Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Measurement of the proton spin structure at long distances


Measuring the spin structure of protons and neutrons tests our understanding of how they arise from quarks and gluons, the fundamental building blocks of nuclear matter. At long distances, the coupling constant of the strong interaction becomes large, requiring non-perturbative methods to calculate quantum chromodynamics processes, such as lattice gauge theory or effective field theories. Here we report proton spin structure measurements from scattering a polarized electron beam off polarized protons. The spin-dependent cross-sections were measured at large distances, corresponding to the region of low momentum transfer squared between 0.012 and 1.0 GeV2. This kinematic range provides unique tests of chiral effective field theory predictions. Our results show that a complete description of the nucleon spin remains elusive, and call for further theoretical works, for example, in lattice quantum chromodynamics. Finally, our data extrapolated to the photon point agree with the Gerasimov–Drell–Hearn sum rule, a fundamental prediction of quantum field theory that relates the anomalous magnetic moment of the proton to its integrated spin-dependent cross-sections.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: The one-photon exchange process in electron–nucleon scattering.
Fig. 2: Results for g1(Q2, W) of the proton.
Fig. 3: Results for Γ1(Q2) for the proton.
Fig. 4: Results for I(Q2) for the proton.
Fig. 5: Results for γ0(Q2) for the proton.

Similar content being viewed by others

Data availability

Experimental data that support the findings of this study will be posted on the CLAS database ( or are available from the corresponding author upon request.

Code availability

The computer codes that support the plots within this paper and the findings of this study are available from X.Z. upon request.


  1. Deur, A., Brodsky, S. J. & de Téramond, G. F. The QCD running coupling. Prog. Part. Nucl. Phys. 90, 1–74 (2016).

    Article  ADS  Google Scholar 

  2. Particle Data Group Review of particle physics. Phys. Rev. D 98, 030001 (2018).

    Article  Google Scholar 

  3. Bernard, V. Chiral perturbation theory and baryon properties. Prog. Part. Nucl. Phys. 60, 82–160 (2008).

    Article  ADS  Google Scholar 

  4. Scherer, S. Chiral perturbation theory: introduction and recent results in the one-nucleon sector. Prog. Part. Nucl. Phys. 64, 1–60 (2010).

    Article  ADS  Google Scholar 

  5. Kuhn, S. E., Chen, J.-P. & Leader, E. Spin structure of the nucleon—status and recent results. Prog. Part. Nucl. Phys. 63, 1–50 (2009).

    Article  ADS  Google Scholar 

  6. Deur, A., Brodsky, S. J. & de Téramond, G. F. The spin structure of the nucleon. Rep. Prog. Phys. 82, 076201 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  7. Wu, C., Ambler, E., Hayward, R., Hoppes, D. & Hudson, R. Experimental test of parity conservation in beta decay. Phys. Rev. 105, 1413–1414 (1957).

    Article  ADS  Google Scholar 

  8. Ellis, J. R. & Jaffe, R. L. A sum rule for deep inelastic electroproduction from polarized protons. Phys. Rev. D 9, 1444–1446 (1974); erratum: 10, 1669–1670(1974).

  9. Brodsky, S. J., Burkardt, M. & Schmidt, I. QCD constraints on the shape of polarized quark and gluon distributions. Nucl. Phys. B 441, 197–214 (1995).

    Article  ADS  Google Scholar 

  10. Chang, C. C. et al. A percent-level determination of the nucleon axial coupling from quantum chromodynamics. Nature 558, 91–94 (2018).

    Article  ADS  Google Scholar 

  11. Crabb, D. G. & Day, D. B. The Virginia/Basel/SLAC polarized target: operation and performance during experiment E143 at SLAC. Nucl. Inst. Meth. A 356, 9–19 (1995).

    Article  ADS  Google Scholar 

  12. Keith, C. D. et al. A polarized target for the CLAS detector. Nucl. Inst. Meth. A 501, 327–339 (2003).

    Article  ADS  Google Scholar 

  13. Mecking, B. A. et al. The CEBAF large acceptance spectrometer (CLAS). Nucl. Inst. Meth. A 503, 513–553 (2003).

    Article  ADS  Google Scholar 

  14. Fersch, R. G. et al. Determination of the proton spin structure functions for 0.05 < Q2< 5 GeV2 using CLAS. Phys. Rev. C 96, 065208 (2017).

    Article  ADS  Google Scholar 

  15. Amarian, M. et al. The Q2 evolution of the generalized Gerasimov–Drell–Hearn integral for the neutron using a 3He target. Phys. Rev. Lett. 89, 242301 (2002).

    Article  ADS  Google Scholar 

  16. Amarian, M. et al. Q2 evolution of the neutron spin structure moments using a 3He target. Phys. Rev. Lett. 92, 022301 (2004).

    Article  ADS  Google Scholar 

  17. Amarian, M. et al. Measurement of the generalized forward spin polarizabilities of the neutron. Phys. Rev. Lett. 93, 152301 (2004).

