computations; the self-citations present (Refs [30,56,57,28]) are minor and corroborated by external benchmarks.

full rationale

The matching functions are genuine computations, not re-labelled inputs. The one-loop coefficients in Eq. (14) and the residual functions in Tab. 1 are obtained by computing partonic matrix elements of the GTMD correlators in Eqs. (3)-(4) with staple-like Wilson lines under the delta-regulator; App. A.2 shows the explicit derivation of the imaginary part in Eq. (21) through the light-cone-gauge boundary condition and the standard identity in Eq. (94). Nothing is fitted: the R^{Y/Gamma,[1]} functions are not tuned to any GTMD data, and the ERBL-region T-even/T-odd mixing in Eqs. (70)-(71) is a consequence of the computed coefficients (Tab. 5, Eq. (67)), not an assumption. The forward limit is checked against published TMD matching (Tab. 6 vs. Refs [16,58]), providing an external anchor. Self-citations are present but not load-bearing: Ref. [30] (same first author) is the predecessor that supplied only C^{U/U}; the GPD splitting functions in Eq. (14) cite Refs [52-56], where Refs [52-55] are external and App. A.3 re-derives the reality of the splitting functions; the regulator-independence remark in App. A.1 cites Ref [28] (co-author Rodini) only as corroboration for a phase already obtained from external Ref [68] plus the algebraic identity Eq. (77). The one genuine weakness - App. A.2 does not demonstrate that the residual-function imaginary part is independent of the delta-regulator - is a physics fragility or correctness risk (the coefficient of delta(y-1/kappa) might differ under another rapidity regulator), not a circular reduction of the target claim to its inputs, and per the review rules it does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The one-loop matching coefficients themselves are parameter-free perturbative results. The numerical demonstration introduces one ad hoc scaling factor (c=10^-1) for linearly polarised gluon GPDs, which does not enter the central claim. The main external inputs are the standard small-b OPE/factorization, the universality of partonic matching functions, the δ-regulator for rapidity divergences, and the proton GTMD parametrisation basis of Ref. [33].

free parameters (1)
  • c (linearly-polarised gluon GPD scaling factor) = 10^-1
    In Sec. 6, eH_T^g = c eH^g, E_T^g = c E^g, eE_T^g = c eE^g with c arbitrarily chosen as 10^-1 because no reliable models exist. This affects numerical illustrations, not the matching coefficients.
assumptions (4)
  • domain assumption OPE factorization of GTMD correlators onto GPD correlators at small |b| (Eq. (11))
    The central framework assumes the b-space GTMD correlators can be expanded in light-like GPD correlators with universal matching coefficients; inherited from TMD factorization [10,11,20] and the double Drell-Yan factorization proof [50].
  • domain assumption Universality of matching coefficients extracted from partonic matrix elements
    The paper computes the matching functions using on-shell parton states with staple-like Wilson lines and assumes the result is target- and process-independent (Sec. 3).
  • standard math δ-regulator for rapidity divergences and its analytic continuation (App. A)
    The imaginary parts in Eq. (14) follow from the δ-regulator and the identity 1/(1-κy+iδ) = PV - iπδ(...); the paper argues (App. A.1) that off-light-cone regulators give the same result, so this is a standard scheme assumption.
  • domain assumption Completeness of the proton GTMD parametrisation in Eqs. (27)-(30) (modified from Ref. [33])
    The matching onto individual GTMDs assumes this basis covers all leading-twist twist-two structures for a proton; any missing structure would alter the matching relations in Sec. 4.

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Cite this review

Pith. "Pith review of One-loop matching for leading-twist generalised transverse-momentum-dependent distributions." pith.science (2026). https://pith.science/paper/SCZV7X27

@misc{pith2026250207576,
  author       = {Pith},
  title        = {Pith review of: One-loop matching for leading-twist generalised transverse-momentum-dependent distributions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SCZV7X27}},
  note         = {Machine review of arXiv:2502.07576}
}
read the original abstract

We present the one-loop matching coefficients necessary to match all of the leading-twist generalised transverse-momentum-dependent distributions (GTMDs) onto generalised parton distributions (GPDs). Matching functions are extracted by computing the first radiative corrections to partonic bilocal correlators with staple-like Wilson lines, as appropriate for high-energy collisions. These correlators are characterised by a transverse displacement and skewed kinematics of external states. Using the proton helicity basis, they are parametrised in terms of GTMDs, which are subsequently related to leading-twist GPDs. Our results provide new insights into the complex dynamics of GTMDs generated by radiative corrections. In particular, we show that time-reversal even and odd contributions to GTMDs in the so-called ERBL region mix both under matching and evolution. Finally, we present a selection of numerical results and comment on the quantitative behaviour of GTMDs.

