- 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

Third moments of nucleon unpolarized, polarized, and transversity parton distribution functions from physical-point lattice QCD

Emilio Taggi1,2,*, Michael Engelhardt3, Jeremy R. Green4, Stefan Krieg1,2, Stefan Meinel5, John W. Negele6, Andrew Pochinsky6, Marcel Rodekamp7, and Sergey Syritsyn8,9

  • *Contact author: e.taggi@fz-juelich.de
PDF

Phys. Rev. D 114, 074507 – Published 14 October, 2026

DOI: https://doi.org/10.1103/kknt-zxsw

Abstract

Using forward matrix elements of local leading-twist operators, we present a determination of the isovector third Mellin moments ⟨x2⟩ of nucleon unpolarized, polarized, and transversity parton distribution functions. Two lattice QCD ensembles at the physical pion mass are used, which were generated using a tree-level Symanzik-improved gauge action and 2+1 flavor tree-level improved Wilson clover fermions coupling via two-level HEX smearing. Leveraging a wide set of operators, two extraction methods for the matrix elements, and the automatic inclusion of model uncertainties via bootstrapped model averages, we extract values of the third Mellin moments. This is the first direct calculation of these observables performed at the physical pion mass.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (127)

  1. A. De Roeck and R. S. Thorne, Structure functions, Prog. Part. Nucl. Phys. 66, 727 (2011).
  2. S. Forte and G. Watt, Progress in the determination of the partonic structure of the proton, Annu. Rev. Nucl. Part. Sci. 63, 291 (2013).
  3. J. Blümlein, The theory of deeply inelastic scattering, Prog. Part. Nucl. Phys. 69, 28 (2013).
  4. E. Perez and E. Rizvi, The quark and gluon structure of the proton, Rep. Prog. Phys. 76, 046201 (2013).
  5. R. D. Ball et al., Parton distribution benchmarking with LHC data, J. High Energy Phys. 04 (2013) 125.
  6. J. Gao, L. Harland-Lang, and J. Rojo, The structure of the proton in the LHC precision era, Phys. Rep. 742, 1 (2018).
  7. J. J. Ethier and E. R. Nocera, Parton distributions in nucleons and nuclei, Annu. Rev. Nucl. Part. Sci. 70, 43 (2020).
  8. H.-W. Lin et al., Parton distributions and lattice QCD calculations: A community white paper, Prog. Part. Nucl. Phys. 100, 107 (2018).
  9. M. Constantinou et al., Lattice QCD calculations of parton physics, arXiv:2202.07193.
  10. H.-W. Lin, Mapping parton distributions of hadrons with lattice QCD, Prog. Part. Nucl. Phys. 144, 104177 (2025).
  11. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, R. Frezzotti, B. Kostrzewa, G. Koutsou, G. Spanoudes, and C. Urbach (Extended Twisted Mass Collaboration), Nucleon axial and pseudoscalar form factors using twisted-mass fermion ensembles at the physical point, Phys. Rev. D 109, 034503 (2024).
  12. Y.-C. Jang, R. Gupta, T. Bhattacharya, B. Yoon, and H.-W. Lin (Precision Neutron Decay Matrix Elements (PNDME) Collaboration), Nucleon isovector axial form factors, Phys. Rev. D 109, 014503 (2024).
  13. A. Walker-Loud et al., Lattice QCD determination of gA, Proc. Sci. CD2018 (2020) 020. [arXiv:1912.08321].
  14. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, and A. Vaquero Aviles-Casco, Nucleon axial, tensor, and scalar charges and σ-terms in lattice QCD, Phys. Rev. D 102, 054517 (2020).
  15. R. Gupta, Y.-C. Jang, B. Yoon, H.-W. Lin, V. Cirigliano, and T. Bhattacharya, Isovector charges of the nucleon from 2+1+1-flavor lattice QCD, Phys. Rev. D 98, 034503 (2018).
  16. C. C. Chang et al., A per-cent-level determination of the nucleon axial coupling from quantum chromodynamics, Nature (London) 558, 91 (2018).
