Gluon and Wilson loop TMDs for hadrons of spin ≤ 1

  • Daniël Boer
  • Sabrina Cotogno
  • Tom van Daal
  • Piet J. Mulders
  • Andrea Signori
  • Ya-Jin Zhou
Open Access
Regular Article - Theoretical Physics


In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small x, related to a correlator of a single Wilson loop. We present the parametrization of Wilson loop correlators in terms of Wilson loop TMDs and discuss the relation between these functions and the small-x ‘dipole’ gluon TMDs. This analysis shows which gluon TMDs are leading or suppressed in the small-x limit. We discuss hadronic targets that are unpolarized, vector polarized (relevant for spin-1/2 and spin-1 hadrons), and tensor polarized (relevant for spin-1 hadrons). The latter are of interest for studies with a future Electron-Ion Collider with polarized deuterons.


QCD Phenomenology 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    A.V. Belitsky, X. Ji and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B 656 (2003) 165 [hep-ph/0208038] [INSPIRE].
  2. [2]
    D. Boer, P.J. Mulders and F. Pijlman, Universality of T odd effects in single spin and azimuthal asymmetries, Nucl. Phys. B 667 (2003) 201 [hep-ph/0303034] [INSPIRE].
  3. [3]
    A.V. Efremov and A.V. Radyushkin, Field theoretic treatment of high momentum transfer processes. 1. Deep inelastic scattering, Theor. Math. Phys. 44 (1980) 573 [Teor. Mat. Fiz. 44 (1980) 17] [INSPIRE].
  4. [4]
    J.C. Collins, D.E. Soper and G.F. Sterman, Factorization for one loop corrections in the Drell-Yan process, Nucl. Phys. B 223 (1983) 381 [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    D. Boer and P.J. Mulders, Color gauge invariance in the Drell-Yan process, Nucl. Phys. B 569 (2000) 505 [hep-ph/9906223] [INSPIRE].
  6. [6]
    J.C. Collins, Leading twist single transverse-spin asymmetries: Drell-Yan and deep inelastic scattering, Phys. Lett. B 536 (2002) 43 [hep-ph/0204004] [INSPIRE].
  7. [7]
    P.J. Mulders and J. Rodrigues, Transverse momentum dependence in gluon distribution and fragmentation functions, Phys. Rev. D 63 (2001) 094021 [hep-ph/0009343] [INSPIRE].
  8. [8]
    S.J. Brodsky, D.S. Hwang and I. Schmidt, Final state interactions and single spin asymmetries in semiinclusive deep inelastic scattering, Phys. Lett. B 530 (2002) 99 [hep-ph/0201296] [INSPIRE].
  9. [9]
    A. Bacchetta, C.J. Bomhof, P.J. Mulders and F. Pijlman, Single spin asymmetries in hadron-hadron collisions, Phys. Rev. D 72 (2005) 034030 [hep-ph/0505268] [INSPIRE].
  10. [10]
    D. Boer and P.J. Mulders, Time reversal odd distribution functions in leptoproduction, Phys. Rev. D 57 (1998) 5780 [hep-ph/9711485] [INSPIRE].
  11. [11]
    D. Boer and W. Vogelsang, Asymmetric jet correlations in pp uparrow scattering, Phys. Rev. D 69 (2004) 094025 [hep-ph/0312320] [INSPIRE].
  12. [12]
    P.J. Mulders and R.D. Tangerman, The complete tree level result up to order 1/Q for polarized deep inelastic leptoproduction, Nucl. Phys. B 461 (1996) 197 [Erratum ibid. B 484 (1997) 538] [hep-ph/9510301] [INSPIRE].
  13. [13]
    J.P. Ralston and D.E. Soper, Production of dimuons from high-energy polarized proton proton collisions, Nucl. Phys. B 152 (1979) 109 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D.W. Sivers, Single spin production asymmetries from the hard scattering of point-like constituents, Phys. Rev. D 41 (1990) 83 [INSPIRE].ADSGoogle Scholar
  15. [15]
    J.C. Collins, Fragmentation of transversely polarized quarks probed in transverse momentum distributions, Nucl. Phys. B 396 (1993) 161 [hep-ph/9208213] [INSPIRE].
