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Moments of parton distribution functions of the pion from lattice QCD using gradient flow

Anthony Francis1, Patrick Fritzsch2, Rohith Karur3,4, Jangho Kim5, Giovanni Pederiva6,7, Dimitra A. Pefkou3,4,*, Antonio Rago8, Andrea Shindler9,4,3,†, André Walker-Loud4,3 et al.

Savvas Zafeiropoulos10

  • *Contact author: dpefkou@berkeley.edu
  • †Contact author: shindler@physik.rwth-aachen.de
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Phys. Rev. D 113, 074520 – Published 29 April, 2026

DOI: https://doi.org/10.1103/vw4v-nyvw

Abstract

We present a nonperturbative determination of the pion valence parton distribution function (PDF) moment ratios ⟨xn−1⟩/⟨x⟩ up to n=6, using the gradient flow in lattice quantum chromodynamics (QCD). As a testing ground, we employ SU(3) isosymmetric gauge configurations generated by the OpenLat initiative with a pseudoscalar mass of mπ≃411  MeV. Our analysis uses four lattice spacings and a nonperturbatively improved action, enabling full control over the continuum extrapolation, and the limit of vanishing flow time, t→0. The flowed ratios exhibit O(a2) scaling across the ensembles, and the continuum-extrapolated results, matched to the MS¯ scheme at μ=2  GeV using next-to-next-to-leading order matching coefficients, show only mild residual flow-time dependence. The resulting ratios, computed with a relatively small number of configurations, are consistent with phenomenological expectations for the pion’s valence distribution, with statistical uncertainties that are competitive with modern global fits. These findings demonstrate that the gradient flow provides an efficient and systematically improvable method to access partonic quantities from first principles. Future extensions of this work will target lighter pion masses toward the physical point, and applications to nucleon structure such as the proton PDFs and the gluon and sea-quark distributions.

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Gradient Flow for Parton Distribution Functions: First Application to the Pion

Anthony Francis, Patrick Fritzsch, Robert V. Harlander, Rohith Karur, Jangho Kim, Jonas T. Kohnen, Giovanni Pederiva, Dimitra A. Pefkou, Antonio Rago, Andrea Shindler, André Walker-Loud, and Savvas Zafeiropoulos
Phys. Rev. Lett. 136, 171903 (2026)

Article Text

References (190)

  1. S. Amoroso et al., Snowmass 2021 whitepaper: Proton structure at the precision frontier, Acta Phys. Pol. B 53, 12 (2022).
  2. R. Abir et al., The case for an EIC theory alliance: Theoretical challenges of the EIC, arXiv:2305.14572.
  3. P. Agostini et al. (LHeC Collaboration and FCC-he Study Group), The large hadron–electron collider at the HL-LHC, J. Phys. G 48, 110501 (2021).
  4. G. Martinelli and C. T. Sachrajda, Pion structure functions from lattice QCD, Phys. Lett. B 196, 184 (1987).
  5. G. Martinelli and C. T. Sachrajda, A lattice calculation of the pion’s form-factor and structure function, Nucl. Phys. B306, 865 (1988).
  6. C. Alexandrou, S. Bacchio, I. Cloet, M. Constantinou, K. Hadjiyiannakou, G. Koutsou, and C. Lauer (ETM Collaboration), Mellin moments ⟨x⟩ and ⟨x2⟩ for the pion and kaon from lattice QCD, Phys. Rev. D 103, 014508 (2021).
  7. C. Alexandrou, S. Bacchio, I. Cloët, M. Constantinou, K. Hadjiyiannakou, G. Koutsou, and C. Lauer (ETM Collaboration), Pion and kaon ⟨x3⟩ from lattice QCD and PDF reconstruction from Mellin moments, Phys. Rev. D 104, 054504 (2021).
  8. U. Aglietti, M. Ciuchini, G. Corbo, E. Franco, G. Martinelli, and L. Silvestrini, Model independent determination of the shape function for inclusive B decays and of the structure functions in DIS, Phys. Lett. B 432, 411 (1998).
  9. W. Detmold and C. J. D. Lin, Deep-inelastic scattering and the operator product expansion in lattice QCD, Phys. Rev. D 73, 014501 (2006).
  10. X. Ji, Parton physics on a euclidean lattice, Phys. Rev. Lett. 110, 262002 (2013).
  11. A. V. Radyushkin, Quasi-parton distribution functions, momentum distributions, and nonparton distribution functions, Phys. Rev. D 96, 034025 (2017).
  12. Y.-Q. Ma and J.-W. Qiu, Extracting parton distribution functions from lattice QCD calculations, Phys. Rev. D 98, 074021 (2018).
  13. V. Braun and D. Müller, Exclusive processes in position space and the pion distribution amplitude, Eur. Phys. J. C 55, 349 (2008).
  14. A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, K. Somfleth, R. D. Young, and J. M. Zanotti, Nucleon structure functions from operator product expansion on the lattice, Phys. Rev. Lett. 118, 242001 (2017).
  15. H.-W. Lin, Calculating the x dependence of hadron parton distribution functions, Proc. Sci. LATTICE2013 (2014) 293.
  16. H.-W. Lin, J.-W. Chen, S. D. Cohen, and X. Ji, Flavor structure of the nucleon sea from lattice QCD, Phys. Rev. D 91, 054510 (2015).
  17. J.-W. Chen, S. D. Cohen, X. Ji, H.-W. Lin, and J.-H. Zhang, Nucleon helicity and transversity parton distributions from lattice QCD, Nucl. Phys. B911, 246 (2016).
  18. H.-W. Lin, J.-W. Chen, T. Ishikawa, and J.-H. Zhang (LP3 Collaboration), Improved parton distribution functions at the physical pion mass, Phys. Rev. D 98, 054504 (2018).
  19. C. Alexandrou, K. Cichy, V. Drach, E. Garcia-Ramos, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Lattice calculation of parton distributions, Phys. Rev. D 92, 014502 (2015).
  20. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, F. Steffens, and C. Wiese, Updated lattice results for parton distributions, Phys. Rev. D 96, 014513 (2017).
  21. C. Alexandrou, K. Cichy, M. Constantinou, K. Hadjiyiannakou, K. Jansen, H. Panagopoulos, and F. Steffens, A complete non-perturbative renormalization prescription for quasi-PDFs, Nucl. Phys. B923, 394 (2017).
  22. J.-W. Chen, T. Ishikawa, L. Jin, H.-W. Lin, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Parton distribution function with nonperturbative renormalization from lattice QCD, Phys. Rev. D 97, 014505 (2018).
  23. J.-H. Zhang, X. Ji, A. Schäfer, W. Wang, and S. Zhao, Accessing gluon parton distributions in large momentum effective theory, Phys. Rev. Lett. 122, 142001 (2019).
  24. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Light-cone parton distribution functions from lattice QCD, Phys. Rev. Lett. 121, 112001 (2018).
  25. J.-H. Zhang, J.-W. Chen, L. Jin, H.-W. Lin, A. Schäfer, and Y. Zhao, First direct lattice-QCD calculation of the x-dependence of the pion parton distribution function, Phys. Rev. D 100, 034505 (2019).
  26. C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato, and F. Steffens, Transversity parton distribution functions from lattice QCD, Phys. Rev. D 98, 091503 (2018).
  27. H.-W. Lin, J.-W. Chen, X. Ji, L. Jin, R. Li, Y.-S. Liu, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Proton isovector helicity distribution on the lattice at physical pion mass, Phys. Rev. Lett. 121, 242003 (2018).
  28. Z.-Y. Fan, Y.-B. Yang, A. Anthony, H.-W. Lin, and K.-F. Liu, Gluon quasi-parton-distribution functions from lattice QCD, Phys. Rev. Lett. 121, 242001 (2018).
  29. W. Wang, J.-H. Zhang, S. Zhao, and R. Zhu, Complete matching for quasidistribution functions in large momentum effective theory, Phys. Rev. D 100, 074509 (2019).
  30. H.-W. Lin and R. Zhang, Lattice finite-volume dependence of the nucleon parton distributions, Phys. Rev. D 100, 074502 (2019).
  31. J.-W. Chen, H.-W. Lin, and J.-H. Zhang, Pion generalized parton distribution from lattice QCD, Nucl. Phys. B952, 114940 (2020).
  32. Y. Chai et al., Parton distribution functions of Δ+ on the lattice, Phys. Rev. D 102, 014508 (2020).
  33. H.-W. Lin, J.-W. Chen, Z. Fan, J.-H. Zhang, and R. Zhang, Valence-quark distribution of the kaon and pion from lattice QCD, Phys. Rev. D 103, 014516 (2021).
  34. R. Zhang, H.-W. Lin, and B. Yoon, Probing nucleon strange and charm distributions with lattice QCD, Phys. Rev. D 104, 094511 (2021).
  35. Z.-Y. Li, Y.-Q. Ma, and J.-W. Qiu, Extraction of next-to-next-to-leading-order parton distribution functions from lattice QCD calculations, Phys. Rev. Lett. 126, 072001 (2021).
  36. 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).
  37. X. Gao, L. Jin, C. Kallidonis, N. Karthik, S. Mukherjee, P. Petreczky, C. Shugert, S. Syritsyn, and Y. Zhao, Valence parton distribution of the pion from lattice QCD: Approaching the continuum limit, Phys. Rev. D 102, 094513 (2020).
  38. K. Zhang, Y.-Y. Li, Y.-K. Huo, A. Schäfer, P. Sun, and Y.-B. Yang (χQCD Collaboration), RI/MOM renormalization of the parton quasidistribution functions in lattice regularization, Phys. Rev. D 104, 074501 (2021).
  39. C. Alexandrou, K. Cichy, M. Constantinou, J. R. Green, K. Hadjiyiannakou, K. Jansen, F. Manigrasso, A. Scapellato, and F. Steffens, Lattice continuum-limit study of nucleon quasi-PDFs, Phys. Rev. D 103, 094512 (2021).
  40. X. Gao, K. Lee, S. Mukherjee, C. Shugert, and Y. Zhao, Origin and resummation of threshold logarithms in the lattice QCD calculations of PDFs, Phys. Rev. D 103, 094504 (2021).
  41. M. Constantinou et al., Parton distributions and lattice-QCD calculations: Toward 3D structure, Prog. Part. Nucl. Phys. 121, 103908 (2021).
  42. 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: Formalism and Wilson coefficients, Phys. Rev. D 104, 074511 (2021).
  43. K.-F. Liu, Evolution equations for connected and disconnected sea parton distributions, Phys. Rev. D 96, 033001 (2017).
  44. K.-F. Liu, PDF in PDFs from hadronic tensor and LaMET, Phys. Rev. D 102, 074502 (2020).
  45. Y.-Q. Ma and J.-W. Qiu, Exploring partonic structure of hadrons using ab initio lattice QCD calculations, Phys. Rev. Lett. 120, 022003 (2018).
  46. G. S. Bali et al., Pion distribution amplitude from Euclidean correlation functions, Eur. Phys. J. C 78, 217 (2018).
  47. G. S. Bali, V. M. Braun, B. Gläßle, M. G”ockeler, M. Gruber, F. Hutzler, P. Korcyl, A. Sch”afer, P. Wein, and J.-H. Zhang, Pion distribution amplitude from Euclidean correlation functions: Exploring universality and higher-twist effects, Phys. Rev. D 98, 094507 (2018).
  48. B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, and S. Zafeiropoulos, Parton distribution functions from Ioffe time pseudodistributions from lattice calculations: Approaching the physical point, Phys. Rev. Lett. 125, 232003 (2020).
  49. R. S. Sufian, J. Karpie, C. Egerer, K. Orginos, J.-W. Qiu, and D. G. Richards, Pion valence quark distribution from matrix element calculated in lattice QCD, Phys. Rev. D 99, 074507 (2019).
  50. R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Joó, Y.-Q. Ma, K. Orginos, J.-W. Qiu, and D. G. Richards, Pion valence quark distribution from current-current correlation in lattice QCD, Phys. Rev. D 102, 054508 (2020).
  51. I. Balitsky, W. Morris, and A. Radyushkin, Gluon pseudo-distributions at short distances: Forward case, Phys. Lett. B 808, 135621 (2020).
  52. K. Orginos, A. Radyushkin, J. Karpie, and S. Zafeiropoulos, Lattice QCD exploration of parton pseudo-distribution functions, Phys. Rev. D 96, 094503 (2017).
  53. J. Karpie, K. Orginos, A. Radyushkin, and S. Zafeiropoulos, Parton distribution functions on the lattice and in the continuum, EPJ Web Conf. 175, 06032 (2018).
  54. J. Karpie, K. Orginos, and S. Zafeiropoulos, Moments of Ioffe time parton distribution functions from non-local matrix elements, J. High Energy Phys. 11 (2018) 178.
  55. J. Karpie, K. Orginos, A. Rothkopf, and S. Zafeiropoulos, Reconstructing parton distribution functions from Ioffe time data: From Bayesian methods to Neural Networks, J. High Energy Phys. 04 (2019) 057.
  56. B. Joó, J. Karpie, K. Orginos, A. Radyushkin, D. Richards, and S. Zafeiropoulos, Parton distribution functions from Ioffe time pseudo-distributions, J. High Energy Phys. 12 (2019) 081.
  57. B. Joó, J. Karpie, K. Orginos, A. V. Radyushkin, D. G. Richards, R. S. Sufian, and S. Zafeiropoulos, Pion valence structure from Ioffe-time parton pseudodistribution functions, Phys. Rev. D 100, 114512 (2019).
  58. A. Radyushkin, One-loop evolution of parton pseudo-distribution functions on the lattice, Phys. Rev. D 98, 014019 (2018).
  59. J.-H. Zhang, J.-W. Chen, and C. Monahan, Parton distribution functions from reduced Ioffe-time distributions, Phys. Rev. D 97, 074508 (2018).
  60. T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao, Factorization theorem relating euclidean and light-cone parton distributions, Phys. Rev. D 98, 056004 (2018).
  61. M. Bhat, K. Cichy, M. Constantinou, and A. Scapellato, Flavor nonsinglet parton distribution functions from lattice QCD at physical quark masses via the pseudodistribution approach, Phys. Rev. D 103, 034510 (2021).
  62. Z. Fan, R. Zhang, and H.-W. Lin, Nucleon gluon distribution function from 2+1+1-flavor lattice QCD, Int. J. Mod. Phys. A 36, 2150080 (2021).
  63. R. S. Sufian, T. Liu, and A. Paul, Gluon distributions and their applications to Ioffe-time distributions, Phys. Rev. D 103, 036007 (2021).
  64. N. Karthik, Quark distribution inside a pion in many-flavor (2+1)-dimensional QCD using lattice computations: UV listens to IR, Phys. Rev. D 103, 074512 (2021).
  65. T. Khan et al. (HadStruc Collaboration), Unpolarized gluon distribution in the nucleon from lattice quantum chromodynamics, Phys. Rev. D 104, 094516 (2021).
  66. Z. Fan and H.-W. Lin, Gluon parton distribution of the pion from lattice QCD, Phys. Lett. B 823, 136778 (2021).
  67. J. Delmar, C. Alexandrou, K. Cichy, M. Constantinou, and K. Hadjiyiannakou, Gluon PDF for the proton using the twisted mass formulation of lattice QCD, Proc. Sci. LATTICE2022 (2023) 099 [arXiv:2212.11399].
  68. A. Salas-Chavira, Z. Fan, and H.-W. Lin, First glimpse into the kaon gluon parton distribution using lattice QCD, Phys. Rev. D 106, 094510 (2022).
  69. J. Karpie, K. Orginos, A. Radyushkin, and S. Zafeiropoulos (HadStruc Collaboration), The continuum and leading twist limits of parton distribution functions in lattice QCD, J. High Energy Phys. 11 (2021) 024.
  70. X. Gao, A. D. Hanlon, N. Karthik, S. Mukherjee, P. Petreczky, P. Scior, S. Shi, S. Syritsyn, Y. Zhao, and K. Zhou, Continuum-extrapolated NNLO valence PDF of the pion at the physical point, Phys. Rev. D 106, 114510 (2022).
  71. X. Xiong, X. Ji, J.-H. Zhang, and Y. Zhao, One-loop matching for parton distributions: Nonsinglet case, Phys. Rev. D 90, 014051 (2014).
  72. X. Ji, Y.-S. Liu, Y. Liu, J.-H. Zhang, and Y. Zhao, Large-momentum effective theory, Rev. Mod. Phys. 93, 035005 (2021).
  73. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Lattice QCD determination of the Bjorken-x dependence of parton distribution functions at next-to-next-to-leading order, Phys. Rev. Lett. 128, 142003 (2022).
  74. L.-B. Chen, W. Wang, and R. Zhu, Next-to-next-to-leading order calculation of quasiparton distribution functions, Phys. Rev. Lett. 126, 072002 (2021).
  75. J. Karpie, R. M. Whitehill, W. Melnitchouk, C. Monahan, K. Orginos, J. W. Qiu, D. G. Richards, N. Sato, and S. Zafeiropoulos (Jefferson Lab Angular Momentum and HadStruc Collaborations), Gluon helicity from global analysis of experimental data and lattice QCD Ioffe time distributions, Phys. Rev. D 109, 036031 (2024).
  76. H. Dutrieux, J. Karpie, C. Monahan, K. Orginos, and S. Zafeiropoulos (HadStruc Collaboration), Evolution of parton distribution functions in the short-distance factorization scheme, J. High Energy Phys. 04 (2024) 061.
  77. 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).
  78. X. Gao, W.-Y. Liu, and Y. Zhao, Parton distributions from boosted fields in the Coulomb gauge, Phys. Rev. D 109, 094506 (2024).
  79. 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).
  80. X. Gao, A. D. Hanlon, S. Mukherjee, P. Petreczky, P. Scior, S. Syritsyn, and Y. Zhao, Lattice QCD determination of the Bjorken-x dependence of PDFs at NNLO, Proc. Sci. LATTICE2022 (2023) 104.
  81. J. Holligan and H.-W. Lin, Nucleon helicity parton distribution function in the continuum limit with self-renormalization, Phys. Lett. B 854, 138731 (2024).
  82. J. Holligan and H.-W. Lin, Pion valence quark distribution at physical pion mass of Nf=2+1+1 lattice QCD, J. Phys. G 51, 065101 (2024).
  83. W. Good, K. Hasan, A. Chevis, and H.-W. Lin, Gluon moment and parton distribution function of the pion from Nf=2+1+1 lattice QCD, Phys. Rev. D 109, 114509 (2024).
  84. Z. Fan, W. Good, and H.-W. Lin, Gluon parton distribution of the nucleon from (2+1+1)-flavor lattice QCD in the physical-continuum limit, Phys. Rev. D 108, 014508 (2023).
  85. J. Liang, T. Draper, K.-F. Liu, A. Rothkopf, and Y.-B. Yang (XQCD Collaboration), Towards the nucleon hadronic tensor from lattice QCD, Phys. Rev. D 101, 114503 (2020).
  86. C. Egerer et al. (HadStruc Collaboration), Transversity parton distribution function of the nucleon using the pseudodistribution approach, Phys. Rev. D 105, 034507 (2022).
  87. H. Dutrieux, R. G. Edwards, C. Egerer, J. Karpie, C. Monahan, K. Orginos, A. Radyushkin, D. Richards, E. Romero, and S. Zafeiropoulos (HadStruc Collaboration), Towards unpolarized GPDs from pseudo-distributions, J. High Energy Phys. 08 (2024) 162.
  88. C. Zimmermann and A. Schäfer, Valence quark PDFs of the proton from two-current correlations in lattice QCD, Phys. Rev. D 110, 074503 (2024).
  89. R. Narayanan and H. Neuberger, Infinite N phase transitions in continuum Wilson loop operators, J. High Energy Phys. 03 (2006) 064.
  90. M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, J. High Energy Phys. 08 (2010) 071.
  91. P. Weisz, Continuum Limit Improved Lattice Action for Pure Yang-Mills Theory. 1, Nucl. Phys. B 212, 1 (1983); M. Lüscher and P. Weisz, On-shell improved lattice gauge theories, Commun. Math. Phys. 98, 433 (1985).
  92. M. Lüscher, Chiral symmetry and the Yang–Mills gradient flow, J. High Energy Phys. 04 (2013) 123.
  93. A. Shindler, Moments of parton distribution functions of any order from lattice QCD, Phys. Rev. D 110, L051503 (2024).
  94. A. Francis et al., companion Letter, Gradient flow for parton distribution functions: First application to the pion, Phys. Rev. Lett. 136, 171903 (2026).
  95. A. Francis, P. Fritzsch, R. Karur, J. Kim, G. Pederiva, D. A. Pefkou, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Probing higher moments of pion parton distribution functions, Proc. Sci. LATTICE2024 (2025) 336 [arXiv:2412.01750].
  96. H. Suzuki, Energy–momentum tensor from the Yang–Mills gradient flow, Prog. Theor. Exp. Phys. 2013, 083B03 (2013); 2015, 079201(E) (2015).
  97. M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, Proc. Sci. LATTICE2013 (2014) 016 [arXiv:1308.5598].
  98. H. Makino and H. Suzuki, Lattice energy–momentum tensor from the Yang-Mills gradient flow–inclusion of fermion fields, Prog. Theor. Exp. Phys. 2014, 063B02 (2014).
  99. R. V. Harlander, Y. Kluth, and F. Lange, The two-loop energy–momentum tensor within the gradient-flow formalism, Eur. Phys. J. C 78, 944 (2018); 79, 858(E) (2019).
  100. J. Artz, R. V. Harlander, F. Lange, T. Neumann, and M. Prausa, Results and techniques for higher order calculations within the gradient-flow formalism, J. High Energy Phys. 06 (2019) 121; 10 (2019) 032(E).
  101. R. V. Harlander, J. T. Kohnen, and A. Shindler, Short-flow-time expansion of non-singlet twist-two operators at next-to-next-to-leading order QCD, arXiv:2511.17145.
  102. F. Cuteri, A. Francis, P. Fritzsch, G. Pederiva, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Benchmark continuum limit results for spectroscopy with stabilized wilson fermions, Proc. Sci. LATTICE2022 (2023) 074 [arXiv:2212.11048].
  103. F. Cuteri, A. S. Francis, P. Fritzsch, G. Pederiva, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Gauge generation and dissemination in OpenLat, Proc. Sci., LATTICE2022 (2023) 426 [arXiv:2212.07314].
  104. A. S. Francis, F. Cuteri, P. Fritzsch, G. Pederiva, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Properties, ensembles and hadron spectra with Stabilised Wilson Fermions, Proc. Sci. LATTICE2021 (2022) 118 [arXiv:2201.03874].
  105. A. Francis, F. Cuteri, P. Fritzsch, G. Pederiva, A. Rago, A. Shindler, A. Walker-Loud, and S. Zafeiropoulos, Progress in generating gauge ensembles with Stabilized Wilson Fermions, Proc. Sci. LATTICE2023 (2024) 048 [arXiv:2312.11298].
  106. A. Francis, P. Fritzsch, M. Lüscher, and A. Rago, Master-field simulations of O(a)-improved lattice QCD: Algorithms, stability and exactness, Comput. Phys. Commun. 255, 107355 (2020).
  107. G. Curci, P. Menotti, and G. Paffuti, Symanzik’s improved Lagrangian for lattice gauge theory, Phys. Lett. 130B, 205 (1983); 135B, 516(E) (1984).
  108. D. Brommel, M. Diehl, M. Gockeler, P. Hagler, R. Horsley, D. Pleiter, P. E. L. Rakow, A. Schafer, G. Schierholz, and J. M. Zanotti, Structure of the pion from full lattice QCD, Proc. Sci. LAT2005 (2006) 360 [arXiv:hep-lat/0509133].
  109. D. Brommel, Pion structure from the lattice, Ph.D. thesis, Regensburg University, 2007.
  110. G. Bali, S. Collins, B. Glässle, M. Göckeler, N. Javadi-Motaghi, J. Najjar, W. Söldner, and A. Sternbeck, Pion structure from lattice QCD, Proc. Sci. LATTICE2013 (2014) 447 [arXiv:1311.7639].
  111. 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).
  112. M. Oehm, C. Alexandrou, M. Constantinou, K. Jansen, G. Koutsou, B. Kostrzewa, F. Steffens, C. Urbach, and S. Zafeiropoulos, ⟨x⟩ and ⟨x2⟩ of the pion PDF from lattice QCD with Nf=2+1+1 dynamical quark flavors, Phys. Rev. D 99, 014508 (2019).
  113. M. Löffler, P. Wein, T. Wurm, S. Weishäupl, D. Jenkins, R. Rödl, A. Schäfer, and L. Walter (RQCD Collaboration), Mellin moments of spin dependent and independent PDFs of the pion and rho meson, Phys. Rev. D 105, 014505 (2022).
  114. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, arXiv:2411.04268.
  115. T. Blum et al. (RBC and UKQCD Collaborations), Domain wall QCD with physical quark masses, Phys. Rev. D 93, 074505 (2016).
  116. G. S. Bali, S. Collins, P. Georg, D. Jenkins, P. Korcyl, A. Schäfer, E. E. Scholz, J. Simeth, W. Söldner, and S. Weishäupl (RQCD Collaboration), Scale setting and the light baryon spectrum in Nf=2+1 QCD with Wilson fermions, J. High Energy Phys. 05 (2023) 035.
  117. M. Bruno, T. Korzec, and S. Schaefer, Setting the scale for the CLS 2+1 flavor ensembles, Phys. Rev. D 95, 074504 (2017).
  118. S. Borsányi, S. Dürr, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, T. Kurth, L. Lellouch, T. Lippert, and C. McNeile (BMW Collaboration), High-precision scale setting in lattice QCD, J. High Energy Phys. 09 (2012) 010.
  119. C. W. Bernard, T. Draper, G. Hockney, and A. Soni, Calculation of weak matrix elements: Some technical aspects, in Lattice Gauge Theory: A Challenge in Large-Scale Computing (Springer US, Boston, MA, 1985).
  120. H. Akaike, Information Theory and an Extension of the Maximum Likelihood Principle (Springer Science+Business Media, New York, 1998).
  121. W. I. Jay and E. T. Neil, Bayesian model averaging for analysis of lattice field theory results, Phys. Rev. D 103, 114502 (2021).
  122. A. Shindler, Chiral Ward identities, automatic O(a) improvement and the gradient flow, Nucl. Phys. B881, 71 (2014).
  123. J. F. Owens, Q2 dependent parametrizations of pion parton distribution functions, Phys. Rev. D 30, 943 (1984).
  124. P. Aurenche, R. Baier, M. Fontannaz, M. Kienzle-Focacci, and M. Werlen, The gluon content of the pion from high-pt photon production, Phys. Lett. B 233, 517 (1989).
  125. P. J. Sutton, A. D. Martin, R. G. Roberts, and W. J. Stirling, Parton distributions for the pion extracted from Drell-Yan and prompt photon experiments, Phys. Rev. D 45, 2349 (1992).
  126. M. Gluck, E. Reya, and A. Vogt, Pionic parton distributions, Z. Phys. C 53, 651 (1992).
  127. M. Gluck, E. Reya, and I. Schienbein, Pionic parton distributions revisited, Eur. Phys. J. C 10, 313 (1999).
  128. K. Wijesooriya, P. E. Reimer, and R. J. Holt, The pion parton distribution function in the valence region, Phys. Rev. C 72, 065203 (2005).
  129. M. Aicher, A. Schafer, and W. Vogelsang, Soft-gluon resummation and the valence parton distribution function of the pion, Phys. Rev. Lett. 105, 252003 (2010).
  130. P. C. Barry, N. Sato, W. Melnitchouk, and C.-R. Ji, First Monte Carlo global QCD analysis of pion parton distributions, Phys. Rev. Lett. 121, 152001 (2018).
  131. P. C. Barry, C.-R. Ji, N. Sato, and W. Melnitchouk (Jefferson Lab Angular Momentum (JAM) Collaboration), Global QCD analysis of pion parton distributions with threshold resummation, Phys. Rev. Lett. 127, 232001 (2021).
  132. I. Novikov et al., Parton distribution functions of the charged pion within the xFitter framework, Phys. Rev. D 102, 014040 (2020).
  133. L. Kotz, A. Courtoy, P. Nadolsky, F. Olness, and M. Ponce-Chavez, Analysis of parton distributions in a pion with Bézier parametrizations, Phys. Rev. D 109, 074027 (2024).
  134. L. Kotz, A. Courtoy, P. Nadolsky, and M. Ponce-Chavez, Fantômas: Epistemic and nuclear uncertainties for the parton distributions of the pion, arXiv:2505.13594.
  135. K. T. McDonald, Pion structure as observed in Fermilab experiment E-615, in 21st Rencontres de Moriond: Strong Interactions and Gauge Theories (Éditions Frontières, Gif-sur-Yvette (France), 1986), pp. 179–186.
  136. M. Werlen, Direct photon production in WA70, in 23rd Rencontres de Moriond: Current Issues in Hadron Physics (Éditions Frontières, Gif-sur-Yvette (France), 1988), pp. 289–294.
  137. A. Brandenburg, S. J. Brodsky, V. V. Khoze, and D. Mueller, Angular distributions in the Drell-Yan process: A closer look at higher twist effects, Phys. Rev. Lett. 73, 939 (1994).
  138. 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).
  139. R. D. Ball, Resummation of hadroproduction cross-sections at high energy, Nucl. Phys. B796, 137 (2008).
  140. V. John, I. Angelov, A. Öncül, and D. Thévenin, Techniques for the reconstruction of a distribution from a finite number of its moments, Chem. Eng. Sci. 62, 2890 (2007).
  141. L. de Souza, G. Janiga, V. John, and D. Thévenin, Reconstruction of a distribution from a finite number of moments with an adaptive spline-based algorithm, Chem. Eng. Sci. 65, 2741 (2010).
  142. G. Athanassoulis and P. Gavriliadis, The truncated hausdorff moment problem solved by using kernel density functions, Prob. Eng. Mech. 17, 273 (2002).
  143. P. Biswas and A. K. Bhattacharya, Function reconstruction as a classical moment problem: A maximum entropy approach, J. Phys. A 43, 405003 (2010).
  144. A. Tagliani, Hausdorff moment problem and maximum entropy: A unified approach, Appl. Math. Comput. 105, 291 (1999).
  145. S. Zhang, X. Wang, T. Lin, and L. Chang, Reconstructing parton distribution function based on maximum entropy method, Chin. Phys. C 48, 033106 (2024).
  146. Z. F. Ezawa, Wide-angle scattering in softened field theory, Nuovo Cimento Soc. Ital. Fis. 23A, 271 (1974).
  147. P. V. Landshoff and J. C. Polkinghorne, Threshold properties of electroproduction and annihilation, Nucl. Phys. B53, 473 (1973).
  148. J. F. Gunion, S. J. Brodsky, and R. Blankenbecler, Large angle scattering and the interchange force, Phys. Rev. D 8, 287 (1973).
  149. G. R. Farrar and D. R. Jackson, The pion form-factor, Phys. Rev. Lett. 43, 246 (1979).
  150. E. L. Berger and S. J. Brodsky, Quark structure functions of mesons and the Drell-Yan process, Phys. Rev. Lett. 42, 940 (1979).
  151. T. Shigetani, K. Suzuki, and H. Toki, Pion structure function in the Nambu and Jona-Lasinio model, Phys. Lett. B 308, 383 (1993).
  152. A. Szczepaniak, C.-R. Ji, and S. R. Cotanch, Generalized relativistic meson wave function, Phys. Rev. D 49, 3466 (1994).
  153. R. M. Davidson and E. Ruiz Arriola, Structure functions of pseudoscalar mesons in the SU(3) NJL model, Phys. Lett. B 348, 163 (1995).
  154. M. B. Hecht, C. D. Roberts, and S. M. Schmidt, Valence quark distributions in the pion, Phys. Rev. C 63, 025213 (2001).
  155. W. Melnitchouk, Quark hadron duality in electron pion scattering, Eur. Phys. J. A 17, 223 (2003).
  156. S. Noguera and S. Scopetta, Pion transverse momentum dependent parton distributions in the Nambu and Jona-Lasinio model, J. High Energy Phys. 11 (2015) 102.
  157. P. T. P. Hutauruk, I. C. Cloet, and A. W. Thomas, Flavor dependence of the pion and kaon form factors and parton distribution functions, Phys. Rev. C 94, 035201 (2016).
  158. T. J. Hobbs, Quantifying finite-momentum effects in the quark quasidistribution functions of mesons, Phys. Rev. D 97, 054028 (2018).
  159. G. F. de Teramond, T. Liu, R. S. Sufian, H. G. Dosch, S. J. Brodsky, and A. Deur (HLFHS Collaboration), Universality of generalized parton distributions in light-front holographic QCD, Phys. Rev. Lett. 120, 182001 (2018).
  160. K. D. Bednar, I. C. Cloët, and P. C. Tandy, Distinguishing quarks and gluons in pion and kaon parton distribution functions, Phys. Rev. Lett. 124, 042002 (2020).
  161. J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Parton distribution functions from a light front Hamiltonian and QCD evolution for light mesons, Phys. Rev. Lett. 122, 172001 (2019).
  162. J. Lan, C. Mondal, S. Jia, X. Zhao, and J. P. Vary, Pion and kaon parton distribution functions from basis light front quantization and QCD evolution, Phys. Rev. D 101, 034024 (2020).
  163. L. Chang, K. Raya, and X. Wang, Pion parton distribution function in light-front holographic QCD, Chin. Phys. C 44, 114105 (2020).
  164. Z.-F. Cui, M. Ding, F. Gao, K. Raya, D. Binosi, L. Chang, C. D. Roberts, J. Rodríguez-Quintero, and S. M. Schmidt, Kaon and pion parton distributions, Eur. Phys. J. C 80, 1064 (2020).
  165. A. Kock, Y. Liu, and I. Zahed, Pion and kaon parton distributions in the QCD instanton vacuum, Phys. Rev. D 102, 014039 (2020).
  166. Z. F. Cui, M. Ding, J. M. Morgado, K. Raya, D. Binosi, L. Chang, F. De Soto, C. D. Roberts, J. Rodríguez-Quintero, and S. M. Schmidt, Emergence of pion parton distributions, Phys. Rev. D 105, L091502 (2022).