Figures

Figures reproduced from arXiv: 2502.07576 by the authors.

Figure 1
Figure 1. The GTMDs Sˆ0;+;u 1,1a (left) and Sˆ0;+;u 1,1b (right) plotted as functions of |kT | at x = 0.15 and µ = √ ζ = 10 GeV. Two different values of ξ are considered: ξ = 10−3 < x (red curves) which probes the DGLAP region, and ξ = 0.3 > x (blue curves) which instead probes the ERBL region. T-even and T-odd components are shown separately as solid and dashed curves, respectively. The plot for Sˆ0;+;u 1,1b is magnified by … view at source ↗
Figure 2
Figure 2. The T-even (solid curves) and T-odd (dashed curves) components to the GTMD [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Ratio between the T-odd and T-even contributions to [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

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Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages · cited by 4 Pith papers

  1. [30]

    Matching generalised transverse-momentum-dependent distributions onto gen- eralised parton distributions at one loop,

    V. Bertone, “Matching generalised transverse-momentum-dependent distributions onto gen- eralised parton distributions at one loop,” Eur. Phys. J. C , vol. 82, no. 10, p. 941, 2022

    work page 2022

  2. [1]

    Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production,

    A. Bacchetta, F. Delcarro, C. Pisano, M. Radici, and A. Signori, “Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production,” JHEP, vol. 06, p. 081, 2017. [Erratum: JHEP 06, 051 (2019)]

    work page 2019

  3. [2]

    Analysis of vector boson production within TMD factor- ization,

    I. Scimemi and A. Vladimirov, “Analysis of vector boson production within TMD factor- ization,” Eur. Phys. J. C , vol. 78, no. 2, p. 89, 2018

    work page 2018

  4. [3]

    Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum,

    I. Scimemi and A. Vladimirov, “Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum,” JHEP, vol. 06, p. 137, 2020

    work page 2020

  5. [4]

    Unpolarized transverse momentum distributions from a global fit of Drell-Yan and semi-inclusive deep-inelastic scattering data,

    A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Piacenza, M. Radici, and A. Signori, “Unpolarized transverse momentum distributions from a global fit of Drell-Yan and semi-inclusive deep-inelastic scattering data,” JHEP, vol. 10, p. 127, 2022

    work page 2022

  6. [5]

    Origin of single transverse-spin asymmetries in high-energy colli- sions,

    J. Cammarota, L. Gamberg, Z.-B. Kang, J. A. Miller, D. Pitonyak, A. Prokudin, T. C. Rogers, and N. Sato, “Origin of single transverse-spin asymmetries in high-energy colli- sions,” Phys. Rev. D , vol. 102, no. 5, p. 054002, 2020

    work page 2020

  7. [6]

    Global analysis of the Sivers functions at NLO+NNLL in QCD,

    M. G. Echevarria, Z.-B. Kang, and J. Terry, “Global analysis of the Sivers functions at NLO+NNLL in QCD,” JHEP, vol. 01, p. 126, 2021

    work page 2021

  8. [7]

    PDF bias and flavor dependence in TMD distributions,

    M. Bury, F. Hautmann, S. Leal-Gomez, I. Scimemi, A. Vladimirov, and P. Zurita, “PDF bias and flavor dependence in TMD distributions,” JHEP, vol. 10, p. 118, 2022. 23

    work page 2022

Show all 70 references
  1. [8]

    Extraction of pion transverse momentum distributions from Drell-Yan data,

    M. Cerutti, L. Rossi, S. Venturini, A. Bacchetta, V. Bertone, C. Bissolotti, and M. Radici, “Extraction of pion transverse momentum distributions from Drell-Yan data,” Phys. Rev. D, vol. 107, no. 1, p. 014014, 2023

    2023

  2. [9]

    Extraction of unpolarized transverse momentum distributions from fit of Drell-Yan data at N 4LL

    V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, “Extraction of unpolarized transverse momentum distributions from fit of Drell-Yan data at N 4LL.” 5 preprint

  3. [10]

    Collins, Foundations of Perturbative QCD, vol

    J. Collins, Foundations of Perturbative QCD, vol. 32 of Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology . Cambridge University Press, 7 2023

    2023

  4. [11]

    Calculation of transverse momentum dependent distributions beyond the leading power,