  17. E. Berkowitz et al., An accurate calculation of the nucleon axial charge with lattice QCD, arXiv:1704.01114.
  18. T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, H.-W. Lin, and B. Yoon, Axial, scalar and tensor charges of the nucleon from 2+1+1-flavor lattice QCD, Phys. Rev. D 94, 054508 (2016).
  19. D. Djukanovic, G. von Hippel, H. B. Meyer, K. Ottnad, and H. Wittig, Improved analysis of isovector nucleon matrix elements with Nf=2+1 flavors of O(a) improved Wilson fermions, Phys. Rev. D 109, 074507 (2024).
  20. R. Tsuji, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, K. Sato, E. Shintani, H. Watanabe, and T. Yamazaki (PACS Collaboration), Nucleon form factors in Nf=2+1 lattice QCD at the physical point: Finite lattice spacing effect on the root-mean-square radii, Phys. Rev. D 109, 094505 (2024).
  21. G. S. Bali, S. Collins, S. Heybrock, M. Löffler, R. Rödl, W. Söldner, and S. Weishäupl (RQCD Collaboration), Octet baryon isovector charges from Nf=2+1 lattice QCD, Phys. Rev. D 108, 034512 (2023).
  22. R. E. Smail et al. (QCDSF/UKQCD/CSSM Collaborations), Constraining beyond the standard model nucleon isovector charges, Phys. Rev. D 108, 094511 (2023).
  23. R. Tsuji, N. Tsukamoto, Y. Aoki, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, E. Shintani, and T. Yamazaki (PACS Collaboration), Nucleon isovector couplings in Nf=2+1 lattice QCD at the physical point, Phys. Rev. D 106, 094505 (2022).
  24. D. Djukanovic, G. von Hippel, J. Koponen, H. B. Meyer, K. Ottnad, T. Schulz, and H. Wittig, Isovector axial form factor of the nucleon from lattice QCD, Phys. Rev. D 106, 074503 (2022).
  25. S. Park, R. Gupta, B. Yoon, S. Mondal, T. Bhattacharya, Y.-C. Jang, B. Joó, and F. Winter (Nucleon Matrix Elements (NME) Collaboration), Precision nucleon charges and form factors using (2+1)-flavor lattice QCD, Phys. Rev. D 105, 054505 (2022).
  26. B. Blossier, P. Boucaud, M. Brinet, F. De Soto, V. Morenas, O. Pène, K. Petrov, and J. Rodríguez-Quintero (ETM Collaboration), High statistics determination of the strong coupling constant in taylor scheme and its OPE Wilson coefficient from lattice QCD with a dynamical charm, Phys. Rev. D 89, 014507 (2014).
  27. N. Hasan, J. Green, S. Meinel, M. Engelhardt, S. Krieg, J. Negele, A. Pochinsky, and S. Syritsyn, Nucleon axial, scalar, and tensor charges using lattice QCD at the physical pion mass, Phys. Rev. D 99, 114505 (2019).
  28. T. Harris, G. von Hippel, P. Junnarkar, H. B. Meyer, K. Ottnad, J. Wilhelm, H. Wittig, and L. Wrang, Nucleon isovector charges and twist-2 matrix elements with Nf=2+1 dynamical Wilson quarks, Phys. Rev. D 100, 034513 (2019).
  29. E. Shintani, K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, and T. Yamazaki, Nucleon form factors and root-mean-square radii on a (10.8  fm)4 lattice at the physical point, Phys. Rev. D 99, 014510 (2019); 102, 019902(E) (2020).
  30. K.-I. Ishikawa, Y. Kuramashi, S. Sasaki, N. Tsukamoto, A. Ukawa, and T. Yamazaki (PACS Collaboration), Nucleon form factors on a large volume lattice near the physical point in 2+1 flavor QCD, Phys. Rev. D 98, 074510 (2018).
  31. J. Liang, Y.-B. Yang, T. Draper, M. Gong, and K.-F. Liu, Quark spins and anomalous ward identity, Phys. Rev. D 98, 074505 (2018).
  32. N. Yamanaka, S. Hashimoto, T. Kaneko, and H. Ohki (JLQCD Collaboration), Nucleon charges with dynamical overlap fermions, Phys. Rev. D 98, 054516 (2018).
