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Gross-Llewellyn Smith sum rule from lattice QCD

K. U. Can1, J. A. Crawford1, R. Horsley2, P. E. L. Rakow3, T. G. Schar1, G. Schierholz4, H. Stüben5, R. D. Young1, J. M. Zanotti1 et al. (QCDSF Collaboration)

J. M. Zanotti1 (QCDSF Collaboration)

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Phys. Rev. D 111, 114505 – Published 10 June, 2025

DOI: https://doi.org/10.1103/jb3r-lmvz

Abstract

We compute the Gross-Llewellyn Smith sum rule, i.e., the lowest odd moment of the parity-violating structure function, F3, of the nucleon from a lattice QCD calculation of the Compton amplitude. Our calculations are performed on 483×96 lattices at the SU(3) symmetric point for two lattice spacings. We extract the moments for several values of the current momenta in the range 0.5≲Q2≲10  GeV2, covering both the nonperturbative and perturbative regimes. We compare our moments to the Gross-Llewellyn Smith sum rule and discuss the implications for higher-twist effects, a determination of αs(Q2) from a hadronic quantity complementing the phenomenological and other lattice approaches, and electroweak box contributions crucial for Cabibbo–Kobayashi–Maskawa matrix unitarity studies.

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References (72)

  1. D. J. Gross and C. H. Llewellyn Smith, High-energy neutrino—nucleon scattering, current algebra and partons, Nucl. Phys. B14, 337 (1969).
  2. A. Retey and J. A. M. Vermaseren, Some higher moments of deep inelastic structure functions at next-to-next-to-leading order of perturbative QCD, Nucl. Phys. B604, 281 (2001).
  3. S. A. Larin and J. A. M. Vermaseren, The αs3 corrections to the Bjorken sum rule for polarized electroproduction and to the Gross-Llewellyn Smith sum rule, Phys. Lett. B 259, 345 (1991).
  4. L. Hayen, Standard model O(α) renormalization of gA and its impact on new physics searches, Phys. Rev. D 103, 113001 (2021).
  5. A. H. Mueller, Combining higher twist terms with finite order perturbative contributions, Phys. Lett. B 308, 355 (1993).
  6. J. H. Kim et al., Measurement of αs(Q2) from the Gross–Llewellyn Smith sum rule, Phys. Rev. Lett. 81, 3595 (1998).
  7. M. L. Mangano, S. I. Alekhin, M. Anselmino, R. D. Ball, M. Boglione, U. D’Alesio, S. Davidson, D. Lellis, J. R. Ellis, S. Forte, P. Gambino, T. Gehrmann, A. L. Kataev, A. Kotzinian, S. A. Kulagin, B. Lehmann-Dronke, P. Migliozzi, F. Murgia, and G. Ridolfi, Physics at the front end of a neutrino factory: A quantitative appraisal, Reports No. CERN-TH-2001-131, No. CERN-TH-2001-131, 2001, 10.5170/CERN-2004-002.185.
  8. A. Deur, S. J. Brodsky, and G. F. de Teramond, The QCD running coupling, Prog. Part. Nucl. Phys. 90, 1 (2016).
  9. A. Deur, S. J. Brodsky, and C. D. Roberts, QCD running couplings and effective charges, Prog. Part. Nucl. Phys. 134, 104081 (2024).
  10. J. C. Hardy and I. S. Towner, Superallowed 0+→0+ nuclear β decays: 2014 critical survey, with precise results for Vud and CKM unitarity, Phys. Rev. C 91, 025501 (2015).
  11. J. C. Hardy and I. S. Towner, Superallowed 0+→0+ nuclear β decays: 2020 critical survey, with implications for Vud and CKM unitarity, Phys. Rev. C 102, 045501 (2020).
  12. K. S. Kumar, S. Mantry, W. J. Marciano, and P. A. Souder, Low energy measurements of the weak mixing angle, Annu. Rev. Nucl. Part. Sci. 63, 237 (2013).
  13. M. Gorchtein and C. Y. Seng, Superallowed nuclear beta decays and precision tests of the standard model, Annu. Rev. Nucl. Part. Sci. 74, 23 (2024).
  14. P. G. Blunden, W. Melnitchouk, and A. W. Thomas, New formulation of γZ box corrections to the weak charge of the proton, Phys. Rev. Lett. 107, 081801 (2011).
  15. W. J. Marciano and A. Sirlin, Improved calculation of electroweak radiative corrections and the value of Vud, Phys. Rev. Lett. 96, 032002 (2006).
  16. C.-Y. Seng, M. Gorchtein, H. H. Patel, and M. J. Ramsey-Musolf, Reduced hadronic uncertainty in the determination of Vud, Phys. Rev. Lett. 121, 241804 (2018).
  17. C. Y. Seng, M. Gorchtein, and M. J. Ramsey-Musolf, Dispersive evaluation of the inner radiative correction in neutron and nuclear β decay, Phys. Rev. D 100, 013001 (2019).
  18. K. Shiells, P. G. Blunden, and W. Melnitchouk, Electroweak axial structure functions and improved extraction of the Vud CKM matrix element, Phys. Rev. D 104, 033003 (2021).
  19. J. Arrington, P. Blunden, and W. Melnitchouk, Review of two-photon exchange in electron scattering, Prog. Part. Nucl. Phys. 66, 782 (2011).
  20. X. Feng, M. Gorchtein, L.-C. Jin, P.-X. Ma, and C.-Y. Seng, First-principles calculation of electroweak box diagrams from lattice QCD, Phys. Rev. Lett. 124, 192002 (2020).
  21. P.-X. Ma, X. Feng, M. Gorchtein, L.-C. Jin, and C.-Y. Seng, Lattice QCD calculation of the electroweak box diagrams for the kaon semileptonic decays, Phys. Rev. D 103, 114503 (2021).
  22. P.-X. Ma, X. Feng, M. Gorchtein, L.-C. Jin, K.-F. Liu, C.-Y. Seng, B.-G. Wang, and Z.-L. Zhang, Lattice QCD calculation of electroweak box contributions to superallowed nuclear and neutron beta decays, Phys. Rev. Lett. 132, 191901 (2024).
  23. J.-S. Yoo, T. Bhattacharya, R. Gupta, S. Mondal, and B. Yoon, Electroweak box diagrams on the lattice for pion and neutron decay, Proc. Sci. LATTICE2022 (2023) 360 [arXiv:2212.12830].
  24. J.-S. Yoo, T. Bhattacharya, R. Gupta, S. Mondal, and B. Yoon, Electroweak box diagram contribution for pion and kaon decay from lattice QCD, Phys. Rev. D 108, 034508 (2023).
  25. 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 (QCDSF Collaboration), Nucleon structure functions from operator product expansion on the lattice, Phys. Rev. Lett. 118, 242001 (2017).
  26. K. U. Can, A. Hannaford-Gunn, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, K. Y. Somfleth, H. Stüben, R. D. Young, and J. M. Zanotti (QCDSF/UKQCD/CSSM Collaborations), Lattice QCD evaluation of the Compton amplitude employing the Feynman-Hellmann theorem, Phys. Rev. D 102, 114505 (2020).
  27. M. Batelaan, K. U. Can, A. Hannaford-Gunn, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, H. Stüben, R. D. Young, and J. M. Zanotti (QCDSF/UKQCD/CSSM Collaborations), Moments and power corrections of longitudinal and transverse proton structure functions from lattice QCD, Phys. Rev. D 107, 054503 (2023).
  28. K. U. Can (QCDSF/UKQCD Collaborations), The Compton amplitude and nucleon structure functions, Proc. Sci. LATTICE2022 (2023) 237 [arXiv:2212.09197].
  29. A. Hannaford-Gunn, K. U. Can, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, H. Stüben, G. Schierholz, R. D. Young, and J. M. Zanotti (CSSM/QCDSF/UKQCD Collaborations), Generalized parton distributions from the off-forward Compton amplitude in lattice QCD, Phys. Rev. D 105, 014502 (2022).
  30. A. Hannaford-Gunn, K. U. Can, J. A. Crawford, R. Horsley, P. E. L. Rakow, G. Schierholz, H. Stüben, R. D. Young, and J. M. Zanotti (CSSM/QCDSF/UKQCD Collaborations), Reconstructing generalized parton distributions from the lattice off-forward Compton amplitude, Phys. Rev. D 110, 014509 (2024).
  31. X. Ji, The nucleon structure functions from deep-inelastic scattering with electroweak currents, Nucl. Phys. B402, 217 (1993).
  32. A. W. Thomas and W. Weise, The Structure of the Nucleon (Wiley, Germany, 2001).
  33. M. Constantinou, R. Horsley, H. Panagopoulos, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, and J. M. Zanotti, Renormalization of local quark-bilinear operators for Nf=3 flavors of stout link nonperturbative clover fermions, Phys. Rev. D 91, 014502 (2015).
