Journal of High Energy Physics

, 2019:137 | Cite as

Parton distributions from lattice data: the nonsinglet case

  • Krzysztof Cichy
  • Luigi Del Debbio
  • Tommaso GianiEmail author
Open Access
Regular Article - Theoretical Physics


We revise the relation between Parton Distribution Functions (PDFs) and matrix elements computable from lattice QCD, focusing on the quasi-Parton Distribution Functions (qPDFs) approach. We exploit the relation between PDFs and qPDFs in the case of the unpolarized isovector parton distribution to obtain a factorization formula relating the real and imaginary part of qPDFs matrix elements to specific nonsinglet distributions, and we propose a general framework to extract PDFs from the available lattice data, treating them on the same footing as experimental data. We implement the proposed approach within the NNPDF framework, and we study the potentiality of such lattice data in constraining PDFs, assuming some plausible scenarios to assess the unknown systematic uncertainties. We finally extract the two nonsinglet distributions involved in our analysis from a selection of the available lattice data.


Lattice field theory simulation QCD Phenomenology 


Open Access

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  1. [1]
    NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J.C 77 (2017) 663 [arXiv:1706.00428] [INSPIRE].
  2. [2]
    S. Dulat et al., New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev.D 93 (2016) 033006 [arXiv:1506.07443] [INSPIRE].
  3. [3]
    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 (2017) 014011 [arXiv:1701.05838] [INSPIRE].
  4. [4]
    A.D. Martin, W.J. Stirling, R.S. Thorne and G. Watt, Parton distributions for the LHC, Eur. Phys. J.C 63 (2009) 189 [arXiv:0901.0002] [INSPIRE].
  5. [5]
    A. Buckley et al., LHAPDF6: parton density access in the LHC precision era, Eur. Phys. J.C 75 (2015) 132 [arXiv:1412.7420] [INSPIRE].
  6. [6]
    X. Ji, Parton Physics on a Euclidean Lattice, Phys. Rev. Lett.110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].
  7. [7]
    X. Ji, Parton Physics from Large-Momentum Effective Field Theory, Sci. China Phys. Mech. Astron.57 (2014) 1407 [arXiv:1404.6680] [INSPIRE].
  8. [8]
    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 (2019) 3036904 [arXiv:1811.07248] [INSPIRE].
  9. [9]
    C. Monahan, Recent Developments in x-dependent Structure Calculations, PoS(LATTICE2018)018 (2018) [arXiv:1811.00678] [INSPIRE].
  10. [10]
    X. Ji and J.-H. Zhang, Renormalization of quasiparton distribution, Phys. Rev.D 92 (2015) 034006 [arXiv:1505.07699] [INSPIRE].
  11. [11]
    T. Ishikawa, Y.-Q. Ma, J.-W. Qiu and S. Yoshida, Practical quasi parton distribution functions, arXiv:1609.02018 [INSPIRE].
  12. [12]
    M. Constantinou and H. Panagopoulos, Perturbative renormalization of quasi-parton distribution functions, Phys. Rev.D 96 (2017) 054506 [arXiv:1705.11193] [INSPIRE].
  13. [13]
    C. Alexandrou et al., A complete non-perturbative renormalization prescription for quasi-PDFs, Nucl. Phys.B 923 (2017) 394 [arXiv:1706.00265] [INSPIRE].
  14. [14]
    X. Ji, J.-H. Zhang and Y. Zhao, More On Large-Momentum Effective Theory Approach to Parton Physics, Nucl. Phys.B 924 (2017) 366 [arXiv:1706.07416] [INSPIRE].
  15. [15]
    X. Ji, J.-H. Zhang and Y. Zhao, Renormalization in Large Momentum Effective Theory of Parton Physics, Phys. Rev. Lett.120 (2018) 112001 [arXiv:1706.08962] [INSPIRE].
  16. [16]
    T. Ishikawa, Y.-Q. Ma, J.-W. Qiu and S. Yoshida, Renormalizability of quasiparton distribution functions, Phys. Rev.D 96 (2017) 094019 [arXiv:1707.03107] [INSPIRE].
  17. [17]