    V. Moos and A. Vladimirov, “Calculation of transverse momentum dependent distributions beyond the leading power,” JHEP, vol. 12, p. 145, 2020

    2020

  5. [12]

    TMD Fragmentation Functions at N 3LO,

    M. A. Ebert, B. Mistlberger, and G. Vita, “TMD Fragmentation Functions at N 3LO,” JHEP, vol. 07, p. 121, 2021

    2021

  6. [13]

    Unpolarized quark and gluon TMD PDFs and FFs at N 3LO,

    M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, “Unpolarized quark and gluon TMD PDFs and FFs at N 3LO,” JHEP, vol. 06, p. 115, 2021

    2021

  7. [14]

    Transverse momentum dependent transversely polarized distributions at next-to-next-to-leading-order,

    D. Gutierrez-Reyes, I. Scimemi, and A. Vladimirov, “Transverse momentum dependent transversely polarized distributions at next-to-next-to-leading-order,” JHEP, vol. 07, p. 172, 2018

    2018

  8. [15]

    Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution,

    D. Gutierrez-Reyes, S. Leal-Gomez, I. Scimemi, and A. Vladimirov, “Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution,” JHEP, vol. 11, p. 121, 2019

    2019

  9. [16]

    Twist-2 matching of transverse momentum dependent distributions,

    D. Guti´ errez-Reyes, I. Scimemi, and A. A. Vladimirov, “Twist-2 matching of transverse momentum dependent distributions,” Phys. Lett. B , vol. 769, pp. 84–89, 2017

    2017

  10. [17]

    Transverse momentum in double parton scattering: factorisation, evolution and matching,

    M. G. A. Buffing, M. Diehl, and T. Kasemets, “Transverse momentum in double parton scattering: factorisation, evolution and matching,” JHEP, vol. 01, p. 044, 2018

    2018

  11. [18]

    Evolution of the helicity and transversity Transverse- Momentum-Dependent parton distributions,

    A. Bacchetta and A. Prokudin, “Evolution of the helicity and transversity Transverse- Momentum-Dependent parton distributions,” Nucl. Phys. B , vol. 875, pp. 536–551, 2013

    2013

  12. [19]

    Operator Constraints for Twist-3 Functions and Lorentz Invariance Properties of Twist-3 Observables,

    K. Kanazawa, Y. Koike, A. Metz, D. Pitonyak, and M. Schlegel, “Operator Constraints for Twist-3 Functions and Lorentz Invariance Properties of Twist-3 Observables,” Phys. Rev. D, vol. 93, no. 5, p. 054024, 2016

    2016

  13. [20]

    Matching of transverse momentum dependent distributions at twist-3,

    I. Scimemi and A. Vladimirov, “Matching of transverse momentum dependent distributions at twist-3,” Eur. Phys. J. C , vol. 78, no. 10, p. 802, 2018

    2018

  14. [21]

    Relations between generalized and transverse mo- mentum dependent parton distributions,

    S. Meissner, A. Metz, and K. Goeke, “Relations between generalized and transverse mo- mentum dependent parton distributions,” Phys. Rev. D , vol. 76, p. 034002, 2007

    2007

  15. [22]

    Universality of T odd effects in single spin and azimuthal asymmetries,

    D. Boer, P. J. Mulders, and F. Pijlman, “Universality of T odd effects in single spin and azimuthal asymmetries,” Nucl. Phys. B , vol. 667, pp. 201–241, 2003

    2003

  16. [23]

    A Unified picture for single transverse-spin asymmetries in hard processes,

    X. Ji, J.-W. Qiu, W. Vogelsang, and F. Yuan, “A Unified picture for single transverse-spin asymmetries in hard processes,” Phys. Rev. Lett. , vol. 97, p. 082002, 2006

    2006

  17. [24]

    Transverse momentum dependent quark distributions and polarized Drell-Yan processes,

    J. Zhou, F. Yuan, and Z.-T. Liang, “Transverse momentum dependent quark distributions and polarized Drell-Yan processes,” Phys. Rev. D , vol. 81, p. 054008, 2010

    2010

  18. [25]

    Next-to-leading order transverse momentum-weighted Sivers asymmetry in semi-inclusive deep inelastic scattering: the role of the three-gluon correlator,

    L.-Y. Dai, Z.-B. Kang, A. Prokudin, and I. Vitev, “Next-to-leading order transverse momentum-weighted Sivers asymmetry in semi-inclusive deep inelastic scattering: the role of the three-gluon correlator,” Phys. Rev. D , vol. 92, no. 11, p. 114024, 2015