  167. L. Albino, I. M. Higuera-Angulo, K. Raya, and A. Bashir, Pseudoscalar mesons: Light front wave functions, GPDs, and PDFs, Phys. Rev. D 106, 034003 (2022).
  168. M. Ahmady, S. Kaur, C. Mondal, and R. Sandapen, Pion spectroscopy and dynamics using the holographic light-front Schrödinger equation and the ’t Hooft equation, Phys. Lett. B 836, 137628 (2023).
  169. B. Pasquini, S. Rodini, and S. Venturini (MAP (Multi-dimensional Analyses of Partonic distributions) Collaboration), Valence quark, sea, and gluon content of the pion from the parton distribution functions and the electromagnetic form factor, Phys. Rev. D 107, 114023 (2023).
  170. Y. Lu, Y.-Z. Xu, K. Raya, C. D. Roberts, and J. Rodríguez-Quintero, Pion distribution functions from low-order Mellin moments, Phys. Lett. B 850, 138534 (2024).
  171. C. Alexandrou, G. Iannelli, K. Jansen, and F. Manigrasso (Extended Twisted Mass Collaboration), Parton distribution functions from lattice QCD using Bayes-Gauss-Fourier transforms, Phys. Rev. D 102, 094508 (2020).
  172. A. Candido, L. Del Debbio, T. Giani, and G. Petrillo, Bayesian inference with Gaussian processes for the determination of parton distribution functions, Eur. Phys. J. C 84, 716 (2024).
  173. H. Dutrieux, J. Karpie, K. Orginos, and S. Zafeiropoulos, Simple nonparametric reconstruction of parton distributions from limited Fourier information, Phys. Rev. D 111, 034515 (2025).
  174. Y. C. Medrano, H. Dutrieux, J. Karpie, K. Orginos, and S. Zafeiropoulos, Gaussian processes for inferring parton distributions, arXiv:2510.21041.
  175. P. C. Barry et al. (Jefferson Lab Angular Momentum (JAM) and HadStruc Collaborations), Complementarity of experimental and lattice QCD data on pion parton distributions, Phys. Rev. D 105, 114051 (2022).
  176. P. C. Barry, C.-R. Ji, W. Melnitchouk, N. Sato, and F. Steffens (JAM Collaboration), First simultaneous global QCD analysis of kaon and pion parton distributions with lattice QCD constraints, arXiv:2510.11979.
  177. M. Lüscher, Code available at, https://luscher.web.cern.ch/luscher/openQCD/.
  178. R. G. Edwards and B. Joó (SciDAC Collaboration, LHPC Collaboration, and UKQCD Collaboration Collaboration), The chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
  179. M. Clark, R. Babich, K. Barros, R. Brower, and C. Rebbi, Solving lattice QCD systems of equations using mixed precision solvers on GPUs, Comput. Phys. Commun. 181, 1517 (2010).
  180. R. Babich, M. Clark, B. Joó, G. Shi, R. Brower, and S. Gottlieb, Scaling lattice QCD beyond 100 GPUs, in SC11 International Conference for High Performance Computing, Networking, Storage and Analysis (Association for Computing Machinery (ACM), New York, NY, USA, 2011); arXiv:1109.2935.
  181. M. A. Clark, B. Joó, A. Strelchenko, M. Cheng, A. Gambhir, and R. C. Brower (QUDA Collaboration), Accelerating lattice QCD multigrid on GPUs using fine-grained parallelization, in International Conference for High Performance Computing, Networking, Storage and Analysis (IEEE, Piscataway, NJ, USA, 2016); arXiv:1612.07873.
  182. F. T. Winter, M. A. Clark, R. G. Edwards, and B. Joó, A framework for lattice QCD calculations on GPUs, in 28th IEEE International Parallel and Distributed Processing Symposium (IEEE, Piscataway, NJ, USA, 2014); arXiv:1408.5925.
  183. A. Gambhir, D. Brantley, J. Chang, B. Hörz, H. Monge-Camacho, P. Vranas, and A. Walker-Loud, lalibe, https://github.com/callat-qcd/lalibe (2018).
  184. C. R. Harris et al., Array programming with numpy, Nature (London) 585, 357 (2020).
  185. P. Virtanen et al. (SciPy 1.0 Contributors), scipy 1.0: Fundamental algorithms for scientific computing in python, Nat. Methods 17, 261 (2020).
  186. G. P. Lepage, gvar v. 11.9.1, 10.5281/zenodo.4290884 (2020).
  187. G. P. Lepage, lsqfit v. 11.7, 10.5281/zenodo.4037174 (2020).
  188. J. D. Hunter, matplotlib: A 2d graphics environment, Comput. Sci. Eng. 9, 90 (2007).
  189. www.gauss-centre.eu.
  190. Jülich Supercomputing Centre, Juwels cluster and booster: Exascale pathfinder with modular supercomputing architecture at Juelich supercomputing centre, J. Large-Scale Res. Facil. 7, A183 (2021).

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Phys. Rev. D 113, 074520– Published 29 April, 2026
  • Received 13 November 2025
  • Accepted 23 March 2026
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