- 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

Heavy quark mass effects in charged-current deep-inelastic scattering at approximate NNLO in the Aivazis-Collins-Olness-Tung scheme

P. Risse1,2,3,*, V. Bertone4, T. Ježo1, K. Kovařík1, A. Kusina5, F. I. Olness2, and I. Schienbein6

  • *Contact author: prisse@smu.edu
PDF

Phys. Rev. D 112, 114004 – Published 1 December, 2025

DOI: https://doi.org/10.1103/hmlr-p7zb

Abstract

The approximate SACOT-χ scheme for heavy quark production in deep-inelastic scattering was initially formulated for the neutral current structure functions F2 and FL. We extend this approach to the charged current case (also including F3), and thereby complete the definitions for the most relevant inclusive structure functions. Furthermore, we implement these structure functions in the open-source code apfel++ which provides fast numerical evaluations over a wide kinematic range; this addition to the apfel++ code is publicly available, with details provided in the Appendix. This SACOT-χ implementation enables detailed numerical insights on the mass dependence of the structure functions and cross sections in the (x,Q2)-plane for both neutral and charged current processes. We consider kinematic regions relevant for the experimental measurements from fixed-target νDIS experiments (NuTeV, CCFR, and Chorus) and HERA, and also projections for the upcoming EIC. In particular, the νDIS experiments reveal a surprisingly strong dependence on the mass effects, offering valuable insights that may help resolve long-standing challenges in accurately describing these datasets.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. H. Abramowicz et al. (H1 and ZEUS Collaborations), Combination and QCD analysis of charm production cross section measurements in deep-inelastic ep scattering at HERA, Eur. Phys. J. C 73, 2311 (2013).
  2. H. Abramowicz et al. (ZEUS Collaboration), Measurement of beauty and charm production in deep inelastic scattering at HERA and measurement of the beauty-quark mass, J. High Energy Phys. 09 (2014) 127.
  3. H. Abramowicz et al. (H1 and 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).
  4. H. Abramowicz et al. (H1 and ZEUS Collaborations), Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA, Eur. Phys. J. C 78, 473 (2018).
  5. M. A. G. Aivazis, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 1. General formalism and kinematics of charged current and neutral current production processes, Phys. Rev. D 50, 3085 (1994).
  6. M. A. G. Aivazis, J. C. Collins, F. I. Olness, and W.-K. Tung, Leptoproduction of heavy quarks. 2. A Unified QCD formulation of charged and neutral current processes from fixed target to collider energies, Phys. Rev. D 50, 3102 (1994).
  7. S. Kretzer and I. Schienbein, Heavy quark initiated contributions to deep inelastic structure functions, Phys. Rev. D 58, 094035 (1998).
  8. W.-K. Tung, S. Kretzer, and C. Schmidt, Open heavy flavor production in QCD: Conceptual framework and implementation issues, J. Phys. G 28, 983 (2002).
  9. B. A. Kniehl, G. Kramer, I. Schienbein, and H. Spiesberger, Inclusive B-meson production at the LHC in the GM-VFN scheme, Phys. Rev. D 84, 094026 (2011).
  10. R. S. Thorne and R. G. Roberts, An ordered analysis of heavy flavor production in deep inelastic scattering, Phys. Rev. D 57, 6871 (1998).
  11. R. S. Thorne, A variable-flavor number scheme for NNLO, Phys. Rev. D 73, 054019 (2006).
  12. M. Cacciari, M. Greco, and P. Nason, The pT spectrum in heavy-flavour hadroproduction., J. High Energy Phys. 05 (1998) 007.
  13. S. Forte, E. Laenen, P. Nason, and J. Rojo, Heavy quarks in deep-inelastic scattering, Nucl. Phys. B834, 116 (2010).
  14. S. Höche, J. Krause, and F. Siegert, Multijet merging in a variable flavor number scheme, Phys. Rev. D 100, 014011 (2019).
