- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Probing gluon TMD models with Drell-Yan structure functions

Jan Ferdyan*

  • Institute of Theoretical Physics, Jagiellonian University, prof. Stanisława Łojasiewicza 11, 30-348 Kraków, Poland

  • *Contact author: jan.ferdyan@doctoral.uj.edu.pl
PDF

Phys. Rev. D 114, 054039 – Published 18 September, 2026

DOI: https://doi.org/10.1103/6syl-hp6j

Abstract

We compute structure functions for the Drell-Yan process in proton-proton collisions at the center-of-mass energy S=8  TeV, both parity conserving and parity breaking. For this calculation, we use the high-energy factorization formalism. The hard scattering matrix elements used in our derivation consist of two channels, qvalg*→qV* and g*g*→qq¯V*, both at the tree level. We consider four types of gluon transverse momentum distribution (TMD) models: Gaussian, Weizsäcker-Williams (WW), Kimber-Martin-Ryskin (KMR), and Jung-Hautmann. We also consider the models with phenomenological adjustments to improve the data description. We additionally compare the KMR model with its next-to-leading-order extension, called the Martin-Ryskin-Watt model, and the Kotikov-Lipatov result for the gluon TMD. We derive and compare the structure functions calculated for different gluon TMD models with the ATLAS 2016 data. Based on this comparison, we calculate χ2 per number of degrees of freedom for each of the predictions. This assessment shows clear differences between the predictions obtained with different TMD models, both in the description of the full dataset and in the case of individual structure functions. The best description of the structure functions’ data is obtained with one of the modified WW models. Our analysis can serve to identify the features of the TMD model that should be considered in future gluon TMD fits.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (90)