  16. [16]
    A. Kotzinian, New quark distributions and semiinclusive electroproduction on the polarized nucleons, Nucl. Phys. B 441 (1995) 234 [hep-ph/9412283] [INSPIRE].
  17. [17]
    R.D. Tangerman and P.J. Mulders, Intrinsic transverse momentum and the polarized Drell-Yan process, Phys. Rev. D 51 (1995) 3357 [hep-ph/9403227] [INSPIRE].
  18. [18]
    A. Bacchetta and P.J. Mulders, Deep inelastic leptoproduction of spin-one hadrons, Phys. Rev. D 62 (2000) 114004 [hep-ph/0007120] [INSPIRE].
  19. [19]
    R.L. Jaffe and A. Manohar, Nuclear gluonometry, Phys. Lett. B 223 (1989) 218 [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    X. Artru and M. Mekhfi, Transversely polarized parton densities, their evolution and their measurement, Z. Phys. C 45 (1990) 669 [INSPIRE].ADSGoogle Scholar
  21. [21]
    C.J. Bomhof, P.J. Mulders and F. Pijlman, The construction of gauge-links in arbitrary hard processes, Eur. Phys. J. C 47 (2006) 147 [hep-ph/0601171] [INSPIRE].
  22. [22]
    M.G.A. Buffing, A. Mukherjee and P.J. Mulders, Generalized universality of higher transverse moments of quark TMD correlators, Phys. Rev. D 86 (2012) 074030 [arXiv:1207.3221] [INSPIRE].ADSGoogle Scholar
  23. [23]
    M.G.A. Buffing, A. Mukherjee and P.J. Mulders, Generalized universality of definite rank gluon transverse momentum dependent correlators, Phys. Rev. D 88 (2013) 054027 [arXiv:1306.5897] [INSPIRE].ADSGoogle Scholar
  24. [24]
    D. Boer, P.J. Mulders, C. Pisano and J. Zhou, Asymmetries in heavy quark pair and dijet production at an EIC, JHEP 08 (2016) 001 [arXiv:1605.07934] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    R.L. Jaffe, Spin, twist and hadron structure in deep inelastic processes, hep-ph/9602236 [INSPIRE].
  26. [26]
    J. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2011).CrossRefGoogle Scholar
  27. [27]
    M.G. Echevarria, T. Kasemets, P.J. Mulders and C. Pisano, QCD evolution of (un)polarized gluon TMDPDFs and the Higgs q T -distribution, JHEP 07 (2015) 158 [arXiv:1502.05354] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    I.O. Cherednikov, T. Mertens and F.F. Van der Veken, Evolution of cusped light-like Wilson loops and geometry of the loop space, Phys. Rev. D 86 (2012) 085035 [arXiv:1208.1631] [INSPIRE].ADSGoogle Scholar
  29. [29]
    D. Boer, M.G. Echevarria, P. Mulders and J. Zhou, Single spin asymmetries from a single Wilson loop, Phys. Rev. Lett. 116 (2016) 122001 [arXiv:1511.03485] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    S. Meissner, A. Metz and K. Goeke, Relations between generalized and transverse momentum dependent parton distributions, Phys. Rev. D 76 (2007) 034002 [hep-ph/0703176] [INSPIRE].
  31. [31]
    A. Bacchetta, Probing the transverse spin of quarks in deep inelastic scattering, Ph.D. thesis, Vrije Universiteit, Amsterdam The Netherlands (2002) [hep-ph/0212025] [INSPIRE].
  32. [32]
    K. Goeke, A. Metz, P.V. Pobylitsa and M.V. Polyakov, Lorentz invariance relations among parton distributions revisited, Phys. Lett. B 567 (2003) 27 [hep-ph/0302028] [INSPIRE].