    J. Green, K. Jansen and F. Steffens, Nonperturbative Renormalization of Nonlocal Quark Bilinears for Parton Quasidistribution Functions on the Lattice Using an Auxiliary Field, Phys. Rev. Lett.121 (2018) 022004 [arXiv:1707.07152] [INSPIRE].
  18. [18]
    A.V. Radyushkin, Structure of parton quasi-distributions and their moments, Phys. Lett.B 788 (2019) 380 [arXiv:1807.07509] [INSPIRE].
  19. [19]
    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 (2019) 142001 [arXiv:1808.10824] [INSPIRE].
  20. [20]
    Z.-Y. Li, Y.-Q. Ma and J.-W. Qiu, Multiplicative Renormalizability of Operators defining Quasiparton Distributions, Phys. Rev. Lett.122 (2019) 062002 [arXiv:1809.01836] [INSPIRE].
  21. [21]
    X. Xiong, X. Ji, J.-H. Zhang and Y. Zhao, One-loop matching for parton distributions: Nonsinglet case, Phys. Rev.D 90 (2014) 014051 [arXiv:1310.7471] [INSPIRE].
  22. [22]
    Y.-Q. Ma and J.-W. Qiu, Extracting Parton Distribution Functions from Lattice QCD Calculations, Phys. Rev.D 98 (2018) 074021 [arXiv:1404.6860] [INSPIRE].
  23. [23]
    R.A. Briceño, M.T. Hansen and C.J. Monahan, Role of the Euclidean signature in lattice calculations of quasidistributions and other nonlocal matrix elements, Phys. Rev.D 96 (2017) 014502 [arXiv:1703.06072] [INSPIRE].
  24. [24]
    Y.-Q. Ma and J.-W. Qiu, Exploring Partonic Structure of Hadrons Using ab initio Lattice QCD Calculations, Phys. Rev. Lett.120 (2018) 022003 [arXiv:1709.03018] [INSPIRE].
  25. [25]
    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 (2018) 056004 [arXiv:1801.03917] [INSPIRE].
  26. [26]
    X. Ji, A. Schäfer, X. Xiong and J.-H. Zhang, One-Loop Matching for Generalized Parton Distributions, Phys. Rev.D 92 (2015) 014039 [arXiv:1506.00248] [INSPIRE].
  27. [27]
    X. Xiong and J.-H. Zhang, One-loop matching for transversity generalized parton distribution, Phys. Rev.D 92 (2015) 054037 [arXiv:1509.08016] [INSPIRE].
  28. [28]
    W. Wang, S. Zhao and R. Zhu, Gluon quasidistribution function at one loop, Eur. Phys. J.C 78 (2018) 147 [arXiv:1708.02458] [INSPIRE].
  29. [29]
    I.W. Stewart and Y. Zhao, Matching the quasiparton distribution in a momentum subtraction scheme, Phys. Rev.D 97 (2018) 054512 [arXiv:1709.04933] [INSPIRE].
  30. [30]
    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 (2018) 112001 [arXiv:1803.02685] [INSPIRE].
  31. [31]
    C. Alexandrou, K. Cichy, M. Constantinou, K. Jansen, A. Scapellato and F. Steffens, Transversity parton distribution functions from lattice QCD, Phys. Rev.D 98 (2018) 091503 [arXiv:1807.00232] [INSPIRE].
  32. [32]
    Y.-S. Liu et al., Unpolarized quark distribution from lattice QCD: A systematic analysis of renormalization and matching, arXiv:1807.06566 [INSPIRE].
  33. [33]
    Y.-S. Liu et al., Nucleon Transversity Distribution at the Physical Pion Mass from Lattice QCD, arXiv:1810.05043 [INSPIRE].
  34. [34]
    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 (2015) 054510 [arXiv:1402.1462] [INSPIRE].
  35. [35]
    C. Alexandrou et al., Lattice calculation of parton distributions, Phys. Rev.D 92 (2015) 014502 [arXiv:1504.07455] [INSPIRE].
  36. [36]
    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.B 911 (2016) 246 [arXiv:1603.06664] [INSPIRE].
  37. [37]
    C. Alexandrou et al., Updated Lattice Results for Parton Distributions, Phys. Rev.D 96 (2017) 014513 [arXiv:1610.03689] [INSPIRE].
  38. [38]
    J.-H. Zhang, J.-W. Chen, X. Ji, L. Jin and H.-W. Lin, Pion Distribution Amplitude from Lattice QCD, Phys. Rev.D 95 (2017) 094514 [arXiv:1702.00008] [INSPIRE].
  39. [39]
    LP3 collaboration, Improved parton distribution functions at the physical pion mass, Phys. Rev.D 98 (2018) 054504 [arXiv:1708.05301] [INSPIRE].
  40. [40]
    LP3 collaboration, Kaon Distribution Amplitude from Lattice QCD and the Flavor SU (3) Symmetry, Nucl. Phys.B 939 (2019) 429 [arXiv:1712.10025] [INSPIRE].
  41. [41]
    J.-W. Chen et al., Lattice Calculation of Parton Distribution Function from LaMET at Physical Pion Mass with Large Nucleon Momentum, arXiv:1803.04393 [INSPIRE].
  42. [42]
    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 (2019) 034505 [arXiv:1804.01483] [INSPIRE].
  43. [43]
    H.-W. Lin et al., Proton Isovector Helicity Distribution on the Lattice at Physical Pion Mass, Phys. Rev. Lett.121 (2018) 242003 [arXiv:1807.07431] [INSPIRE].
  44. [44]
    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 (2018) 242001 [arXiv:1808.02077] [INSPIRE].
  45. [45]
    C. Alexandrou et al., Systematic uncertainties in parton distribution functions from lattice QCD simulations at the physical point, Phys. Rev.D 99 (2019) 114504 [arXiv:1902.00587] [INSPIRE].
  46. [46]
    T. Izubuchi et al., Valence parton distribution function of pion from fine lattice, Phys. Rev.D 100 (2019) 034516 [arXiv:1905.06349] [INSPIRE].
  47. [47]
    J. Karpie, K. Orginos, A. Rothkopf and S. Zafeiropoulos, Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to Neural Networks, JHEP04 (2019) 057 [arXiv:1901.05408] [INSPIRE].
  48. [48]
    A.V. Radyushkin, Quasi-parton distribution functions, momentum distributions and pseudo-parton distribution functions, Phys. Rev.D 96 (2017) 034025 [arXiv:1705.01488] [INSPIRE].
  49. [49]
    K. Orginos, A. Radyushkin, J. Karpie and S. Zafeiropoulos, Lattice QCD exploration of parton pseudo-distribution functions, Phys. Rev.D 96 (2017) 094503 [arXiv:1706.05373] [INSPIRE].
  50. [50]
    J. Karpie, K. Orginos and S. Zafeiropoulos, Moments of Ioffe time parton distribution functions from non-local matrix elements, JHEP11 (2018) 178 [arXiv:1807.10933] [INSPIRE].
  51. [51]
    W. Detmold and C.J.D. Lin, Deep-inelastic scattering and the operator product expansion in lattice QCD, Phys. Rev.D 73 (2006) 014501 [hep-lat/0507007] [INSPIRE].
  52. [52]
    V. Braun and D. Müller, Exclusive processes in position space and the pion distribution amplitude, Eur. Phys. J.C 55 (2008) 349 [arXiv:0709.1348] [INSPIRE].
  53. [53]
    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 (2019) 074507 [arXiv:1901.03921] [INSPIRE].
  54. [54]
    J.C. Collins, Intrinsic transverse momentum. 1. Nongauge theories, Phys. Rev.D 21 (1980) 2962 [INSPIRE].
  55. [55]
    J.C. Collins and D.E. Soper, Parton Distribution and Decay Functions, Nucl. Phys.B 194 (1982) 445 [INSPIRE].
  56. [56]
    A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS, Comput. Phys. Commun.170 (2005) 65 [hep-ph/0408244] [INSPIRE].
  57. [57]
    J. Gao, L. Harland-Lang and J. Rojo, The Structure of the Proton in the LHC Precision Era, Phys. Rept.742 (2018) 1 [arXiv:1709.04922] [INSPIRE].
  58. [58]
    H.-W. Lin et al., Parton distributions and lattice QCD calculations: a community white paper, Prog. Part. Nucl. Phys.100 (2018) 107 [arXiv:1711.07916] [INSPIRE].
  59. [59]
    V. Braun, P. Gornicki and L. Mankiewicz, Ioffe-time distributions instead of parton momentum distributions in description of deep inelastic scattering, Phys. Rev.D 51 (1995) 6036 [hep-ph/9410318] [INSPIRE].
  60. [60]
    G. Martinelli, C. Pittori, C.T. Sachrajda, M. Testa and A. Vladikas, A General method for nonperturbative renormalization of lattice operators, Nucl. Phys.B 445 (1995) 81 [hep-lat/9411010] [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    A. Radyushkin, Nonperturbative Evolution of Parton Quasi-Distributions, Phys. Lett.B 767 (2017) 314 [arXiv:1612.05170] [INSPIRE].
  62. [62]
    ETM collaboration, First physics results at the physical pion mass from Nf = 2 Wilson twisted mass fermions at maximal twist, Phys. Rev.D 95 (2017) 094515 [arXiv:1507.05068] [INSPIRE].
  63. [63]
    C. Alexandrou and C. Kallidonis, Low-lying baryon masses using Nf = 2 twisted mass clover-improved fermions directly at the physical pion mass, Phys. Rev.D 96 (2017) 034511 [arXiv:1704.02647] [INSPIRE].
  64. [64]
    NNPDF collaboration, A First Determination of Parton Distributions with Theoretical Uncertainties, arXiv:1905.04311 [INSPIRE].
  65. [65]
    NNPDF collaboration, Parton Distributions with Theory Uncertainties: General Formalism and First Phenomenological Studies, arXiv:1906.10698 [INSPIRE].
  66. [66]
    S. Syritsyn, Review of Hadron Structure Calculations on a Lattice, PoS(LATTICE2013)009 (2014) [arXiv:1403.4686] [INSPIRE].
  67. [67]
    M. Constantinou, Hadron Structure, PoS(LATTICE2014)001 (2015) [arXiv:1411.0078] [INSPIRE].
  68. [68]
    M. Constantinou, Recent progress in hadron structure from Lattice QCD, PoS(CD15)009 (2015) [arXiv:1511.00214] [INSPIRE].
  69. [69]
    C. Alexandrou, Selected results on hadron structure using state-of-the-art lattice QCD simulations, in Proceedings, 45th International Symposium on Multiparticle Dynamics (ISMD 2015), Kreuth, Germany, 4–9 October 2015 (2015) [arXiv:1512.03924] [INSPIRE].
  70. [70]
    J. Green, Systematics in nucleon matrix element calculations, PoS(LATTICE2018)016 (2018) [arXiv:1812.10574] [INSPIRE].
  71. [71]
    R.A. Briceño, J.V. Guerrero, M.T. Hansen and C.J. Monahan, Finite-volume effects due to spatially nonlocal operators, Phys. Rev.D 98 (2018) 014511 [arXiv:1805.01034] [INSPIRE].
  72. [72]
    V.M. Braun, A. Vladimirov and J.-H. Zhang, Power corrections and renormalons in parton quasidistributions, Phys. Rev.D 99 (2019) 014013 [arXiv:1810.00048] [INSPIRE].
  73. [73]
    G.S. Bali et al., Pion distribution amplitude from Euclidean correlation functions, Eur. Phys. J.C 78 (2018) 217 [arXiv:1709.04325] [INSPIRE].
  74. [74]
    NNPDF collaboration, Neural network determination of parton distributions: The Nonsinglet case, JHEP03 (2007) 039 [hep-ph/0701127] [INSPIRE].
  75. [75]
    NNPDF collaboration, A Determination of parton distributions with faithful uncertainty estimation, Nucl. Phys.B 809 (2009) 1 [Erratum ibid.B 816 (2009) 293] [arXiv:0808.1231] [INSPIRE].
  76. [76]
    R.D. Ball et al., A first unbiased global NLO determination of parton distributions and their uncertainties, Nucl. Phys.B 838 (2010) 136 [arXiv:1002.4407] [INSPIRE].
  77. [77]
    N. Hansen, The CMA evolution strategy: A tutorial, arXiv:1604.00772.
  78. [78]
    NNPDF collaboration, A determination of the fragmentation functions of pions, kaons and protons with faithful uncertainties, Eur. Phys. J.C 77 (2017) 516 [arXiv:1706.07049] [INSPIRE].
  79. [79]
    V. Bertone, S. Carrazza and N.P. Hartland, APFELgrid: a high performance tool for parton density determinations, Comput. Phys. Commun.212 (2017) 205 [arXiv:1605.02070] [INSPIRE].
  80. [80]
    Z. Kassabov, Reportengine: A framework for declarative data analysis, (2019).
  81. [81]
    NNPDF collaboration, Parton distributions for the LHC Run II, JHEP04 (2015) 040 [arXiv:1410.8849] [INSPIRE].

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© The Author(s) 2019

Authors and Affiliations

  1. 1.Faculty of PhysicsAdam Mickiewicz UniversityPozna’nPoland
  2. 2.The Higgs Centre for Theoretical PhysicsThe University of EdinburghEdinburghUnited Kingdom

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