  1. S. D. Drell and Tung-Mow Yan, Massive lepton pair production in Hadron-Hadron collisions at high-energies, Phys. Rev. Lett. 25, 316 (1970); 25, 902(E) (1970).
  2. John C. Collins and Davison E. Soper, Angular distribution of dileptons in high-energy hadron collisions, Phys. Rev. D 16, 2219 (1977).
  3. K. Kajantie, J. Lindfors, and Risto Raitio, QCD angular correlations for Muon pair production, Phys. Lett. 74B, 384 (1978).
  4. J. Cleymans and M. Kuroda, Angular distribution of dileptons in hadronic collisions, Nucl. Phys. B155, 480 (1979); B160, 510(E) (1979).
  5. J. Cleymans and M. Kuroda, Angular distribution of lepton pairs and QCD, Phys. Lett. 80B, 385 (1979); 86B, 426(E) (1979).
  6. C. S. Lam and Wu-Ki Tung, A systematic approach to inclusive lepton pair production in hadronic collisions, Phys. Rev. D 18, 2447 (1978).
  7. C. S. Lam and Wu-Ki Tung, Structure function relations at large transverse momenta in lepton pair production processes, Phys. Lett. 80B, 228 (1979).
  8. C. S. Lam and Wu-Ki Tung, A parton model relation sans QCD modifications in lepton pair productions, Phys. Rev. D 21, 2712 (1980).
  9. Georges Aad et al., Measurement of the low-mass Drell-Yan differential cross section at s=7  TeV using the ATLAS detector, J. High Energy Phys. 06 (2014) 112.
  10. Georges Aad et al., Measurement of the angular coefficients in Z-boson events using electron and muon pairs from data taken at s=8  TeV with the ATLAS detector, J. High Energy Phys. 08 (2016) 159.
  11. Pietro Faccioli, Carlos Lourenco, Joao Seixas, and Hermine K. Wohri, Model-independent constraints on the shape parameters of dilepton angular distributions, Phys. Rev. D 83, 056008 (2011).
  12. Jen-Chieh Peng, Wen-Chen Chang, Randall Evan McClellan, and Oleg Teryaev, Interpretation of angular distributions of Z-boson production at colliders, Phys. Lett. B 758, 384 (2016).
  13. John C. Collins, Davison E. Soper, and George F. Sterman, Does the Drell-Yan cross-section factorize?, Phys. Lett. 109B, 388 (1982).
  14. John C. Collins, Davison E. Soper, and George F. Sterman, All order factorization for Drell-Yan cross-sections, Phys. Lett. 134B, 263 (1984).
  15. John C. Collins, Davison E. Soper, and George F. Sterman, Factorization for short distance Hadron—Hadron scattering, Nucl. Phys. B261, 104 (1985).
  16. John C. Collins, Davison E. Soper, and George F. Sterman, Soft gluons and factorization, Nucl. Phys. B308, 833 (1988).
  17. M. Chaichian, M. Hayashi, and K. Yamagishi, Angular distributions of high mass dileptons with finite transverse momentum in high-energy hadronic collisions, Phys. Rev. D 25, 130 (1982); 26, 2534(E) (1982).
  18. E. Mirkes, Angular decay distribution of leptons from W bosons at NLO in hadronic collisions, Nucl. Phys. B387, 3 (1992).
  19. E. Mirkes and J. Ohnemus, Angular distributions of Drell-Yan lepton pairs at the Tevatron: Order αs2 corrections and Monte Carlo studies, Phys. Rev. D 51, 4891 (1995).
  20. E. Mirkes and J. Ohnemus, W and Z polarization effects in hadronic collisions, Phys. Rev. D 50, 5692 (1994).
  21. R. Gauld, A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. Glover, and A. Huss, Precise predictions for the angular coefficients in Z-boson production at the LHC, J. High Energy Phys. 11 (2017) 003.
  22. Rikkert Frederix and Timea Vitos, Electroweak corrections to the angular coefficients in finite-pTZ-boson production and dilepton decay, Eur. Phys. J. C 80, 939 (2020).
  23. Mathieu Pellen, Rene Poncelet, Andrei Popescu, and Timea Vitos, Angular coefficients in W+j production at the LHC with high precision, Eur. Phys. J. C 82, 693 (2022).
  24. Simone Alioli, Paolo Nason, Carlo Oleari, and Emanuele Re, NLO vector-boson production matched with shower in POWHEG, J. High Energy Phys. 07 (2008) 060.
  25. Daniel Boer and Werner Vogelsang, Drell-Yan lepton angular distribution at small transverse momentum, Phys. Rev. D 74, 014004 (2006).
  26. S. Catani, M. Ciafaloni, and F. Hautmann, Gluon contributions to small x heavy flavour production, Phys. Lett. B 242, 97 (1990).
  27. S. Catani, M. Ciafaloni, and F. Hautmann, High-energy factorization and small x heavy flavor production, Nucl. Phys. B366, 135 (1991).
  28. S. J. Brodsky, Arthur Hebecker, and E. Quack, The Drell-Yan process and factorization in impact parameter space, Phys. Rev. D 55, 2584 (1997).
  29. B. Z. Kopeliovich, J. Raufeisen, and A. V. Tarasov, The color dipole picture of the Drell-Yan process, Phys. Lett. B 503, 91 (2001).
  30. Francois Gelis and Jamal Jalilian-Marian, Dilepton production from the color glass condensate, Phys. Rev. D 66, 094014 (2002).