  33. L. Liu, T. Chen, T. Draper, J. Liang, K.-F. Liu, G. Wang, and Y.-B. Yang χQCD Collaboration), Nucleon isovector scalar charge from overlap fermions, Phys. Rev. D 104, 094503 (2021).
  34. M. Abramczyk, T. Blum, T. Izubuchi, C. Jung, M. Lin, A. Lytle, S. Ohta, and E. Shintani, Nucleon mass and isovector couplings in 2+1-flavor dynamical domain-wall lattice QCD near physical mass, Phys. Rev. D 101, 034510 (2020).
  35. C. Alexandrou et al., Moments of the nucleon transverse quark spin densities using lattice QCD, Phys. Rev. D 107, 054504 (2023).
  36. T. Bhattacharya, V. Cirigliano, S. Cohen, R. Gupta, A. Joseph, H.-W. Lin, and B. Yoon (PNDME Collaboration), Iso-vector and Iso-scalar tensor charges of the nucleon from lattice QCD, Phys. Rev. D 92, 094511 (2015).
  37. T. Bhattacharya, V. Cirigliano, R. Gupta, H.-W. Lin, and B. Yoon, Neutron electric dipole moment and tensor charges from lattice QCD, Phys. Rev. Lett. 115, 212002 (2015).
  38. S. Mondal, R. Gupta, S. Park, B. Yoon, T. Bhattacharya, and H.-W. Lin, Moments of nucleon isovector structure functions in 2+1+1-flavor QCD, Phys. Rev. D 102, 054512 (2020).
  39. C. Alexandrou, S. Bacchio, M. Constantinou, J. Finkenrath, K. Hadjiyiannakou, K. Jansen, G. Koutsou, H. Panagopoulos, and G. Spanoudes, Complete flavor decomposition of the spin and momentum fraction of the proton using lattice QCD simulations at physical pion mass, Phys. Rev. D 101, 094513 (2020).
  40. C. Alexandrou et al., Moments of nucleon generalized parton distributions from lattice QCD simulations at physical pion mass, Phys. Rev. D 101, 034519 (2020).
  41. M. Rodekamp, M. Engelhardt, J. R. Green, S. Krieg, S. Liuti, S. Meinel, J. W. Negele, A. Pochinsky, and S. Syritsyn, Moments of nucleon unpolarized, polarized, and transversity parton distribution functions from lattice QCD at the physical point, Phys. Rev. D 109, 074508 (2024).
  42. S. Mondal, T. Bhattacharya, R. Gupta, B. Joó, H.-W. Lin, S. Park, F. Winter, and B. Yoon, Nucleon isovector momentum fraction, helicity and transversity moment using lattice QCD, Proc. Sci. LATTICE2021 (2021) 513. [arXiv:2201.00067].
  43. S. Mondal, R. Gupta, S. Park, B. Yoon, T. Bhattacharya, B. Joó, and F. Winter (Nucleon Matrix Elements (NME) Collaboration), Nucleon momentum fraction, helicity and transversity from 2+1-flavor lattice QCD, J. High Energy Phys. 04 (2021) 044.
  44. Y.-B. Yang, J. Liang, Y.-J. Bi, Y. Chen, T. Draper, K.-F. Liu, and Z. Liu, Proton mass decomposition from the QCD energy momentum tensor, Phys. Rev. Lett. 121, 212001 (2018).
  45. J. R. Green, M. Engelhardt, S. Krieg, J. W. Negele, A. V. Pochinsky, and S. N. Syritsyn, Nucleon structure from lattice QCD using a nearly physical pion mass, Phys. Lett. B 734, 290 (2014).
  46. J. D. Bratt et al. (LHPC Collaboration), Nucleon structure from mixed action calculations using 2+1 flavors of asqtad sea and domain wall valence fermions, Phys. Rev. D 82, 094502 (2010).
  47. Y. Aoki, T. Blum, H.-W. Lin, S. Ohta, S. Sasaki, R. Tweedie, J. Zanotti, and T. Yamazaki, Nucleon isovector structure functions in (2+1)-flavor QCD with domain wall fermions, Phys. Rev. D 82, 014501 (2010).
  48. G. S. Bali, S. Collins, M. Göckeler, R. Rödl, A. Schäfer, and A. Sternbeck, Nucleon generalized form factors from two-flavor lattice QCD, Phys. Rev. D 100, 014507 (2019).
  49. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, C. Kallidonis, G. Koutsou, A. Vaquero Avilés-Casco, and C. Wiese, Nucleon spin and momentum decomposition using lattice QCD simulations, Phys. Rev. Lett. 119, 142002 (2017).
  50. A. Abdel-Rehim et al., Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass, Phys. Rev. D 92, 114513 (2015); 93, 039904(E) (2016).
  51. G. S. Bali, S. Collins, B. Gläßle, M. Göckeler, J. Najjar, R. H. Rödl, A. Schäfer, R. W. Schiel, A. Sternbeck, and W. Söldner, The moment ⟨x⟩u−d of the nucleon from Nf=2 lattice QCD down to nearly physical quark masses, Phys. Rev. D 90, 074510 (2014).
  52. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG) Collaboration), FLAG review 2024, Phys. Rev. D 113, 014508 (2026).
  53. S. Bürger, T. Wurm, M. Löffler, M. Göckeler, G. Bali, S. Collins, A. Schäfer, and A. Sternbeck (RQCD Collaboration), Lattice results for the longitudinal spin structure and color forces on quarks in a nucleon, Phys. Rev. D 105, 054504 (2022).
  54. M. Bruno et al., Simulation of QCD with Nf=2+1 flavors of non-perturbatively improved Wilson fermions, J. High Energy Phys. 02 (2015) 043.
  55. M. Göckeler, P. Hägler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, and J. M. Zanotti (QCDSF Collaboration), Generalized parton distributions and structure functions from full lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 399 (2005).
  56. D. Dolgov et al. (LHPC, TXL Collaboration), Moments of nucleon light cone quark distributions calculated in full lattice QCD, Phys. Rev. D 66, 034506 (2002).
  57. M. Göckeler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, H. Stüben, and J. M. Zanotti, Investigation of the second moment of the nucleon’s g1 and g2 structure functions in two-flavor lattice QCD, Phys. Rev. D 72, 054507 (2005).
  58. A. Francis et al., Gradient flow for parton distribution functions: First application to the pion, Phys. Rev. Lett. 136, 171903 (2026).
  59. A. Francis, P. Fritzsch, R. Karur, J. Kim, G. Pederiva, D. A. Pefkou, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Moments of parton distributions functions of the pion from lattice QCD using gradient flow, Phys. Rev. D 113, 074520 (2026).
  60. A. Shindler, Moments of parton distribution functions of any order from lattice QCD, Phys. Rev. D 110, L051503 (2024).
  61. W. Detmold, A. V. Grebe, I. Kanamori, C. J. D. Lin, R. J. Perry, and Y. Zhao (HOPE Collaboration), Parton physics from a heavy-quark operator product expansion: Lattice QCD calculation of the fourth moment of the pion distribution amplitude, Phys. Rev. D 113, 014510 (2026).
  62. K. Cichy and M. Constantinou, A guide to light-cone PDFs from lattice QCD: An overview of approaches, techniques and results, Adv. High Energy Phys. 2019, 3036904 (2019).
  63. C. Alexandrou, M. Constantinou, K. Hadjiyiannakou, K. Jansen, and F. Manigrasso, Flavor decomposition of the nucleon unpolarized, helicity, and transversity parton distribution functions from lattice QCD simulations, Phys. Rev. D 104, 054503 (2021).
  64. X. Gao, A. D. Hanlon, J. Holligan, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, and Y. Zhao, Unpolarized proton PDF at NNLO from lattice QCD with physical quark masses, Phys. Rev. D 107, 074509 (2023).
  65. Z. Fan, X. Gao, R. Li, H.-W. Lin, N. Karthik, S. Mukherjee, P. Petreczky, S. Syritsyn, Y.-B. Yang, and R. Zhang, Isovector parton distribution functions of the proton on a superfine lattice, Phys. Rev. D 102, 074504 (2020).
  66. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, H.-T. Shu, F. Yao, R. Zhang, and Y. Zhao, Proton isovector helicity PDF at NNLO and the twist-3 moment d˜2 from lattice QCD at physical quark masses, Phys. Rev. D 114, 014505 (2026).
  67. C. Egerer et al. (HadStruc Collaboration), Transversity parton distribution function of the nucleon using the pseudodistribution approach, Phys. Rev. D 105, 034507 (2022).
  68. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, Q. Shi, S. Syritsyn, and Y. Zhao, Transversity PDFs of the proton from lattice QCD with physical quark masses, Phys. Rev. D 109, 054506 (2024).
  69. Z. Pang, J.-H. Zhang, and D.-J. Zhao, Moments from momentum derivatives in lattice QCD, Chin. Phys. 49, 101001 (2025).
  70. I. Sakata, A general method for obtaining Clebsch-Gordan coefficients of finite groups. I. Its application to point and space groups, J. Math. Phys. (N.Y.) 15, 1702 (1974).
  71. M. Baake, B. Gemünden, and R. Oedingen, Structure and representations of the symmetry group of the four-dimensional cube, J. Math. Phys. (N.Y.) 23, 944 (1982); 23, 2595(E) (1982).
  72. M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller, Lattice operators for moments of the structure functions and their transformation under the hypercubic group, Phys. Rev. D 54, 5705 (1996).
  73. S. Wandzura and F. Wilczek, Sum rules for spin dependent electroproduction: Test of relativistic constituent quarks, Phys. Lett. B 72, 195 (1977).
  74. R. L. Jaffe, g2—The Nucleon’s Other Spin-Dependent Structure Function, Comments Nucl. Part. Phys. 19, 239 (1990).
  75. R. L. Jaffe and X.-D. Ji, Chiral odd parton distributions and Drell-Yan processes, Nucl. Phys. B375, 527 (1992).
  76. S. Capitani and G. Rossi, Deep inelastic scattering in improved lattice QCD. 1. The first moment of structure functions, Nucl. Phys. B433, 351 (1995).
  77. M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller, Polarized and unpolarized nucleon structure functions from lattice QCD, Phys. Rev. D 53, 2317 (1996).
  78. M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller, Perturbative renormalization of lattice bilinear quark operators, Nucl. Phys. B472, 309 (1996).
  79. M. Göckeler, P. Hägler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, and J. M. Zanotti (QCDSF, UKQCD Collaborations), Quark helicity flip generalized parton distributions from two-flavor lattice QCD, Phys. Lett. B 627, 113 (2005).
  80. C. Alexandrou, Parton distribution functions from lattice QCD, Few Body Syst. 57, 621 (2016).
  81. B. Knippschild, H. Wittig, S. Capitani, and M. Della Morte, Systematic errors in extracting nucleon properties from lattice QCD, Proc. Sci. LATTICE2010 (2011) 147.
  82. M. Donnellan, J. Bulava, and R. Sommer (ALPHA Collaboration), The B*Bπ Coupling in the Static Limit, Proc. Sci. LATTICE2010 (2011) 303. [arXiv:1011.4393].
  83. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  84. E. T. Neil and J. W. Sitison, Improved information criteria for Bayesian model averaging in lattice field theory, Phys. Rev. D 109, 014510 (2024).
  85. E. T. Neil and J. W. Sitison, Model averaging approaches to data subset selection, Phys. Rev. E 108, 045308 (2023).
  86. S. Dürr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabó, and G. Vulvert (BMW Collaboration), Lattice QCD at the physical point: Simulation and analysis details, J. High Energy Phys. 08 (2010) 148.
  87. S. Dürr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, K. K. Szabó, and G. Vulvert (BMW Collaboration), Lattice QCD at the physical point: Light quark masses, Phys. Lett. B 701, 265 (2011).
  88. S. Güsken, A study of smearing techniques for hadron correlation functions, Nucl. Phys. B, Proc. Suppl. 17, 361 (1990).
  89. G. S. Bali, B. Lang, B. U. Musch, and A. Schäfer, Novel quark smearing for hadrons with high momenta in lattice QCD, Phys. Rev. D 93, 094515 (2016).
