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  • Open Access

Evolution of unpolarized transverse momentum dependent parton distributions of twist-3

Xue-Tao Liu1, Ping-An Liu1, Yu-Lu Liu1, and An-Ping Chen1,2,*

  • 1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China
  • 2Jiangxi Provincial Key Laboratory of Advanced Electronic Materials and Devices, Nanchang 330022, China

  • *Contact author: chenanping@jxnu.edu.cn
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Phys. Rev. D 111, 114025 – Published 23 June, 2025

DOI: https://doi.org/10.1103/sz46-x2sk

Abstract

We revisit the calculation of the evolution equations for four unpolarized twist-3 transverse momentum dependent (TMD) parton distribution functions (PDFs) at O(αs). Unlike the existing calculations in the literature, which are based on the background field method, we derive the evolution equations directly through diagram expansion. Additionally, instead of using the δ regulator, we employ the exponential regulator to regularize the rapidity divergences. We find that the evolutions of the four twist-3 TMD PDFs are governed by eight homogeneous equations. Our evolution kernels agree with one set of results in the literature, but differ from another by a sign. Considering the advantages of the exponential regulator in high-order perturbative calculations, our results are crucial for computing the twist-3 TMD PDFs at O(αs2).

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References (49)

  1. X.-d. Ji, J.-p. Ma, and F. Yuan, QCD factorization for semi-inclusive deep-inelastic scattering at low transverse momentum, Phys. Rev. D 71, 034005 (2005).
  2. X.-d. Ji, J.-P. Ma, and F. Yuan, QCD factorization for spin-dependent cross sections in DIS and Drell-Yan processes at low transverse momentum, Phys. Lett. B 597, 299 (2004).
  3. J. C. Collins and A. Metz, Universality of soft and collinear factors in hard-scattering factorization, Phys. Rev. Lett. 93, 252001 (2004).
  4. J. C. Collins, D. E. Soper, and G. F. Sterman, Transverse momentum distribution in Drell-Yan pair and W and Z boson production, Nucl. Phys. B250, 199 (1985).
  5. J. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology Vol. 32 (Cambridge University Press, Cambridge, England, 2013), pp. 1–624.
  6. J. C. Collins and D. E. Soper, Parton distribution and decay functions, Nucl. Phys. B194, 445 (1982).
  7. T. Becher and M. Neubert, Drell-Yan production at small qT, transverse parton distributions and the collinear anomaly, Eur. Phys. J. C 71, 1665 (2011).
  8. T. Becher and G. Bell, Analytic regularization in soft-collinear effective theory, Phys. Lett. B 713, 41 (2012).
  9. J.-y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, The rapidity renormalization group, Phys. Rev. Lett. 108, 151601 (2012).
  10. J.-Y. Chiu, A. Jain, D. Neill, and I. Z. Rothstein, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, J. High Energy Phys. 05 (2012) 084.
  11. M. G. Echevarria, A. Idilbi, and I. Scimemi, Factorization theorem for Drell-Yan at low qT and transverse momentum distributions on-the-light-cone, J. High Energy Phys. 07 (2012) 002.
  12. M. G. Echevarría, A. Idilbi, and I. Scimemi, Soft and collinear factorization and transverse momentum dependent parton distribution functions, Phys. Lett. B 726, 795 (2013).
  13. Z.-F. Deng, W. Wang, and J. Zeng, Transverse-momentum-dependent wave functions and soft functions at one-loop in large momentum effective theory, J. High Energy Phys. 09 (2022) 046.
  14. Y. Li, D. Neill, and H. X. Zhu, An exponential regulator for rapidity divergences, Nucl. Phys. B960, 115193 (2020).
  15. R. Boussarie et al., TMD handbook, arXiv:2304.03302.
  16. M. A. Ebert, I. W. Stewart, and Y. Zhao, Towards quasi-transverse momentum dependent PDFs computable on the lattice, J. High Energy Phys. 09 (2019) 037.
  17. T. Gehrmann, T. Lubbert, and L. L. Yang, Transverse parton distribution functions at next-to-next-to-leading order: The quark-to-quark case, Phys. Rev. Lett. 109, 242003 (2012).
  18. T. Gehrmann, T. Luebbert, and L. L. Yang, Calculation of the transverse parton distribution functions at next-to-next-to-leading order, J. High Energy Phys. 06 (2014) 155.
  19. M. G. Echevarria, I. Scimemi, and A. Vladimirov, Unpolarized transverse momentum dependent parton distribution and fragmentation functions at next-to-next-to-leading order, J. High Energy Phys. 09 (2016) 004.
  20. M.-X. Luo, X. Wang, X. Xu, L. L. Yang, T.-Z. Yang, and H. X. Zhu, Transverse parton distribution and fragmentation functions at NNLO: The quark case, J. High Energy Phys. 10 (2019) 083.
  21. M.-X. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Transverse parton distribution and fragmentation functions at NNLO: The gluon case, J. High Energy Phys. 01 (2020) 040.
  22. M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Quark transverse parton distribution at the next-to-next-to-next-to-leading order, Phys. Rev. Lett. 124, 092001 (2020).
  23. M.-x. Luo, T.-Z. Yang, H. X. Zhu, and Y. J. Zhu, Unpolarized quark and gluon TMD PDFs and FFs at N3LO, J. High Energy Phys. 06 (2021) 115.
  24. M. A. Ebert, B. Mistlberger, and G. Vita, Transverse momentum dependent PDFs at N3LO, J. High Energy Phys. 09 (2020) 146.
  25. M. A. Ebert, B. Mistlberger, and G. Vita, TMD fragmentation functions at N3LO, J. High Energy Phys. 07 (2021) 121.
  26. T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, J. High Energy Phys. 06 (2010) 094.
  27. C. Duhr, B. Mistlberger, and G. Vita, Four-loop rapidity anomalous dimension and event shapes to fourth logarithmic order, Phys. Rev. Lett. 129, 162001 (2022).
  28. I. Moult, H. X. Zhu, and Y. J. Zhu, The four loop QCD rapidity anomalous dimension, J. High Energy Phys. 08 (2022) 280.
  29. P. J. Mulders and R. D. Tangerman, The complete tree level result up to order 1/Q for polarized deep inelastic leptoproduction, Nucl. Phys. B461, 197 (1996); B484, 538(E) (1997).
  30. D. Boer, P. J. Mulders, and F. Pijlman, Universality of T odd effects in single spin and azimuthal asymmetries, Nucl. Phys. B667, 201 (2003).
  31. A. Bacchetta, P. J. Mulders, and F. Pijlman, New observables in longitudinal single-spin asymmetries in semi-inclusive DIS, Phys. Lett. B 595, 309 (2004).
  32. A. Bacchetta, M. Diehl, K. Goeke, A. Metz, P. J. Mulders, and M. Schlegel, Semi-inclusive deep inelastic scattering at small transverse momentum, J. High Energy Phys. 02 (2007) 093.
  33. D. Boer, R. Jakob, and P. J. Mulders, Asymmetries in polarized hadron production in e+e− annihilation up to order 1/Q, Nucl. Phys. B504, 345 (1997).
  34. Z. Lu and I. Schmidt, Transverse momentum dependent twist-three result for polarized Drell-Yan processes, Phys. Rev. D 84, 114004 (2011).
  35. A. Bacchetta, D. Boer, M. Diehl, and P. J. Mulders, Matches and mismatches in the descriptions of semi-inclusive processes at low and high transverse momentum, J. High Energy Phys. 08 (2008) 023.
  36. A. P. Chen and J. P. Ma, Light-cone singularities and transverse-momentum-dependent factorization at twist-3, Phys. Lett. B 768, 380 (2017).
  37. A. Bacchetta, G. Bozzi, M. G. Echevarria, C. Pisano, A. Prokudin, and M. Radici, Azimuthal asymmetries in unpolarized SIDIS and Drell-Yan processes: A case study towards TMD factorization at subleading twist, Phys. Lett. B 797, 134850 (2019).
  38. A. Vladimirov, V. Moos, and I. Scimemi, Transverse momentum dependent operator expansion at next-to-leading power, J. High Energy Phys. 01 (2022) 110.
  39. M. A. Ebert, A. Gao, and I. W. Stewart, Factorization for azimuthal asymmetries in SIDIS at next-to-leading power, J. High Energy Phys. 06 (2022) 007; 07 (2023) 96.
  40. L. Gamberg, Z.-B. Kang, D. Y. Shao, J. Terry, and F. Zhao, Transverse-momentum-dependent factorization at next-to-leading power, arXiv:2211.13209.
  41. S. Rodini and A. Vladimirov, Definition and evolution of transverse momentum dependent distribution of twist-three, J. High Energy Phys. 08 (2022) 031; 12 (2022) 48.
  42. S. Rodini and A. Vladimirov, Transverse momentum dependent factorization for SIDIS at next-to-leading power, Phys. Rev. D 110, 034009 (2024).
  43. X.-d. Ji and F. Yuan, Parton distributions in light cone gauge: Where are the final state interactions?, Phys. Lett. B 543, 66 (2002).
  44. A. V. Belitsky, X. Ji, and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B656, 165 (2003).
  45. I. O. Cherednikov and N. G. Stefanis, Renormalization, Wilson lines, and transverse-momentum dependent parton distribution functions, Phys. Rev. D 77, 094001 (2008).
  46. A. Idilbi and I. Scimemi, Singular and regular gauges in soft collinear effective theory: The introduction of the new Wilson line T, Phys. Lett. B 695, 463 (2011).
  47. D. Boer, L. Gamberg, B. Musch, and A. Prokudin, Bessel-weighted asymmetries in semi inclusive deep inelastic scattering, J. High Energy Phys. 10 (2011) 021.
  48. J. P. Ma and G. P. Zhang, Evolution of chirality-odd twist-3 fragmentation functions, Phys. Lett. B 772, 559 (2017).
  49. A. V. Manohar and I. W. Stewart, The zero-bin and mode factorization in quantum field theory, Phys. Rev. D 76, 074002 (2007).

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Phys. Rev. D 111, 114025– Published 23 June, 2025
  • Received 18 March 2025
  • Accepted 23 May 2025
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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.

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