  31. Francois Gelis and Jamal Jalilian-Marian, Drell-Yan production and Lam-Tung relation in the color glass condensate formalism, Phys. Rev. D 76, 074015 (2007).
  32. A. V. Lipatov, M. A. Malyshev, and N. P. Zotov, Drell-Yan lepton pair production at high energies in the kt-factorization approach, J. High Energy Phys. 12 (2011) 117.
  33. S. P. Baranov, A. V. Lipatov, and N. P. Zotov, Drell-Yan lepton pair production at the LHC and transverse momentum dependent quark densities of the proton, Phys. Rev. D 89, 094025 (2014).
  34. Wolfgang Schäfer and Antoni Szczurek, Low mass Drell-Yan production of lepton pairs at forward directions at the LHC: A hybrid approach, Phys. Rev. D 93, 074014 (2016).
  35. S. P. Baranov, H. Jung, A. V. Lipatov, and M. A. Malyshev, Associated production of Z bosons and b-jets at the LHC in the combined kT+ collinear QCD factorization approach, Eur. Phys. J. C 77, 772 (2017).
  36. Pieter Taels, Forward production of a Drell-Yan pair and a jet at small x at next-to-leading order, J. High Energy Phys. 01 (2024) 005.
  37. Leszek Motyka, Mariusz Sadzikowski, and Tomasz Stebel, Lam-Tung relation breaking in Z0 hadroproduction as a probe of parton transverse momentum, Phys. Rev. D 95, 114025 (2017).
  38. M. A. Nefedov, N. N. Nikolaev, and V. A. Saleev, Drell-Yan lepton pair production at high energies in the Parton Reggeization approach, Phys. Rev. D 87, 014022 (2013).
  39. Maxim A. Nefedov and Vladimir A. Saleev, High-energy factorization for Drell-Yan process in pp and pp¯ collisions with new Unintegrated PDFs, Phys. Rev. D 102, 114018 (2020).
  40. S. S. Omelyanchuk, V. A. Saleev, and A. V. Shipilova, Spectra of Z bosons and lepton angular coefficients in the leading approximation of the Parton Reggeization approach with allowance for the next-to-leading-order corrections, Phys. At. Nucl. 87, 237 (2024).
  41. V. A. Saleev and S. S. Omelyanchuk, Lepton angular coefficients in the Z-boson production in the high-energy factorization approach, Phys. Part. Nucl. 56, 423 (2025).
  42. Yan B. Bandeira, Victor P. Goncalves, and Wolfgang Schäfer, Dilepton angular distributions in the color-dipole S-matrix framework, J. High Energy Phys. 10 (2025) 010.
  43. Leszek Motyka, Mariusz Sadzikowski, and Tomasz Stebel, Twist expansion of Drell-Yan structure functions in color dipole approach, J. High Energy Phys. 05 (2015) 087.
  44. Dawid Brzeminski, Leszek Motyka, Mariusz Sadzikowski, and Tomasz Stebel, Twist decomposition of Drell-Yan structure functions: Phenomenological implications, J. High Energy Phys. 01 (2017) 005.
  45. John C. Collins, Davison E. Soper, and George F. Sterman, Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys. B250, 199 (1985).
  46. Sara Piloneta and Alexey Vladimirov, Angular distributions of Drell-Yan leptons in the TMD factorization approach, J. High Energy Phys. 12 (2024) 059.
  47. Alexey Vladimirov, Angular structure of Drell-Yan reaction in the TMD factorization approach, in Proceedings of the 59th Rencontres de Moriond: QCD and High Energy Interactions (ARISF, 2025).
  48. Victor S. Fadin, E. A. Kuraev, and L. N. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. 60B, 50 (1975).
  49. E. A. Kuraev, L. N. Lipatov, and Victor S. Fadin, Multiregge processes in the Yang-Mills theory, Sov. Phys. JETP 44, 443 (1976).
  50. E. A. Kuraev, L. N. Lipatov, and Victor S. Fadin, The Pomeranchuk singularity in Nonabelian gauge theories, Sov. Phys. JETP 45, 199 (1977).
  51. I. I. Balitsky and L. N. Lipatov, The Pomeranchuk Singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28, 822 (1978).
  52. Marcello Ciafaloni, Coherence effects in initial jets at small q**2/s, Nucl. Phys. B296, 49 (1988).
  53. S. Catani, F. Fiorani, and G. Marchesini, Small x behavior of initial state radiation in perturbative QCD, Nucl. Phys. B336, 18 (1990).
  54. S. Catani, F. Fiorani, and G. Marchesini, QCD coherence in initial state radiation, Phys. Lett. B 234, 339 (1990).
  55. Giuseppe Marchesini, QCD coherence in the structure function and associated distributions at small x, Nucl. Phys. B445, 49 (1995).
  56. I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996).
  57. Yuri V. Kovchegov, Small x F(2) structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999).
  58. Jamal Jalilian-Marian, Alex Kovner, Andrei Leonidov, and Heribert Weigert, The Wilson renormalization group for low x physics: Towards the high density regime, Phys. Rev. D 59, 014014 (1998).
  59. Jamal Jalilian-Marian, Alex Kovner, Andrei Leonidov, and Heribert Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B504, 415 (1997).
  60. Heribert Weigert, Unitarity at small Bjorken x, Nucl. Phys. A703, 823 (2002).