  90. T. Blum, T. Izubuchi, and E. Shintani, New class of variance-reduction techniques using lattice symmetries, Phys. Rev. D 88, 094503 (2013).
  91. E. Shintani, R. Arthur, T. Blum, T. Izubuchi, C. Jung, and C. Lehner, Covariant approximation averaging, Phys. Rev. D 91, 114511 (2015).
  92. S. Spiegel and C. Lehner, High-precision continuum limit study of the HVP short-distance window, Phys. Rev. D 111, 114517 (2025).
  93. M. Göckeler, R. Horsley, D. Pleiter, P. E. L. Rakow, and G. Schierholz (QCDSF Collaboration), A lattice determination of moments of unpolarised nucleon structure functions using improved Wilson fermions, Phys. Rev. D 71, 114511 (2005).
  94. A. Buckley, J. Ferrando, S. Lloyd, K. Nordström, B. Page, M. Rüfenacht, M. Schönherr, and G. Watt, lhapdf6: Parton density access in the LHC precision era, Eur. Phys. J. C 75, 132 (2015).
  95. R. D. Ball et al. (NNPDF Collaboration), The path to N3LO parton distributions, Eur. Phys. J. C 84, 659 (2024).
  96. J. McGowan, T. Cridge, L. A. Harland-Lang, and R. S. Thorne, Approximate N3LO parton distribution functions with theoretical uncertainties: MSHT20aN3LO PDFs, Eur. Phys. J. C 83, 185 (2023); 83, 302(E) (2023).
  97. E. R. Nocera, R. D. Ball, S. Forte, G. Ridolfi, and J. Rojo (NNPDF Collaboration), A first unbiased global determination of polarized PDFs and their uncertainties, Nucl. Phys. B887, 276 (2014).
  98. J. Cruz-Martinez, T. Hasenack, F. Hekhorn, G. Magni, E. R. Nocera, T. R. Rabemananjara, J. Rojo, T. Sharma, and G. van Seeventer, NNPDFpol2.0: A global determination of polarised PDFs and their uncertainties at next-to-next-to-leading order, J. High Energy Phys. 07 (2025) 168.
  99. V. Bertone, A. Chiefa, and E. R. Nocera (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Helicity-dependent parton distribution functions at next-to-next-to-leading order accuracy from inclusive and semi-inclusive deep-inelastic scattering data, Phys. Lett. B 865, 139497 (2025).
  100. R. D. Ball et al. (NNPDF Collaboration), Parton distributions from high-precision collider data, Eur. Phys. J. C 77, 663 (2017).
  101. S. Dulat, T.-J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump, and C. P. Yuan, New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D 93, 033006 (2016).
  102. L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C 75, 204 (2015).
  103. S. Alekhin, J. Blümlein, S. Moch, and R. Placakyte, Parton distribution functions, αs, and heavy-quark masses for LHC Run II, Phys. Rev. D 96, 014011 (2017).
  104. A. Accardi, L. T. Brady, W. Melnitchouk, J. F. Owens, and N. Sato, Constraints on large-x parton distributions from new weak boson production and deep-inelastic scattering data, Phys. Rev. D 93, 114017 (2016).
  105. H. Abramowicz et al. (H1, ZEUS Collaborations), Combination of measurements of inclusive deep inelastic e±p scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75, 580 (2015).
  106. J. Butterworth et al., PDF4LHC recommendations for LHC run II, J. Phys. G 43, 023001 (2016).
  107. L. Gamberg, M. Malda, J. A. Miller, D. Pitonyak, A. Prokudin, and N. Sato (JAM Collaboration), Updated QCD global analysis of single transverse-spin asymmetries: Extracting H˜, and the role of the Soffer bound and lattice QCD, Phys. Rev. D 106, 034014 (2022).
  108. L. Gamberg, M. Malda, J. Miller, D. Pitonyak, A. Prokudin, and N. Sato, Jam3d library notebook, https://colab.research.google.com/github/pitonyak25/jam3d_dev_lib/blob/main/JAM3D_Library.ipynb (accessed: 2026-04-13).
  109. C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl (JAM Collaboration), Transversity distributions and tensor charges of the nucleon: Extraction from dihadron production and their universal nature, Phys. Rev. Lett. 132, 091901 (2024).
  110. C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl (Jefferson Lab Angular Momentum (JAM) Collaboration), First simultaneous global QCD analysis of dihadron fragmentation functions and transversity parton distribution functions, Phys. Rev. D 109, 034024 (2024).
  111. C. Cocuzza, A. Metz, D. Pitonyak, A. Prokudin, N. Sato, and R. Seidl, Google colab notebook, https://colab.research.google.com/github/prokudin/JAMDiFF_library/blob/main/JAMDiFF_Library.ipynb (2023) (accessed: 2023).
  112. A. Pochinsky, QLua, https://github.com/usqcd-software/qlua (accessed: 2026-04-08).
  113. R. Babich, J. Brannick, R. C. Brower, M. A. Clark, T. A. Manteuffel, S. F. McCormick, J. C. Osborn, and C. Rebbi, Adaptive multigrid algorithm for the lattice Wilson-Dirac operator, Phys. Rev. Lett. 105, 201602 (2010).
  114. J. Osborn et al., QOPQDP, https://github.com/usqcd-software/qopqdp (accessed: 2026-04-08).
  115. Jülich Supercomputing Centre, JURECA: Modular supercomputer at Jülich Supercomputing Centre, J. Large-Scale Res. Facil. 4, A132 (2018).
  116. Jülich Supercomputing Centre, JUQUEEN: IBM blue gene/Q supercomputer system at the Jülich Supercomputing Centre, J. Large-Scale Res. Facil. 1, A1 (2015).
  117. Jülich Supercomputing Centre, JUWELS: Modular Tier-0/1 supercomputer at the Jülich Supercomputing Centre, J. Large-Scale Res. Facil. 5, A171 (2019).
  118. G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa, and A. Vladikas, A general method for non-perturbative renormalization of lattice operators, Nucl. Phys. B445, 81 (1995).
  119. M. Göckeler, R. Horsley, H. Oelrich, H. Perlt, D. Petters, P. E. L. Rakow, A. Schäfer, G. Schierholz, and A. Schiller, Non-perturbative renormalization of composite operators in lattice QCD, Nucl. Phys. B544, 699 (1999).
  120. C. Sturm, Y. Aoki, N. H. Christ, T. Izubuchi, C. T. C. Sachrajda, and A. Soni, Renormalization of quark bilinear operators in a momentum-subtraction scheme with a nonexceptional subtraction point, Phys. Rev. D 80, 014501 (2009).
  121. J. A. Gracey, RI′/SMOM scheme amplitudes for deep inelastic scattering operators at one loop in QCD, Phys. Rev. D 83, 054024 (2011).
  122. J. A. Gracey, Three loop anomalous dimensions of higher moments of the non-singlet twist-2 Wilson and transversity operators in the MS¯ and RI′ schemes, J. High Energy Phys. 10 (2006) 040.
  123. V. M. Braun et al., The ρ-meson light-cone distribution amplitudes from lattice QCD, J. High Energy Phys. 04 (2016) 082.
  124. V. N. Velizhanin, Four-loop anomalous dimension of the third and fourth moments of the nonsinglet twist-2 operator in QCD, Int. J. Mod. Phys. A 35, 2050199 (2020).
  125. P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Massless Propagators, R(s) and Multiloop QCD, Nucl. Part. Phys. Proc. 261–262, 3 (2015).
  126. B. A. Kniehl and O. L. Veretin, Moments n=2 and n=3 of the Wilson twist-two operators at three loops in the RI′/SMOM scheme, Nucl. Phys. B961, 115229 (2020).
  127. P. Boucaud, F. de Soto, J. P. Leroy, A. Le Yaouanc, J. Micheli, H. Moutarde, O. Pène, and J. Rodriguez-Quintero, Artifacts and ⟨A2⟩ power corrections: Reexamining Zψ(p2) and ZV in the momentum-subtraction scheme, Phys. Rev. D 74, 034505 (2006).

Outline

Information

Phys. Rev. D 114, 074507– Published 14 October, 2026
  • Received 3 June 2026
  • Accepted 24 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