    2015

  19. [26]

    Collinear matching for Sivers function at next-to-leading order,

    I. Scimemi, A. Tarasov, and A. Vladimirov, “Collinear matching for Sivers function at next-to-leading order,” JHEP, vol. 05, p. 125, 2019. 24

    2019

  20. [27]

    Sivers, Boer-Mulders and worm-gear distributions at next-to-leading order,

    F. Rein, S. Rodini, A. Sch¨ afer, and A. Vladimirov, “Sivers, Boer-Mulders and worm-gear distributions at next-to-leading order,” JHEP, vol. 01, p. 116, 2023

    2023

  21. [28]

    Definition and evolution of transverse momentum dependent distribution of twist-three,

    S. Rodini and A. Vladimirov, “Definition and evolution of transverse momentum dependent distribution of twist-three,” JHEP, vol. 08, p. 031, 2022. [Erratum: JHEP 12, 048 (2022)]

    2022

  22. [29]

    Collinear matching for next-to-leading power transverse-momentum distributions,

    S. Rodini, A. C. Alvaro, and B. Pasquini, “Collinear matching for next-to-leading power transverse-momentum distributions,” Phys. Lett. B , vol. 845, p. 138163, 2023

    2023

  23. [31]

    Generalized parton correlation functions for a spin-0 hadron,

    S. Meissner, A. Metz, M. Schlegel, and K. Goeke, “Generalized parton correlation functions for a spin-0 hadron,” JHEP, vol. 08, p. 038, 2008

    2008

  24. [32]

    Generalized parton correlation functions for a spin-1/2 hadron,

    S. Meissner, A. Metz, and M. Schlegel, “Generalized parton correlation functions for a spin-1/2 hadron,” JHEP, vol. 08, p. 056, 2009

    2009

  25. [33]

    Structure analysis of the generalized correlator of quark and gluon for a spin-1/2 target,

    C. Lorc´ e and B. Pasquini, “Structure analysis of the generalized correlator of quark and gluon for a spin-1/2 target,” JHEP, vol. 09, p. 138, 2013

    2013

  26. [34]

    Twist-2 generalized transverse-momentum dependent parton distributions and the spin/orbital structure of the nucleon,

    K. Kanazawa, C. Lorc´ e, A. Metz, B. Pasquini, and M. Schlegel, “Twist-2 generalized transverse-momentum dependent parton distributions and the spin/orbital structure of the nucleon,” Phys. Rev. D , vol. 90, no. 1, p. 014028, 2014

    2014

  27. [35]

    Proper definition and evolution of generalized transverse momentum dependent distribu- tions,

    M. G. Echevarria, A. Idilbi, K. Kanazawa, C. Lorc´ e, A. Metz, B. Pasquini, and M. Schlegel, “Proper definition and evolution of generalized transverse momentum dependent distribu- tions,” Phys. Lett. B , vol. 759, pp. 336–341, 2016

    2016

  28. [36]

    Quark Wigner Distributions and Orbital Angular Momentum,

    C. Lorce and B. Pasquini, “Quark Wigner Distributions and Orbital Angular Momentum,” Phys. Rev. D , vol. 84, p. 014015, 2011

    2011

  29. [37]

    Notes on the orbital angular momentum of quarks in the nucleon,

    Y. Hatta, “Notes on the orbital angular momentum of quarks in the nucleon,” Phys. Lett. B, vol. 708, pp. 186–190, 2012

    2012

  30. [38]

    Wigner, Husimi, and generalized transverse momen- tum dependent distributions in the color glass condensate,

    Y. Hagiwara, Y. Hatta, and T. Ueda, “Wigner, Husimi, and generalized transverse momen- tum dependent distributions in the color glass condensate,” Phys. Rev. D , vol. 94, no. 9, p. 094036, 2016

    2016

  31. [39]

    Accessing the gluon Wigner distribution in ultraperipheral pA collisions,

    Y. Hagiwara, Y. Hatta, R. Pasechnik, M. Tasevsky, and O. Teryaev, “Accessing the gluon Wigner distribution in ultraperipheral pA collisions,” Phys. Rev. D, vol. 96, no. 3, p. 034009, 2017

    2017

  32. [40]

    Probing the Small- x Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering,

    Y. Hatta, B.-W. Xiao, and F. Yuan, “Probing the Small- x Gluon Tomography in Correlated Hard Diffractive Dijet Production in Deep Inelastic Scattering,” Phys. Rev. Lett., vol. 116, no. 20, p. 202301, 2016

    2016

  33. [41]

    Gluon orbital angular momentum at small-x,

    Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao, and B. Xiao, “Gluon orbital angular momentum at small-x,” Phys. Rev. D , vol. 95, no. 11, p. 114032, 2017

    2017

  34. [42]

    Elliptic Flow in Small Systems due to Elliptic Gluon Distributions?,

    Y. Hagiwara, Y. Hatta, B.-W. Xiao, and F. Yuan, “Elliptic Flow in Small Systems due to Elliptic Gluon Distributions?,” Phys. Lett. B , vol. 771, pp. 374–378, 2017

    2017

  35. [43]

    Hunting the Gluon Orbital Angular Momentum at the Electron-Ion Collider,

    X. Ji, F. Yuan, and Y. Zhao, “Hunting the Gluon Orbital Angular Momentum at the Electron-Ion Collider,” Phys. Rev. Lett. , vol. 118, no. 19, p. 192004, 2017

    2017

  36. [44]

    Generalized TMDs and the exclusive double Drell–Yan process,

    S. Bhattacharya, A. Metz, and J. Zhou, “Generalized TMDs and the exclusive double Drell–Yan process,” Phys. Lett. B, vol. 771, pp. 396–400, 2017. [Erratum: Phys.Lett.B 810, 135866 (2020)]. 25

    2020

  37. [45]

    Exclusive double quarko- nium production and generalized TMDs of gluons,

    S. Bhattacharya, A. Metz, V. K. Ojha, J.-Y. Tsai, and J. Zhou, “Exclusive double quarko- nium production and generalized TMDs of gluons,” Phys. Lett. B , vol. 833, p. 137383, 2022

    2022

  38. [46]

    GTMD model predictions for diffractive dijet production at EIC,

    D. Boer and C. Setyadi, “GTMD model predictions for diffractive dijet production at EIC,” Phys. Rev. D , vol. 104, no. 7, p. 074006, 2021

    2021

  39. [47]

    Probing gluon GTMDs through exclusive coherent diffractive processes,

    D. Boer and C. Setyadi, “Probing gluon GTMDs through exclusive coherent diffractive processes,” Eur. Phys. J. C , vol. 83, no. 10, p. 890, 2023

    2023

  40. [48]

    Accessing the gluon GTMD F1,4 in exclusive π0 production in ep collisions,

    S. Bhattacharya, D. Zheng, and J. Zhou, “Accessing the gluon GTMD F1,4 in exclusive π0 production in ep collisions,” Phys. Rev. D , vol. 109, no. 9, p. 096029, 2024

    2024

  41. [49]

    Probing the Quark Orbital Angular Momentum at Electron-Ion Colliders Using Exclusive π0 Production,

    S. Bhattacharya, D. Zheng, and J. Zhou, “Probing the Quark Orbital Angular Momentum at Electron-Ion Colliders Using Exclusive π0 Production,” Phys. Rev. Lett., vol. 133, no. 5, p. 051901, 2024

    2024

  42. [50]

    GTMDs and the factorization of exclusive double Drell-Yan,

    M. G. Echevarria, P. A. Gutierrez Garcia, and I. Scimemi, “GTMDs and the factorization of exclusive double Drell-Yan,” Phys. Lett. B , vol. 840, p. 137881, 2023

    2023

  43. [51]

    Bootstrapping Rapidity Anomalous Dimensions for Transverse- Momentum Resummation,

    Y. Li and H. X. Zhu, “Bootstrapping Rapidity Anomalous Dimensions for Transverse- Momentum Resummation,” Phys. Rev. Lett. , vol. 118, no. 2, p. 022004, 2017

    2017

  44. [52]

    Off forward parton distributions,

    X.-D. Ji, “Off forward parton distributions,” J. Phys. G , vol. 24, pp. 1181–1205, 1998

    1998

  45. [53]

    Generalized parton distributions,

    M. Diehl, “Generalized parton distributions,” Phys. Rept., vol. 388, pp. 41–277, 2003

    2003

  46. [54]

    Evolution equation for generalized parton distri- butions,

    M. Kirch, A. Manashov, and A. Schafer, “Evolution equation for generalized parton distri- butions,” Phys. Rev. D , vol. 72, p. 114006, 2005

    2005

  47. [55]

    Some numerical studies of the evolution of generalized parton distributions,

    M. Diehl and W. Kugler, “Some numerical studies of the evolution of generalized parton distributions,” Phys. Lett. B , vol. 660, pp. 202–211, 2008

    2008

  48. [56]

    One-loop evo- lution of twist-2 generalized parton distributions,

    V. Bertone, R. F. del Castillo, M. G. Echevarria, O. del R ´ ıo, and S. Rodini, “One-loop evo- lution of twist-2 generalized parton distributions,” Phys. Rev. D, vol. 109, no. 3, p. 034023, 2024

    2024

  49. [57]

    Revisiting evolu- tion equations for generalised parton distributions,

    V. Bertone, H. Dutrieux, C. Mezrag, J. M. Morgado, and H. Moutarde, “Revisiting evolu- tion equations for generalised parton distributions,” Eur. Phys. J. C , vol. 82, no. 10, p. 888, 2022

    2022

  50. [58]

    Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order,

    M. G. Echevarria, I. Scimemi, and A. Vladimirov, “Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order,” JHEP, vol. 09, p. 004, 2016

    2016

  51. [59]

    NUCLEAR GLUONOMETRY,

    R. L. Jaffe and A. Manohar, “NUCLEAR GLUONOMETRY,” Phys. Lett. B , vol. 223, pp. 218–224, 1989

    1989

  52. [60]

    Q**2 evolution of spin dependent parton densities,

    W. Vogelsang, “Q**2 evolution of spin dependent parton densities,” Acta Phys. Polon. B , vol. 29, pp. 1189–1200, 1998

    1998

  53. [61]

    Transverse-momentum-dependent parton distributions up to N3LL from Drell-Yan data,

    A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, F. Delcarro, F. Piacenza, and M. Radici, “Transverse-momentum-dependent parton distributions up to N3LL from Drell-Yan data,” JHEP, vol. 07, p. 117, 2020

    2020

  54. [62]

    Vector meson electroproduction at small Bjorken-x and generalized parton distributions,

    S. V. Goloskokov and P. Kroll, “Vector meson electroproduction at small Bjorken-x and generalized parton distributions,” Eur. Phys. J. C , vol. 42, pp. 281–301, 2005

    2005

  55. [63]

    The Role of the quark and gluon GPDs in hard vector- meson electroproduction,

    S. V. Goloskokov and P. Kroll, “The Role of the quark and gluon GPDs in hard vector- meson electroproduction,” Eur. Phys. J. C , vol. 53, pp. 367–384, 2008. 26

    2008

  56. [64]

    An Attempt to understand exclusive pi+ electroproduc- tion,

    S. V. Goloskokov and P. Kroll, “An Attempt to understand exclusive pi+ electroproduc- tion,” Eur. Phys. J. C , vol. 65, pp. 137–151, 2010

    2010

  57. [65]

    PARTONS: PARtonic Tomography Of Nucleon Software: A computing framework for the phenomenology of Generalized Parton Distributions,

    B. Berthou et al., “PARTONS: PARtonic Tomography Of Nucleon Software: A computing framework for the phenomenology of Generalized Parton Distributions,” Eur. Phys. J. C , vol. 78, no. 6, p. 478, 2018

    2018

  58. [66]

    APFEL: A PDF Evolution Library with QED corrections,

    V. Bertone, S. Carrazza, and J. Rojo, “APFEL: A PDF Evolution Library with QED corrections,” Comput. Phys. Commun. , vol. 185, pp. 1647–1668, 2014

    2014

  59. [67]

    APFEL++: A new PDF evolution library in C++,

    V. Bertone, “APFEL++: A new PDF evolution library in C++,” PoS, vol. DIS2017, p. 201, 2018

    2018

  60. [68]

    Transverse momentum dependent operator ex- pansion at next-to-leading power,

    A. Vladimirov, V. Moos, and I. Scimemi, “Transverse momentum dependent operator ex- pansion at next-to-leading power,” JHEP, vol. 01, p. 110, 2022

    2022

  61. [69]

    Factorization Theorem For Drell-Yan At Low qT And Transverse Momentum Distributions On-The-Light-Cone,

    M. G. Echevarria, A. Idilbi, and I. Scimemi, “Factorization Theorem For Drell-Yan At Low qT And Transverse Momentum Distributions On-The-Light-Cone,” JHEP, vol. 07, p. 002, 2012

    2012

  62. [70]

    Two-loop splitting in double parton distributions: the colour non-singlet case,

    M. Diehl, J. R. Gaunt, and P. Ploessl, “Two-loop splitting in double parton distributions: the colour non-singlet case,” JHEP, vol. 08, p. 040, 2021. 27

    2021

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