  61. Edmond Iancu, Andrei Leonidov, and Larry D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1, Nucl. Phys. A692, 583 (2001).
  62. Edmond Iancu, Andrei Leonidov, and Larry D. McLerran, The Renormalization group equation for the color glass condensate, Phys. Lett. B 510, 133 (2001).
  63. Krzysztof J. Golec-Biernat and M. Wusthoff, Saturation effects in deep inelastic scattering at low Q**2 and its implications on diffraction, Phys. Rev. D 59, 014017 (1998).
  64. M. A. Kimber, Alan D. Martin, and M. G. Ryskin, Unintegrated parton distributions and prompt photon hadroproduction, Eur. Phys. J. C 12, 655 (2000).
  65. M. A. Kimber, Alan D. Martin, and M. G. Ryskin, Unintegrated parton distributions, Phys. Rev. D 63, 114027 (2001).
  66. Alan D. Martin and M. G. Ryskin, Unintegrated generalized parton distributions, Phys. Rev. D 64, 094017 (2001).
  67. V. N. Gribov and L. N. Lipatov, e+e− pair annihilation and deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15, 675 (1972).
  68. L. N. Lipatov, The parton model and perturbation theory, Yad. Fiz. 20, 181 (1974).
  69. Guido Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B126, 298 (1977).
  70. Yuri L. Dokshitzer, Calculation of the structure functions for deep inelastic scattering and e+e− annihilation by perturbation theory in quantum chromodynamics, Sov. Phys. JETP 46, 641 (1977).
  71. Fabio Dominguez, Cyrille Marquet, Bo-Wen Xiao, and Feng Yuan, Universality of unintegrated gluon distributions at small x, Phys. Rev. D 83, 105005 (2011).
  72. A. D. Martin, M. G. Ryskin, and G. Watt, NLO prescription for unintegrated parton distributions, Eur. Phys. J. C 66, 163 (2010).
  73. A. V. Kotikov and A. V. Lipatov, Updating TMD parton densities in a proton within the Kimber-Martin-Ryskin approach, Phys. Rev. D 111, 094009 (2025).
  74. John C. Collins and R. Keith Ellis, Heavy quark production in very high-energy hadron collisions, Nucl. Phys. B360, 3 (1991).
  75. Michal Deak and Florian Schwennsen, Z and W± production associated with quark-antiquark pair in kT-factorization at the LHC, J. High Energy Phys. 09 (2008) 035.
  76. F. Hautmann and H. Jung, Transverse momentum dependent gluon density from DIS precision data, Nucl. Phys. B883, 1 (2014).
  77. Krzysztof J. Golec-Biernat and M. Wusthoff, Saturation in diffractive deep inelastic scattering, Phys. Rev. D 60, 114023 (1999).
  78. F. Hautmann, H. Jung, M. Krämer, P. J. Mulders, E. R. Nocera, T. C. Rogers, and A. Signori, TMDlib and TMDplotter: Library and plotting tools for transverse-momentum-dependent parton distributions, Eur. Phys. J. C 74, 3220 (2014).
  79. A. Bermudez Martinez, P. Connor, H. Jung, A. Lelek, R. Žlebčík, F. Hautmann, and V. Radescu, Collinear and TMD parton densities from fits to precision DIS measurements in the parton branching method, Phys. Rev. D 99, 074008 (2019).
  80. I. Bubanja et al., The small kT region in Drell–Yan production at next-to-leading order with the parton branching method, Eur. Phys. J. C 84, 154 (2024).
  81. F. Hautmann, H. Jung, A. Lelek, V. Radescu, and R. Zlebcik, Soft-gluon resolution scale in QCD evolution equations, Phys. Lett. B 772, 446 (2017).
  82. Krzysztof Golec-Biernat and Anna M. Stasto, On the use of the KMR unintegrated parton distribution functions, Phys. Lett. B 781, 633 (2018).
  83. Hung-Liang Lai, Marco Guzzi, Joey Huston, Zhao Li, Pavel M. Nadolsky, Jon Pumplin, and C. P. Yuan, New parton distributions for collider physics, Phys. Rev. D 82, 074024 (2010).
  84. John Campbell, Joey Huston, and Frank Krauss, The Black Book of Quantum Chromodynamics: A Primer for the LHC Era (Oxford University Press, New York, 2018).
  85. G. Peter Lepage, Adaptive multidimensional integration: VEGAS enhanced, J. Comput. Phys. 439, 110386 (2021).
  86. S. Navas et al., Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  87. Anna Kulesza and W. James Stirling, Sudakov logarithm resummation in transverse momentum space for electroweak boson production at hadron colliders, Nucl. Phys. B555, 279 (1999).
  88. G. Watt, A. D. Martin, and M. G. Ryskin, Unintegrated parton distributions and electroweak boson production at hadron colliders, Phys. Rev. D 70, 014012 (2004); 70, 079902(E) (2004).
  89. Jan Ferdyan and Błażej Ruba, Spinor-helicity calculation of the g*g*→qq¯V* amplitude at the tree level, Acta Phys. Pol. B 55, 9-A2 (2024).
  90. L. N. Lipatov, Gauge invariant effective action for high-energy processes in QCD, Nucl. Phys. B452, 369 (1995).

Outline

Information

Phys. Rev. D 114, 054039– Published 18 September, 2026
  • Received 20 April 2026
  • Accepted 19 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation