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Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to neural networks
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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 05 April 2019

Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to neural networks

  • Joseph Karpie1,2,
  • Kostas Orginos1,2,
  • Alexander Rothkopf3 &
  • …
  • Savvas Zafeiropoulos4 

Journal of High Energy Physics volume 2019, Article number: 57 (2019) Cite this article

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A preprint version of the article is available at arXiv.

Abstract

The computation of the parton distribution functions (PDF) or distribution amplitudes (DA) of hadrons from first principles lattice QCD constitutes a central open problem in high energy nuclear physics. In this study, we present and evaluate the efficiency of several numerical methods, well established in the study of inverse problems, to reconstruct the full x-dependence of PDFs. Our starting point are the so called Ioffe time PDFs, which are accessible from Euclidean time simulations in conjunction with a matching procedure. Using realistic mock data tests, we find that the ill-posed incomplete Fourier transform underlying the reconstruction requires careful regularization, for which both the Bayesian approach as well as neural networks are efficient and flexible choices.

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References

  1. R. Feynman, Photon-hadron interactions, Advanced Books Classics, Avalon Publishing (1998).

  2. K.-F. Liu, Parton degrees of freedom from the path integral formalism, Phys. Rev. D 62 (2000) 074501 [hep-ph/9910306] [INSPIRE].

  3. 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].

  4. V. Braun and D. Mueller, Exclusive processes in position space and the pion distribution amplitude, Eur. Phys. J. C 55 (2008) 349 [arXiv:0709.1348] [INSPIRE].

    Article  ADS  Google Scholar 

  5. X. Ji, Parton physics on a Euclidean lattice, Phys. Rev. Lett. 110 (2013) 262002 [arXiv:1305.1539] [INSPIRE].

    Article  ADS  Google Scholar 

  6. 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].

    ADS  Google Scholar 

  7. A.V. Radyushkin, Quasi-parton distribution functions, momentum distributions and pseudo-parton distribution functions, Phys. Rev. D 96 (2017) 034025 [arXiv:1705.01488] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  8. A.J. Chambers et al., Nucleon structure functions from operator product expansion on the lattice, Phys. Rev. Lett. 118 (2017) 242001 [arXiv:1703.01153] [INSPIRE].

    Article  ADS  Google Scholar 

  9. C. Best et al., Pion and rho structure functions from lattice QCD, Phys. Rev. D 56 (1997) 2743 [hep-lat/9703014] [INSPIRE].

  10. Zeuthen-Rome (ZeRo) collaboration, Non-perturbative pion matrix element of a twist-2 operator from the lattice, Eur. Phys. J. C 40 (2005) 69 [hep-lat/0405027] [INSPIRE].

  11. C. Alexandrou et al., Nucleon Spin and Momentum Decomposition Using Lattice QCD Simulations, Phys. Rev. Lett. 119 (2017) 142002 [arXiv:1706.02973] [INSPIRE].

    Article  ADS  Google Scholar 

  12. M. Oehm et al., 〈x〉 and 〈x 2〉 of the pion PDF from lattice QCD with N f = 2 + 1 + 1 dynamical quark flavors, Phys. Rev. D 99 (2019) 014508 [arXiv:1810.09743] [INSPIRE].

  13. 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].

    ADS  MathSciNet  Google Scholar 

  14. A. Radyushkin, One-loop evolution of parton pseudo-distribution functions on the lattice, Phys. Rev. D 98 (2018) 014019 [arXiv:1801.02427] [INSPIRE].

    ADS  Google Scholar 

  15. J.-H. Zhang, J.-W. Chen and C. Monahan, Parton distribution functions from reduced Ioffe-time distributions, Phys. Rev. D 97 (2018) 074508 [arXiv:1801.03023] [INSPIRE].

    ADS  Google Scholar 

  16. T. Izubuchi et al., Factorization theorem relating euclidean and light-cone parton distributions, Phys. Rev. D 98 (2018) 056004 [arXiv:1801.03917] [INSPIRE].

    ADS  Google Scholar 

  17. J. Karpie, K. Orginos, A. Radyushkin and S. Zafeiropoulos, Parton distribution functions on the lattice and in the continuum, EPJ Web Conf. 175 (2018) 06032 [arXiv:1710.08288] [INSPIRE].

    Article  Google Scholar 

  18. 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].

    ADS  Google Scholar 

  19. 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].

    ADS  MathSciNet  Google Scholar 

  20. 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].

    ADS  Google Scholar 

  21. C. Alexandrou et al., Lattice calculation of parton distributions, Phys. Rev. D 92 (2015) 014502 [arXiv:1504.07455] [INSPIRE].

    ADS  Google Scholar 

  22. J.-W. Chen et al., Nucleon helicity and transversity parton distributions from lattice QCD, Nucl. Phys. B 911 (2016) 246 [arXiv:1603.06664] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  23. C. Alexandrou et al., Light-cone parton distribution functions from lattice QCD, Phys. Rev. Lett. 121 (2018) 112001 [arXiv:1803.02685] [INSPIRE].

    Article  ADS  Google Scholar 

  24. J.-H. Zhang et al., Pion distribution amplitude from lattice QCD, Phys. Rev. D 95 (2017) 094514 [arXiv:1702.00008] [INSPIRE].

    ADS  Google Scholar 

  25. 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].

    Article  ADS  MATH  Google Scholar 

  26. A.V. Radyushkin, Structure of parton quasi-distributions and their moments, Phys. Lett. B 788 (2019) 380 [arXiv:1807.07509] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  27. J. Karpie, K. Orginos and S. Zafeiropoulos, Moments of Ioffe time parton distribution functions from non-local matrix elements, JHEP 11 (2018) 178 [arXiv:1807.10933] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. J.-W. Chen et al., Parton distribution function with nonperturbative renormalization from lattice QCD, Phys. Rev. D 97 (2018) 014505 [arXiv:1706.01295] [INSPIRE].

    ADS  Google Scholar 

  29. W. Broniowski and E. Ruiz Arriola, Partonic quasidistributions of the proton and pion from transverse-momentum distributions, Phys. Rev. D 97 (2018) 034031 [arXiv:1711.03377] [INSPIRE].

    ADS  Google Scholar 

  30. 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].

    Article  ADS  Google Scholar 

  31. K.-F. Liu and S.-J. Dong, Origin of difference between \( \overline{d} \) and Å« partons in the nucleon, Phys. Rev. Lett. 72 (1994) 1790 [hep-ph/9306299] [INSPIRE].

  32. G.S. Bali et al., Pion distribution amplitude from Euclidean correlation functions: exploring universality and higher-twist effects, Phys. Rev. D 98 (2018) 094507 [arXiv:1807.06671] [INSPIRE].

    ADS  Google Scholar 

  33. R.S. Sufian et al., Pion valence quark distribution from matrix element calculated in lattice QCD, arXiv:1901.03921 [INSPIRE].

  34. 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].

    Article  ADS  Google Scholar 

  35. K. Cichy and M. Constantinou, A guide to light-cone PDFs from lattice QCD: an overview of approaches, techniques and results, arXiv:1811.07248 [INSPIRE].

  36. C. Monahan, Recent developments in x-dependent structure calculations, PoS(LATTICE 2018)018 [arXiv:1811.00678] [INSPIRE].

  37. LP3 collaboration, Improved parton distribution functions at the physical pion mass, Phys. Rev. D 98 (2018) 054504 [arXiv:1708.05301] [INSPIRE].

  38. G. Backus and F. Gilbert, The resolving power of gross Earth data, Geophys. J. Int. 16 (1968) 169.

    Article  ADS  MATH  Google Scholar 

  39. S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes in C: the art of scientific computing, Cambridge University Press, Cambridge U.K. (1992).

    MATH  Google Scholar 

  40. B.B. Brandt, A. Francis, H.B. Meyer and D. Robaina, Pion quasiparticle in the low-temperature phase of QCD, Phys. Rev. D 92 (2015) 094510 [arXiv:1506.05732] [INSPIRE].

    ADS  Google Scholar 

  41. R.-A. Tripolt, P. Gubler, M. Ulybyshev and L. Von Smekal, Numerical analytic continuation of Euclidean data, Comput. Phys. Commun. 237 (2019) 129 [arXiv:1801.10348] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  42. J. Liang, K.-F. Liu and Y.-B. Yang, Lattice calculation of hadronic tensor of the nucleon, EPJ Web Conf. 175 (2018) 14014 [arXiv:1710.11145] [INSPIRE].

    Article  Google Scholar 

  43. M.V. Ulybyshev, C. Winterowd and S. Zafeiropoulos, Direct detection of metal-insulator phase transitions using the modified Backus-Gilbert method, EPJ Web Conf. 175 (2018) 03008 [arXiv:1710.06675] [INSPIRE].

    Article  Google Scholar 

  44. M. Ulybyshev, C. Winterowd and S. Zafeiropoulos, Collective charge excitations and the metal-insulator transition in the square lattice Hubbard-Coulomb model, Phys. Rev. B 96 (2017) 205115 [arXiv:1707.04212] [INSPIRE].

    Article  ADS  Google Scholar 

  45. S. Forte, L. Garrido, J.I. Latorre and A. Piccione, Neural network parametrization of deep inelastic structure functions, JHEP 05 (2002) 062 [hep-ph/0204232] [INSPIRE].

  46. NNPDF collaboration, Parton distributions from high-precision collider data, Eur. Phys. J. C 77 (2017) 663 [arXiv:1706.00428] [INSPIRE].

  47. NNPDF collaboration, Parton distributions for the LHC Run II, JHEP 04 (2015) 040 [arXiv:1410.8849] [INSPIRE].

  48. 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].

    Article  ADS  MATH  Google Scholar 

  49. NNPDF collaboration, Reweighting NNPDFs: the W lepton asymmetry, Nucl. Phys. B 849 (2011) 112 [Erratum ibid. B 854 (2012) 926] [arXiv:1012.0836] [INSPIRE].

  50. J. Rojo, Machine Learning tools for global PDF fits, talk given at the 13th Conference on Quark Confinement and the Hadron Spectrum (Confinement XIII), July 31–August 6, Maynooth, Ireland (2018), arXiv:1809.04392 [INSPIRE].

  51. J. Skilling and S.F. Gull, Bayesian maximum entropy image reconstruction, Lecture Notes — Monograph Series volume 20, Institute of Mathematical Statistics, Hayward, U.S.A. (1991).

  52. M. Asakawa, T. Hatsuda and Y. Nakahara, Maximum entropy analysis of the spectral functions in lattice QCD, Prog. Part. Nucl. Phys. 46 (2001) 459 [hep-lat/0011040] [INSPIRE].

  53. A. Rothkopf, Improved maximum entropy analysis with an extended search space, J. Comput. Phys. 238 (2013) 106 [arXiv:1110.6285] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  54. Y. Burnier and A. Rothkopf, Bayesian approach to spectral function reconstruction for euclidean quantum field theories, Phys. Rev. Lett. 111 (2013) 182003 [arXiv:1307.6106] [INSPIRE].

    Article  ADS  Google Scholar 

  55. A. Buckley et al., LHAPDF6: parton density access in the LHC precision era, Eur. Phys. J. C 75 (2015) 132 [arXiv:1412.7420] [INSPIRE].

    Article  ADS  Google Scholar 

  56. B. Carpenter et al., Stan: a probabilistic programming language, J. Stat. Softw. 76 (2017).

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Authors and Affiliations

  1. Department of Physics, The College of William & Mary, Williamsburg, VA, 23187, U.S.A.

    Joseph Karpie & Kostas Orginos

  2. Thomas Jefferson National Accelerator Facility, Newport News, VA, 23606, U.S.A.

    Joseph Karpie & Kostas Orginos

  3. Faculty of Science and Technology, University of Stavanger, 4021, Stavanger, Norway

    Alexander Rothkopf

  4. Institute for Theoretical Physics, Heidelberg University, Philosophenweg 12, 69120, Heidelberg, Germany

    Savvas Zafeiropoulos

Authors
  1. Joseph Karpie
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  2. Kostas Orginos
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  4. Savvas Zafeiropoulos
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Corresponding author

Correspondence to Savvas Zafeiropoulos.

Additional information

ArXiv ePrint: 1901.05408

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Cite this article

Karpie, J., Orginos, K., Rothkopf, A. et al. Reconstructing parton distribution functions from Ioffe time data: from Bayesian methods to neural networks. J. High Energ. Phys. 2019, 57 (2019). https://doi.org/10.1007/JHEP04(2019)057

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  • Received: 22 January 2019

  • Revised: 04 March 2019

  • Accepted: 14 March 2019

  • Published: 05 April 2019

  • DOI: https://doi.org/10.1007/JHEP04(2019)057

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Keywords

  • Lattice QCD
  • Lattice Quantum Field Theory
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