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  • Open Access

Tomography of the rho meson in the QCD instanton vacuum: Transverse momentum dependent parton distribution functions

Wei-Yang Liu* and Ismail Zahed†

  • Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA

  • *Contact author: wei-yang.liu@stonybrook.edu
  • †Contact author: ismail.zahed@stonybrook.edu
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Phys. Rev. D 112, 034028 – Published 26 August, 2025

DOI: https://doi.org/10.1103/6ffp-qs8p

Abstract

We analyze the rho meson unpolarized and polarized transverse momentum dependent parton distribution functions (TMDPDFs) in the instanton liquid model (ILM). The corresponding TMDs in ILM are approximated by a constituent quark beam function in the leading Fock state multiplied by a rapidity-dependent factor resulting from the staple-shaped Wilson lines, for fixed longitudinal momentum, transverse separation, and rapidity. At the resolution of the ILM, all of the rho meson TMDs are symmetric in parton x for fixed transverse momentum, and Gaussian-like in the transverse momentum for fixed parton x. The latter is a direct consequence of the profiling of the quark zero modes in the ILM. The evolved TMDs at higher rapidity using the Collins-Soper kernel, and higher resolution using the renormalization group, show substantial skewness towards low parton x.

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References (82)

  1. E. C. Aschenauer, I. Borsa, R. Sassot, and C. Van Hulse, Phys. Rev. D 99, 094004 (2019).
  2. R. Boussarie et al., arXiv:2304.03302.
  3. J. Blumlein, Prog. Part. Nucl. Phys. 69, 28 (2013).
  4. A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, F. Delcarro, F. Piacenza, and M. Radici, J. High Energy Phys. 07 (2020) 117.
  5. A. Vladimirov, J. High Energy Phys. 10 (2019) 090.
  6. I. Scimemi and A. Vladimirov, Eur. Phys. J. C 78, 89 (2018).
  7. M. Cerutti, L. Rossi, S. Venturini, A. Bacchetta, V. Bertone, C. Bissolotti, and M. Radici (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Phys. Rev. D 107, 014014 (2023).
  8. A. Bacchetta, F. Delcarro, C. Pisano, M. Radici, and A. Signori, J. High Energy Phys. 06 (2017) 081; 06 (2019) 51.
  9. A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Piacenza, M. Radici, and A. Signori (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), J. High Energy Phys. 10 (2022) 127.
  10. V. Moos, I. Scimemi, A. Vladimirov, and P. Zurita, J. High Energy Phys. 05 (2024) 036.
  11. W.-Y. Liu and I. Zahed, arXiv:2502.09789.
  12. W.-Y. Liu, I. Zahed, and Y. Zhao, Phys. Rev. D 111, 074022 (2025).
  13. M.-H. Chu et al. (Lattice Parton Collaboration), Phys. Rev. D 109, L091503 (2024).
  14. M.-H. Chu et al. (Lattice Parton (LPC) Collaboration), Phys. Rev. D 106, 034509 (2022).
  15. Z.-F. Deng, W. Wang, and J. Zeng, J. High Energy Phys. 09 (2022) 046.
  16. M. A. Ebert, I. W. Stewart, and Y. Zhao, Phys. Rev. D 99, 034505 (2019).
  17. A. Avkhadiev, P. E. Shanahan, M. L. Wagman, and Y. Zhao, Phys. Rev. D 108, 114505 (2023).
  18. M. Schlemmer, A. Vladimirov, C. Zimmermann, M. Engelhardt, and A. Schäfer, J. High Energy Phys. 08 (2021) 004.
  19. T. C. Rogers, Eur. Phys. J. A 52, 153 (2016).
  20. Q.-A. Zhang et al. (Lattice Parton Collaboration), Phys. Rev. Lett. 125, 192001 (2020).
  21. B. Pasquini, S. Rodini, and A. Bacchetta, Phys. Rev. D 100, 054039 (2019).
  22. Y. Ninomiya, W. Bentz, and I. C. Cloët, Phys. Rev. C 96, 045206 (2017).
  23. C. Michael and P. S. Spencer, Nucl. Phys. B, Proc. Suppl. 42, 261 (1995).
  24. C. Michael and P. S. Spencer, Phys. Rev. D 52, 4691 (1995).
  25. D. B. Leinweber, in Workshop on Light-Cone QCD and Nonperturbative Hadron Physics (1999), pp. 138–143, arXiv:hep-lat/0004025.
  26. I. Bakas, J. Phys. Conf. Ser. 283, 012004 (2011).
  27. J. C. Biddle, W. Kamleh, and D. B. Leinweber, Proc. Sci., LATTICE2018 (2018) 256 [arXiv:1903.07767].
  28. A. Hasenfratz and O. Witzel, Phys. Rev. D 101, 034514 (2020).
  29. A. Athenodorou, P. Boucaud, F. De Soto, J. Rodríguez-Quintero, and S. Zafeiropoulos, J. High Energy Phys. 02 (2018) 140.
  30. J. C. Biddle, W. Kamleh, and D. B. Leinweber, EPJ Web Conf. 245, 06010 (2020).