  33. [33]
    A. Hebecker, Diffraction in deep inelastic scattering, Phys. Rept. 331 (2000) 1 [hep-ph/9905226] [INSPIRE].
  34. [34]
    F. Dominguez, C. Marquet, B.-W. Xiao and F. Yuan, Universality of unintegrated gluon distributions at small x, Phys. Rev. D 83 (2011) 105005 [arXiv:1101.0715] [INSPIRE].ADSGoogle Scholar
  35. [35]
    S. Catani, M. Ciafaloni and F. Hautmann, High-energy factorization and small x heavy flavor production, Nucl. Phys. B 366 (1991) 135 [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    A. Metz and J. Zhou, Distribution of linearly polarized gluons inside a large nucleus, Phys. Rev. D 84 (2011) 051503 [arXiv:1105.1991] [INSPIRE].ADSGoogle Scholar
  37. [37]
    M. Maul, A polarized version of the CCFM equation for gluons, Phys. Rev. D 65 (2002) 094010 [hep-ph/0111031] [INSPIRE].
  38. [38]
    J. Bartels, B.I. Ermolaev and M.G. Ryskin, Nonsinglet contributions to the structure function g 1 at small x, Z. Phys. C 70 (1996) 273 [hep-ph/9507271] [INSPIRE].
  39. [39]
    J. Bartels, B.I. Ermolaev and M.G. Ryskin, Flavor singlet contribution to the structure function g 1 at small x, Z. Phys. C 72 (1996) 627 [hep-ph/9603204] [INSPIRE].
  40. [40]
    Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity evolution at small-x, JHEP 01 (2016) 072 [arXiv:1511.06737] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    Y. Hatta, E. Iancu, K. Itakura and L. McLerran, Odderon in the color glass condensate, Nucl. Phys. A 760 (2005) 172 [hep-ph/0501171] [INSPIRE].
  42. [42]
    J. Zhou, Transverse single spin asymmetries at small x and the anomalous magnetic moment, Phys. Rev. D 89 (2014) 074050 [arXiv:1308.5912] [INSPIRE].ADSGoogle Scholar
  43. [43]
    L. Szymanowski and J. Zhou, The spin dependent odderon in the diquark model, Phys. Lett. B 760 (2016) 249 [arXiv:1604.03207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  44. [44]
    M.G. Ryskin, Odderon and polarization phenomena in QCD, Sov. J. Nucl. Phys. 46 (1987) 337 [Yad. Fiz. 46 (1987) 611] [INSPIRE].
  45. [45]
    N.H. Buttimore, B.Z. Kopeliovich, E. Leader, J. Soffer and T.L. Trueman, The spin dependence of high-energy proton scattering, Phys. Rev. D 59 (1999) 114010 [hep-ph/9901339] [INSPIRE].
  46. [46]
    E. Leader and T.L. Trueman, The odderon and spin dependence of high-energy proton proton scattering, Phys. Rev. D 61 (2000) 077504 [hep-ph/9908221] [INSPIRE].
  47. [47]
    S.J. Brodsky, F. Fleuret, C. Hadjidakis and J.P. Lansberg, Physics opportunities of a fixed-target experiment using the LHC beams, Phys. Rept. 522 (2013) 239 [arXiv:1202.6585] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    D. Boer et al., Gluons and the quark sea at high energies: distributions, polarization, tomography, arXiv:1108.1713 [INSPIRE].
  49. [49]
    D. Boer and C. Pisano, Polarized gluon studies with charmonium and bottomonium at LHCb and AFTER, Phys. Rev. D 86 (2012) 094007 [arXiv:1208.3642] [INSPIRE].ADSGoogle Scholar
  50. [50]
    A. Signori, Gluon TMDs in quarkonium production, Few Body Syst. 57 (2016) 651 [arXiv:1602.03405] [INSPIRE].ADSCrossRefGoogle Scholar
  51. [51]
    W.J. den Dunnen, J.P. Lansberg, C. Pisano and M. Schlegel, Accessing the transverse dynamics and polarization of gluons inside the proton at the LHC, Phys. Rev. Lett. 112 (2014) 212001 [arXiv:1401.7611] [INSPIRE].ADSCrossRefGoogle Scholar
  52. [52]
    C. Pisano, D. Boer, S.J. Brodsky, M.G.A. Buffing and P.J. Mulders, Linear polarization of gluons and photons in unpolarized collider experiments, JHEP 10 (2013) 024 [arXiv:1307.3417] [INSPIRE].ADSCrossRefGoogle Scholar
  53. [53]
    A. Dumitru, T. Lappi and V. Skokov, Distribution of linearly polarized gluons and elliptic azimuthal anisotropy in deep inelastic scattering dijet production at high energy, Phys. Rev. Lett. 115 (2015) 252301 [arXiv:1508.04438] [INSPIRE].ADSCrossRefGoogle Scholar
  54. [54]
    A. Bacchetta, C. Bomhof, U. D’Alesio, P.J. Mulders and F. Murgia, The Sivers single-spin asymmetry in photon-jet production, Phys. Rev. Lett. 99 (2007) 212002 [hep-ph/0703153] [INSPIRE].
  55. [55]
    J.-P. Lansberg and H.-S. Shao, Double-quarkonium production at a fixed-target experiment at the LHC (AFTER@LHC), Nucl. Phys. B 900 (2015) 273 [arXiv:1504.06531] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    D. Boer, C. Lorcé, C. Pisano and J. Zhou, The gluon Sivers distribution: status and future prospects, Adv. High Energy Phys. 2015 (2015) 371396 [arXiv:1504.04332] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  57. [57]
    J.-W. Qiu, M. Schlegel and W. Vogelsang, Probing gluonic spin-orbit correlations in photon pair production, Phys. Rev. Lett. 107 (2011) 062001 [arXiv:1103.3861] [INSPIRE].ADSCrossRefGoogle Scholar
  58. [58]
    F. Dominguez, Unintegrated gluon distributions at small-x, Ph.D. thesis, Columbia U., New York U.S.A. (2011) [INSPIRE].
  59. [59]
    R. Angeles-Martinez et al., Transverse momentum dependent (TMD) parton distribution functions: status and prospects, Acta Phys. Polon. B 46 (2015) 2501 [arXiv:1507.05267] [INSPIRE].ADSCrossRefGoogle Scholar
  60. [60]
    S. Abeyratne et al., Science requirements and conceptual design for a polarized medium energy electron-ion collider at Jefferson lab, arXiv:1209.0757 [INSPIRE].
  61. [61]
    S. Abeyratne et al., MEIC design summary, arXiv:1504.07961 [INSPIRE].
  62. [62]
    J. Ball et al., On the large COMPASS polarized deuteron target, Czech. J. Phys. 56 (2006) F295 [INSPIRE].Google Scholar
  63. [63]
    D. Boer, L. Gamberg, B. Musch and A. Prokudin, Bessel-weighted asymmetries in semi inclusive deep inelastic scattering, JHEP 10 (2011) 021 [arXiv:1107.5294] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2016

Authors and Affiliations

  • Daniël Boer
    • 1
  • Sabrina Cotogno
    • 2
    • 3
  • Tom van Daal
    • 2
    • 3
  • Piet J. Mulders
    • 2
    • 3
  • Andrea Signori
    • 2
    • 3
  • Ya-Jin Zhou
    • 2
    • 3
    • 4
  1. 1.Van Swinderen Institute for Particle Physics and GravityUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Physics and AstronomyVU University AmsterdamAmsterdamThe Netherlands
  3. 3.NikhefAmsterdamThe Netherlands
  4. 4.School of Physics & Key Laboratory of Particle Physics and Particle Irradiation (MOE)Shandong UniversityJinanChina

Personalised recommendations