  34. A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, H. Stüben, R. D. Young, and J. M. Zanotti (CSSM and QCDSF/UKQCD Collaborations), Disconnected contributions to the spin of the nucleon, Phys. Rev. D 92, 114517 (2015).
  35. A. Agadjanov, U.-G. Meißner, and A. Rusetsky, Nucleon in a periodic magnetic field, Phys. Rev. D 95, 031502 (2017).
  36. C.-Y. Seng and U.-G. Meißner, Toward a first-principles calculation of electroweak box diagrams, Phys. Rev. Lett. 122, 211802 (2019).
  37. A. Hannaford-Gunn, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, K. Somfleth, H. Stüben, R. D. Young, and J. M. Zanotti (QCDSF-UKQCD-CSSM Collaborations), Scaling and higher twist in the nucleon Compton amplitude, Proc. Sci. LATTICE2019 (2020) 278 [arXiv:2001.05090].
  38. N. Cundy, M. Göckeler, R. Horsley, T. Kaltenbrunner, A. D. Kennedy, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, A. Schäfer, G. Schierholz, A. Schiller, H. Stüben, and J. M. Zanotti, Non-perturbative improvement of stout-smeared three flavour clover fermions, Phys. Rev. D 79, 094507 (2009).
  39. W. Bietenholz et al. (QCDSF-UKQCD Collaborations), Tuning the strange quark mass in lattice simulations, Phys. Lett. B 690, 436 (2010).
  40. W. Bietenholz, V. Bornyakov, M. Göckeler, R. Horsley, W. G. Lockhart, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, A. Schiller, T. Streuer, H. Stüben, F. Winter, and J. M. Zanotti (QCDSF-UKQCD Collaborations), Flavour blindness and patterns of flavour symmetry breaking in lattice simulations of up, down and strange quarks, Phys. Rev. D 84, 054509 (2011).
  41. S. R. Beane, W. Detmold, R. Horsley, M. Illa, M. Jafry, D. J. Murphy, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, P. E. Shanahan, H. Stüben, M. L. Wagman, F. Winter, R. D. Young, and J. M. Zanotti (NPLQCD and QCDSF Collaborations), Charged multihadron systems in lattice QCD+QED, Phys. Rev. D 103, 054504 (2021).
  42. M. Göckeler, R. Horsley, H. Perlt, P. E. L. Rakow, G. Schierholz, and A. Schiller (QCDSF Collaboration), Operator product expansion on the lattice: Analytic Wilson coefficients, Proc. Sci. LAT2006 (2006) 119 [arXiv:hep-lat/0610064].
  43. K. U. Can, A. Hannaford-Gunn, R. Horsley, P. E. L. Rakow, T. Schar, G. Schierholz, H. Stüben, R. D. Young, and J. M. Zanotti, A lattice QCD calculation of the Compton amplitude subtraction function, Phys. Rev. D 111, 094513 (2025).
  44. S. Navas et al. (Particle Data Group), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  45. J. T. Londergan and A. W. Thomas, Additional corrections to the Gross-Llewellyn Smith sum rule, Phys. Rev. D 82, 113001 (2010).
  46. J. Blümlein, G. Falcioni, and A. De Freitas, The complete O(αs2) non-singlet heavy flavor corrections to the structure functions g1,2ep(x,Q2), F1,2,Lep(x,Q2), F1,2,3ν(ν¯)(x,Q2) and the associated sum rules, Nucl. Phys. B910, 568 (2016).
  47. Y. Aoki et al. (Flavour Lattice Averaging Group (FLAG), FLAG review 2024, arXiv:2411.04268.
  48. K. Maltman, D. Leinweber, P. Moran, and A. Sternbeck, The realistic lattice determination of αs(MZ) revisited, Phys. Rev. D 78, 114504 (2008).
  49. S. Aoki, K. I. Ishikawa, N. Ishizuka, T. Izubuchi, D. Kadoh, K. Kanaya, Y. Kuramashi, K. Murano, Y. Namekawa, M. Okawa, Y. Taniguchi, A. Ukawa, N. Ukita, and T. Yoshié (PACS-CS Collaboration), Precise determination of the strong coupling constant in Nf=2+1 lattice QCD with the Schrodinger functional scheme, J. High Energy Phys. 10 (2009) 053.
  50. C. McNeile, C. T. H. Davies, E. Follana, K. Hornbostel, and G. P. Lepage (HPQCD Collaboration), High-precision c and b masses, and QCD coupling from current-current correlators in lattice and continuum QCD, Phys. Rev. D 82, 034512 (2010).
  51. B. Chakraborty, C. T. H. Davies, B. Galloway, P. Knecht, J. Koponen, G. C. Donald, R. J. Dowdall, G. P. Lepage, and C. McNeile (HPQCD Collaboration), High-precision quark masses and QCD coupling from nf=4 lattice QCD, Phys. Rev. D 91, 054508 (2015).
  52. M. Bruno, M. Dalla Brida, P. Fritzsch, T. Korzec, A. Ramos, S. Schaefer, H. Simma, S. Sint, and R. Sommer (ALPHA Collaboration), QCD coupling from a nonperturbative determination of the three-flavor Λ parameter, Phys. Rev. Lett. 119, 102001 (2017).
  53. A. Bazavov, N. Brambilla, X. Garcia i Tormo, P. Petreczky, J. Soto, A. Vairo, and J. H. Weber (TUMQCD Collaboration), Determination of the QCD coupling from the static energy and the free energy, Phys. Rev. D 100, 114511 (2019).
  54. C. Ayala, X. Lobregat, and A. Pineda, Determination of α(Mz) from an hyperasymptotic approximation to the energy of a static quark-antiquark pair, J. High Energy Phys. 09 (2020) 016.
  55. S. Cali, K. Cichy, P. Korcyl, and J. Simeth, Running coupling constant from position-space current-current correlation functions in three-flavor lattice QCD, Phys. Rev. Lett. 125, 242002 (2020).
  56. M. Dalla Brida, R. Höllwieser, F. Knechtli, T. Korzec, A. Nada, A. Ramos, S. Sint, and R. Sommer (ALPHA Collaboration), Determination of αs(mZ) by the non-perturbative decoupling method, Eur. Phys. J. C 82, 1092 (2022).
  57. P. Petreczky and J. H. Weber, Strong coupling constant from moments of quarkonium correlators revisited, Eur. Phys. J. C 82, 64 (2022).
  58. V. M. Braun and A. V. Kolesnichenko, Power corrections to Bjorken and Gross-Llewellyn Smith sum rules in QCD, Nucl. Phys. B283, 723 (1987).
  59. G. G. Ross and R. G. Roberts, Improved QCD sum rule estimates of the higher twist contributions to polarized and unpolarized nucleon structure functions, Phys. Lett. B 322, 425 (1994).
  60. M. Dasgupta and B. R. Webber, Power corrections and renormalons in deep inelastic structure functions, Phys. Lett. B 382, 273 (1996).
  61. J. Balla, M. V. Polyakov, and C. Weiss, Nucleon matrix elements of higher twist operators from the instanton vacuum, Nucl. Phys. B510, 327 (1998).
  62. M. Batelaan, R. Horsley, Y. Nakamura, H. Perlt, D. Pleiter, P. E. L. Rakow, G. Schierholz, H. Stuben, R. D. Young, and J. Zanotti (QCDSF-UKQCD-CSSM Collaborations), Nucleon form factors from the Feynman-Hellmann method in lattice QCD, Proc. Sci. LATTICE2021 (2022) 426, [arXiv:2202.01366].
  63. R. E. Smail et al. (QCDSF/UKQCD/CSSM Collaborations), Constraining beyond the standard model nucleon isovector charges, Phys. Rev. D 108, 094511 (2023).
  64. J. J. Kelly, Simple parametrization of nucleon form factors, Phys. Rev. C 70, 068202 (2004).
  65. G. Martinelli and C. T. Sachrajda, On the difficulty of computing higher twist corrections, Nucl. Phys. B478, 660 (1996).
  66. M. Beneke, Renormalons, Phys. Rep. 317, 1 (1999).
  67. M. Beneke and V. M. Braun, Renormalons and power corrections, in At the Frontier of Particle Physics/Handbook of QCD, edited by M. Shifman (World Scientific, Singapore, 2001).
  68. V. M. Braun, Higher twists, EPJ Web Conf. 274, 01012 (2022).
  69. A. S. Kronfeld, Factorial growth at low orders in perturbative QCD: Control over truncation uncertainties, J. High Energy Phys. 12 (2023) 108.
  70. T. R. Haar, Y. Nakamura, and H. Stuben, An update on the BQCD hybrid Monte Carlo program, EPJ Web Conf. 175, 14011 (2018).
  71. R. G. Edwards and B. Joó (SciDAC, LHPC, and UKQCD Collaborations), The chroma software system for lattice QCD, Nucl. Phys. B, Proc. Suppl. 140, 832 (2005).
  72. M. Batelaan, K. U. Can, R. Horsley, Y. Nakamura, P. E. L. Rakow, G. Schierholz, H. Stüben, R. D. Young, and J. M. Zanotti (QCDSF-UKQCD-CSSM Collaborations), Feynman-Hellmann approach to transition matrix elements and quasidegenerate energy states, Phys. Rev. D 108, 034507 (2023).

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Phys. Rev. D 111, 114505– Published 10 June, 2025
  • Received 6 March 2025
  • Accepted 15 May 2025
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