  31. D. Diakonov and V. Y. Petrov, Nucl. Phys. B272, 457 (1986).
  32. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 109, 074029 (2024).
  33. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 107, 094024 (2023).
  34. E. Shuryak and I. Zahed, Phys. Rev. D 107, 034027 (2023).
  35. E. Shuryak and I. Zahed, Phys. Rev. D 107, 034026 (2023).
  36. E. Shuryak and I. Zahed, Phys. Rev. D 107, 034024 (2023).
  37. E. Shuryak and I. Zahed, Phys. Rev. D 107, 034023 (2023).
  38. W.-Y. Liu, E. Shuryak, and I. Zahed, Phys. Rev. D 110, 054005 (2024).
  39. W.-Y. Liu, E. Shuryak, C. Weiss, and I. Zahed, Phys. Rev. D 110, 054021 (2024).
  40. I. Zahed, Symmetry 14, 932 (2022).
  41. I. Zahed, Phys. Rev. D 104, 054031 (2021).
  42. D. Diakonov, Prog. Part. Nucl. Phys. 51, 173 (2003).
  43. T. Schäfer and E. V. Shuryak, Rev. Mod. Phys. 70, 323 (1998).
  44. W.-Y. Liu, arXiv:2501.07776.
  45. S. Kumano and Q.-T. Song, Phys. Rev. D 103, 014025 (2021).
  46. R. Angeles-Martinez et al., Acta Phys. Pol. B 46, 2501 (2015).
  47. M. Grosse Perdekamp and F. Yuan, Annu. Rev. Nucl. Part. Sci. 65, 429 (2015).
  48. C. A. Aidala, S. D. Bass, D. Hasch, and G. K. Mallot, Rev. Mod. Phys. 85, 655 (2013).
  49. V. Barone, F. Bradamante, and A. Martin, Prog. Part. Nucl. Phys. 65, 267 (2010).
  50. U. D’Alesio and F. Murgia, Prog. Part. Nucl. Phys. 61, 394 (2008).
  51. C. Shi, J. Li, M. Li, X. Chen, and W. Jia, Phys. Rev. D 106, 014026 (2022).
  52. V. Moos and A. Vladimirov, J. High Energy Phys. 12 (2020) 145.
  53. S. Kaur, C. Mondal, and H. Dahiya, J. High Energy Phys. 01 (2021) 136.
  54. P. Hoodbhoy, R. L. Jaffe, and A. Manohar, Nucl. Phys. B312, 571 (1989).
  55. G. A. Miller, Phys. Rev. C 89, 045203 (2014).
  56. S. Liuti and K. Kathuria, J. Phys. Conf. Ser. 543, 012005 (2014).
  57. J.-L. Zhang and J. Wu, Eur. Phys. J. C 85, 13 (2025).
  58. F. E. Close and S. Kumano, Phys. Rev. D 42, 2377 (1990).
  59. A. Efremov and O. Teryaev, Sov. J. Nucl. Phys. 36, 557 (1982).
  60. A. Y. Umnikov, Phys. Lett. B 391, 177 (1997).
  61. H. He and X.-D. Ji, Phys. Rev. D 52, 2960 (1995).
  62. J. Collins, Foundations of Perturbative QCD, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology Vol. 32 (Cambridge University Press, Cambridge, England, 2023).
  63. X. Ji, J. Ma, and F. Yuan, Phys. Rev. D 71, 034005 (2005).
  64. D. Boer, Int. J. Mod. Phys. Conf. Ser. 40, 1660014 (2016).
  65. J. Collins, EPJ Web Conf. 85, 01002 (2015).
  66. J. Collins and T. Rogers, Phys. Rev. D 91, 074020 (2015).
  67. I. Scimemi and A. Vladimirov, J. High Energy Phys. 03 (2017) 002.
  68. U. D’Alesio, M. G. Echevarria, S. Melis, and I. Scimemi, J. High Energy Phys. 11 (2014) 098.
  69. J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl. Phys. B250, 199 (1985).
  70. T. Rogers, F. Aslan, M. Boglione, T. Rainaldi, A. Simonelli, and J. O. Gonzalez-Hernandez, Proc. Sci., Transversity2024 (2024) 030 [arXiv:2408.07170].
  71. D. Boer and W. J. den Dunnen, Nucl. Phys. B886, 421 (2014).
  72. J. Collins, L. Gamberg, A. Prokudin, T. C. Rogers, N. Sato, and B. Wang, Phys. Rev. D 94, 034014 (2016).
  73. R. L. Jaffe, H. Meyer, and G. Piller, Spin, twist and hadron structure in deep inelastic processes, in Lectures on QCD: Applications, edited by F. Lenz, H. Grießhammer, and D. Stoll (Springer, Berlin, Heidelberg, 1997), pp. 178–249.
  74. R. L. Jaffe and A. Manohar, Nucl. Phys. B321, 343 (1989).
  75. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
  76. S. Moch, J. A. M. Vermaseren, and A. Vogt, Nucl. Phys. B889, 351 (2014).
  77. W. Vogelsang, Phys. Rev. D 57, 1886 (1998).
  78. S. V. Mikhailov and A. A. Vladimirov, Phys. Lett. B 671, 111 (2009).
  79. X. Artru and M. Mekhfi, Z. Phys. C 45, 669 (1990).
  80. C. T. H. Davies and W. J. Stirling, Nucl. Phys. B244, 337 (1984).
  81. J. Collins and T. C. Rogers, Phys. Rev. D 96, 054011 (2017).
  82. D. Gutiérrez-Reyes, I. Scimemi, and A. A. Vladimirov, Phys. Lett. B 769, 84 (2017).

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Phys. Rev. D 112, 034028– Published 26 August, 2025
  • Received 7 April 2025
  • Accepted 21 July 2025
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