  15. R. Gauld, A massive variable flavour number scheme for the Drell-Yan process, SciPost Phys. 12, 024 (2022).
  16. M. Guzzi, P. M. Nadolsky, H.-L. Lai, and C. P. Yuan, General-mass treatment for deep inelastic scattering at two-loop accuracy, Phys. Rev. D 86, 053005 (2012).
  17. J. Gao, T. J. Hobbs, P. M. Nadolsky, C. Sun, and C. P. Yuan, General heavy-flavor mass scheme for charged-current DIS at NNLO and beyond, Phys. Rev. D 105, L011503 (2022).
  18. M. Guzzi, P. Nadolsky, L. Reina, D. Wackeroth, and K. Xie, General mass variable flavor number scheme for Z boson production in association with a heavy quark at hadron colliders, Phys. Rev. D 110, 114030 (2024).
  19. A. Barontini, A. Candido, F. Hekhorn, G. Magni, and R. Stegeman, An FONLL prescription with coexisting flavor number PDFs, J. High Energy Phys. 10 (2024) 004.
  20. J. C. Collins and W.-K. Tung, Calculating heavy quark distributions, Nucl. Phys. B278, 934 (1986).
  21. R. S. Thorne and W. K. Tung, PQCD formulations with heavy quark masses and global analysis, in HERA and the LHC: 4th Workshop on the Implications of HERA for LHC Physics (2008), pp. 332–351, arXiv:0809.0714.
  22. P. M. Nadolsky and W.-K. Tung, Improved formulation of global QCD analysis with Zero-mass matrix elements, Phys. Rev. D 79, 113014 (2009).
  23. T. Stavreva, F. I. Olness, I. Schienbein, T. Jezo, A. Kusina, K. Kovarik, and J. Y. Yu, Heavy quark production in the ACOT Scheme at NNLO and N3LO, Phys. Rev. D 85, 114014 (2012).
  24. K. Xie, Massive elementary particles in the Standard Model and its supersymmetric triplet Higgs extension, Ph.D. thesis, Southern Methodist University, 2019.
  25. V. Bertone, apfel++: A new PDF evolution library in c++, Proc. Sci. DIS2017 (2018) 201 [arXiv:1708.00911].
  26. V. Bertone, apfelxx, https://github.com/vbertone/apfelxx (2017).
  27. V. Bertone, S. Carrazza, and J. Rojo (APFEL Collaboration), APFEL: A PDF evolution library with QED corrections, Comput. Phys. Commun. 185, 1647 (2014).
  28. E. Spezzano et al., Heavy quark contributions to the DIS structure functions F4 and F5 in the ACOT scheme at NLO (to be published), see https://indico.cern.ch/event/1436959/contributions/6352265/.
  29. M. Tzanov et al. (NuTeV Collaboration), Precise measurement of neutrino and anti-neutrino differential cross sections, Phys. Rev. D 74, 012008 (2006).
  30. U.-K. Yang et al. (CCFR/NuTeV Collaboration), Measurements of F2 and xF3ν−xF3ν¯ from CCFR νμ−Fe and ν¯μ−Fe data in a physics model independent way, Phys. Rev. Lett. 86, 2742 (2001).
  31. U.-K. Yang, A measurement of differential cross sections in charged current neutrino interactions on iron and a global structure functions analysis, Ph.D. thesis, Rochester University, 2001.
  32. A. Kayis-Topaksu et al. (CHORUS Collaboration), Leading order analysis of neutrino induced dimuon events in the CHORUS experiment, Nucl. Phys. B798, 1 (2008).
  33. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider: EIC Yellow report, Nucl. Phys. A1026, 122447 (2022).
  34. R. A. Khalek, J. J. Ethier, E. R. Nocera, and J. Rojo, Self-consistent determination of proton and nuclear PDFs at the electron ion collider, Phys. Rev. D 103, 096005 (2021).
  35. M. Krämer, F. I. Olness, and D. E. Soper, Treatment of heavy quarks in deeply inelastic scattering, Phys. Rev. D 62, 096007 (2000).
  36. T. Binoth et al. (SM, NLO Multileg Working Group), The SM and NLO multileg working group: Summary report, in Proceedings of the 6th Les Houches Workshop on Physics at TeV Colliders (2010), pp. 21–189, arXiv:1003.1241.
  37. W. L. van Neerven and E. B. Zijlstra, Order α−s2 contributions to the deep inelastic Wilson coefficient, Phys. Lett. B 272, 127 (1991).
  38. E. B. Zijlstra and W. L. van Neerven, Contribution of the second order gluonic Wilson coefficient to the deep inelastic structure function, Phys. Lett. B 273, 476 (1991).
  39. E. B. Zijlstra and W. L. van Neerven, Order α−s2 QCD corrections to the deep inelastic proton structure functions F2 and F(L), Nucl. Phys. B383, 525 (1992).
  40. W. L. van Neerven and A. Vogt, NNLO evolution of deep inelastic structure functions: The nonsinglet case, Nucl. Phys. B568, 263 (2000).
  41. W. L. van Neerven and A. Vogt, NNLO evolution of deep inelastic structure functions: The singlet case, Nucl. Phys. B588, 345 (2000).
  42. T.-J. Hou et al. (CTEQ-TEA Collaboration), New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC, Phys. Rev. D 103, 014013 (2021).
  43. R. Ruiz et al., Target mass corrections in lepton–nucleus DIS: Theory and applications to nuclear PDFs, Prog. Part. Nucl. Phys. 136, 104096 (2024).
  44. P. Duwentäster, T. Ježo, M. Klasen, K. Kovařík, A. Kusina, K. F. Muzakka, F. I. Olness, R. Ruiz, I. Schienbein, and J. Y. Yu, Impact of heavy quark and quarkonium data on nuclear gluon PDFs, Phys. Rev. D 105, 114043 (2022).
  45. K. F. Muzakka et al., Compatibility of neutrino DIS data and its impact on nuclear parton distribution functions, Phys. Rev. D 106, 074004 (2022).
  46. M. Klasen and H. Paukkunen, Nuclear PDFs after the first decade of LHC data, Annu. Rev. Nucl. Part. Sci. 74, 49 (2024).
  47. R. Abdul Khalek, R. Gauld, T. Giani, E. R. Nocera, T. R. Rabemananjara, and J. Rojo, nNNPDF3.0: Evidence for a modified partonic structure in heavy nuclei, Eur. Phys. J. C 82, 507 (2022).
  48. K. J. Eskola, P. Paakkinen, H. Paukkunen, and C. A. Salgado, EPPS21: A global QCD analysis of nuclear PDFs, Eur. Phys. J. C 82, 413 (2022).
  49. I. Schienbein, J. Y. Yu, K. Kovarik, C. Keppel, J. G. Morfin, F. Olness, and J. F. Owens, PDF nuclear corrections for charged and neutral current processes, Phys. Rev. D 80, 094004 (2009).
  50. H. Paukkunen and C. A. Salgado, Compatibility of neutrino DIS data and global analyses of parton distribution functions, J. High Energy Phys. 07 (2010) 032.
  51. K. Kovarik, I. Schienbein, F. I. Olness, J. Y. Yu, C. Keppel, J. G. Morfin, J. F. Owens, and T. Stavreva, Nuclear corrections in neutrino-nucleus DIS and their compatibility with global NPDF analyses, Phys. Rev. Lett. 106, 122301 (2011).
  52. S. Alekhin et al., HERAFitter, Eur. Phys. J. C 75, 304 (2015).
  53. H. Abdolmaleki et al. (xFitter Collaboration), xfitter: An Open Source QCD Analysis Framework. A resource and reference document for the Snowmass study, arXiv:2206.12465.
  54. S. Carrazza, E. R. Nocera, C. Schwan, and M. Zaro, PineAPPL: combining EW and QCD corrections for fast evaluation of LHC processes, J. High Energy Phys. 12 (2020) 108.
  55. C. Schwan, T. R. Rabemananjara, A. Candido, F. Hekhorn, T. Sharma, S. Carrazza, A. Barontini, J. Wissmann, and J. M. Cruz-Martinez, Nnpdf/pineappl: v1.0.0., 10.5281/zenodo.15635174 (2025).
  56. V. Andreev et al. (H1 Collaboration), Measurement of inclusive ep cross sections at high Q2 at s=225 and 252 GeV and of the longitudinal proton structure function FL at HERA, Eur. Phys. J. C 74, 2814 (2014).
  57. F. I. Olness and W.-K. Tung, Factorization of helicity amplitudes and angular correlations for electroweak processes, Phys. Rev. D 35, 833 (1987).
  58. M. A. G. Aivazis, F. I. Olness, and W.-K. Tung, QCD formulation of charm production in deep inelastic scattering and the sea quark—gluon dichotomy, Phys. Rev. Lett. 65, 2339 (1990).
  59. 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).

Outline

Information

Phys. Rev. D 112, 114004– Published 1 December, 2025
  • Received 27 June 2025
  • Accepted 24 October 2025
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