    Article  ADS  Google Scholar 

  18. Deur, A. et al. Experimental determination of the evolution of the Bjorken integral at low Q2. Phys. Rev. Lett. 93, 212001 (2004).

    Article  ADS  Google Scholar 

  19. Deur, A. et al. Experimental study of isovector spin sum rules. Phys. Rev. D 78, 032001 (2008).

    Article  ADS  Google Scholar 

  20. Dharmawardane, K. V. et al. Measurement of the x- and Q2-dependence of the asymmetry A1 on the nucleon. Phys. Lett. B 641, 11–17 (2006).

    Article  ADS  Google Scholar 

  21. Prok, Y. et al. Moments of the spin structure functions \({g}_{1}^{\rm{p}}\) and \({g}_{1}^{\rm{d}}\) for 0.05 < Q2 < 3.0 GeV2. Phys. Lett. B 672, 12–16 (2009).

  22. Guler, N. et al. Precise determination of the deuteron spin structure at low to moderate Q2 with CLAS and extraction of the neutron contribution. Phys. Rev. C 92, 055201 (2015).

    Article  ADS  Google Scholar 

  23. Chambers, A. J. et al. Nucleon structure functions from operator product expansion on the lattice. Phys. Rev. Lett. 118, 242001 (2017).

    Article  ADS  Google Scholar 

  24. Liang, J., Draper, T., Liu, K.-F., Rothkopf, A. & Yang, Y.-B. Towards the nucleon hadronic tensor from lattice QCD. Phys. Rev. D 101, 114503 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  25. Gerasimov, S. B. A sum rule for magnetic moments and the damping of the nucleon magnetic moment in nuclei. Sov. J. Nucl. Phys. 2, 430–433 (1966); Yad. Fiz. 2, 598–602 (1965).

  26. Drell, S. D. & Hearn, A. C. Exact sum rule for nucleon magnetic moments. Phys. Rev. Lett. 16, 908–911 (1966).

    Article  ADS  Google Scholar 

  27. Dutz, H. et al. Experimental check of the Gerasimov–Drell–Hearn sum rule for 1H. Phys. Rev. Lett. 93, 032003 (2004).

    Article  ADS  Google Scholar 

  28. Hoblit, S. et al. Measurements of \(\overrightarrow{H}\overrightarrow{D}(\overrightarrow{\gamma },\pi )\) and implications for convergence of the Gerasimov–Drell–Hearn integral. Phys. Rev. Lett. 102, 172002 (2009).

  29. Ji, X.-D. & Osborne, J. Generalized sum rules for spin-dependent structure functions of the nucleon. J. Phys. G 27, 127–146 (2001).

    Article  ADS  Google Scholar 

  30. Bernard, V., Kaiser, N. & Meissner, U. G. Small momentum evolution of the extended Drell–Hearn–Gerasimov sum rule. Phys. Rev. D 48, 3062–3069 (1993).

    Article  ADS  Google Scholar 

  31. Ji, X. D., Kao, C.-W. & Osborne, J. Generalized Drell–Hearn–Gerasimov sum rule at order O(p4) in chiral perturbation theory. Phys. Lett. B 472, 1–4 (2000).

    Article  ADS  Google Scholar 

  32. Ji, X. D., Kao, C.-W. & Osborne, J. The nucleon spin polarizability at order O(p4) in chiral perturbation theory. Phys. Rev. D 61, 074003 (2000).

    Article  ADS  Google Scholar 

  33. Bernard, V., Hemmert, T. R. & Meissner, U. G. Novel analysis of chiral loop effects in the generalized Gerasimov–Drell–Hearn sum rule. Phys. Lett. B 545, 105–111 (2002).

    Article  ADS  Google Scholar 

  34. Bernard, V., Hemmert, T. R. & Meissner, U. G. Spin structure of the nucleon at low energies. Phys. Rev. D 67, 076008 (2003).

    Article  ADS  Google Scholar 

  35. Kao, C. W., Spitzenberg, T. & Vanderhaeghen, M. Burkhardt–Cottingham sum rule and forward spin polarizabilities in heavy baryon chiral perturbation theory. Phys. Rev. D 67, 016001 (2003).

    Article  ADS  Google Scholar 

  36. Bernard, V., Epelbaum, E., Krebs, H. & Meissner, U. G. New insights into the spin structure of the nucleon. Phys. Rev. D 87, 054032 (2013).

    Article  ADS  Google Scholar 

  37. Alarcón, J. M., Hagelstein, F., Lensky, V. & Pascalutsa, V. Forward doubly-virtual Compton scattering off the nucleon in chiral perturbation theory: II. Spin polarizabilities and moments of polarized structure functions. Phys. Rev. D 102, 114026 (2020).

    Article  ADS  Google Scholar 

  38. Lensky, V., Alarcón, J. M. & Pascalutsa, V. Moments of nucleon structure functions at next-to-leading order in baryon chiral perturbation theory. Phys. Rev. C 90, 055202 (2014).

    Article  ADS  Google Scholar 

  39. Lensky, V., Pascalutsa, V. & Vanderhaeghen, M. Generalized polarizabilities of the nucleon in baryon chiral perturbation theory. Eur. Phys. J. C 77, 119 (2017).

    Article  ADS  Google Scholar 

  40. Drechsel, D., Pasquini, B. & Vanderhaeghen, M. Dispersion relations in real, virtual Compton scattering. Phys. Rep. 378, 99–205 (2003).

    Article  ADS  Google Scholar 

  41. Burkert, V. D. & Ioffe, B. L. Polarized structure functions of proton and neutron and the Gerasimov–Drell–Hearn and Bjorken sum rules. J. Exp. Theor. Phys. 78, 619–622 (1994).

    ADS  Google Scholar 

  42. Pasechnik, R. S., Soffer, J. & Teryaev, O. V. Nucleon spin structure at low momentum transfers. Phys. Rev. D 82, 076007 (2010).

    Article  ADS  Google Scholar 

  43. Guichon, P. A. M., Liu, G. Q. & Thomas, A. W. Virtual Compton scattering and generalized polarizabilities of the proton. Nucl. Phys. A 591, 606–638 (1995).

    Article  ADS  Google Scholar 

  44. Gurevich, G. M. & Lisin, V. P. Measurement of the proton spin polarizabilities at MAMI. Phys. Part. Nucl. 48, 111–116 (2017).

    Article  Google Scholar 

  45. Adhikari, K. P. et al. Measurement of the Q2 dependence of the deuteron spin structure function g1 and its moments at low Q2 with CLAS. Phys. Rev. Lett. 120, 062501 (2018).

    Article  ADS  Google Scholar 

  46. Sulkosky, V. et al. Measurement of the 3He spin-structure functions and of neutron (3He) spin-dependent sum rules at 0.035 ≤Q2≤0.24 GeV2. Phys. Lett. B 805, 135428 (2020).

    Article  Google Scholar 

  47. Anderson, P. W. More is different. Science 177, 393–396 (1972).

    Article  ADS  Google Scholar 

  48. Arrington, J., Melnitchouk, W. & Tjon, J. A. Global analysis of proton elastic form factor data with two-photon exchange corrections. Phys. Rev. C 76, 035205 (2007).

    Article  ADS  Google Scholar 

  49. Abe, K. et al. Measurements of the proton and deuteron spin structure functions g1 and g2. Phys. Rev. D 58, 112003 (1998).

    Article  ADS  Google Scholar 

Download references


All authors are members of The Jefferson Lab CLAS Collaboration. We thank the personnel of Jefferson Lab for their efforts that resulted in the successful completion of the experiment. We thank Kovacs for her contribution to the early analysis of the data. We are grateful to U.-G. Meißner and V. Pascalutsa for useful discussions on the theoretical χEFT calculations. This work was supported by the US Department of Energy (DOE), the US National Science Foundation, the US Jeffress Memorial Trust, the UK Science and Technology Facilities Council (STFC), the Italian Istituto Nazionale di Fisica Nucleare, the French Institut National de Physique Nucléaire et de Physique des Particules, the French Centre National de la Recherche Scientifique and the National Research Foundation of Korea. This material is based on work supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under contract no. DE-AC05-06OR23177.

Author information

Authors and Affiliations



The members of the Jefferson Lab CLAS Collaboration constructed and operated the experimental equipment used in this experiment. A large number of collaboration members participated in the data collection. The following authors provided various contributions to the experiment design and commissioning, data processing, data analysis and Monte Carlo simulations: M. Battaglieri, R. De Vita, V. A. Drozdov, L. El Fassi, H. Kang, E. Long, M. Osipenko, S. K. Phillips and K. Slifer. The authors who performed the final data analysis and Monte Carlo simulations were A. Deur, S. E. Kuhn, M. Ripani, J. Zhang and X. Zheng. The manuscript was reviewed by the entire CLAS Collaboration before publication, and all authors approved the final version of the manuscript.

Corresponding author

Correspondence to A. Deur.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature Physics thanks Mohammad Ahmed and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Two tables providing all the data shown in the manuscript. One figure displaying the full dataset for \({{{{g}}}^{p}_{1}}({{{Q}}}^{2},{{W}}).\)

Supplementary Data 1

Results for \({{\varGamma }^{p}_{1}},{{{{I}}}^{p}_{{\rm{TT}}}}\) and \({{\gamma }^{p}_{{\rm{0}}}}\), for the measured and the full ranges, along with their statistical and systematic uncertainties.

Supplementary Data 2

Results for \({{{{g}}}^{p}_{{\rm{1}}}}\) and \({{{{A}}}^{p}_{{\rm{1}}}}{{{{F}}}^{p}_{{\rm{1}}}}\) along with their statistical and systematic uncertainties.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zheng, X., Deur, A., Kang, H. et al. Measurement of the proton spin structure at long distances. Nat. Phys. 17, 736–741 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing