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Proton spin structure and generalized polarizabilities in the strong quantum chromodynamics regime


The strong interaction is not well understood at low energies or for interactions with low momentum transfer. Chiral perturbation theory gives testable predictions for the nucleonic generalized polarizabilities, which are fundamental quantities describing the nucleon’s response to an external field. We report a measurement of the proton’s generalized spin polarizabilities extracted with a polarized electron beam and a polarized solid ammonia target in the region where chiral perturbation theory is expected to be valid. The investigated structure function g2 characterizes the internal spin structure of the proton. From its moments, we extract the longitudinal–transverse spin polarizability δLT and twist-3 matrix element and polarizability \(\overline{{d}_{2}}\). Our results provide discriminating power between existing chiral perturbation theory calculations and will help provide a better understanding of this strong quantum chromodynamics regime.

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Fig. 1: Proton spin structure function g2 as a function of invariant mass W.
Fig. 2: Longitudinal–transverse spin polarizability δLT (10−4 fm4) for the proton.
Fig. 3: Unitless scaling of the longitudinal–transverse spin polarizability \(\frac{{Q}^{6}}{{(2M)}^{2}}{\delta }_{\mathrm{LT}}\) for the proton.
Fig. 4: Higher moment \(\overline{{d}_{2}}\) for the proton.

Data availability

The primary experimental data involved in the findings of this analysis are available in the tables and source data files of this Article, with additional data available on request from K.S. (, D.R. ( and J.P.C. ( Source data are provided with this paper.

Code availability

All the computer codes that support this analysis are available on request from K.S. (, D.R. ( and J.-P.C. (


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We would like to thank the Hall A technical staff, polarized target group, and the accelerator operators for their efforts and dedication. We would also like to thank J. M. Alarcón, V. Bernard, P. Bosted, E. Christy, A. Deur, E. Epelbaum, F. Hagelstein, H. Krebs, S. Kuhn, V. Lensky, U.-G. Meißner and V. Pascalutsa for their very helpful discussion and suggestions on this publication. This work was supported by the Department of Energy (DOE) under grants DE-FG02-88ER40410 (funding the University of New Hampshire Nuclear Physics group, received by K.S.), DE-FG02-96ER40950 (funding the University of Virginia Polarized Target Group, received by D.D.) and DE-AC02-06CH11357 (funding the Argonne National Lab group, received by P.R.). The Southeastern Universities Research Association operates the Thomas Jefferson National Accelerator Facility for the DOE under contract DE-AC05-06OR23177.

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Authors and Affiliations



All the authors were involved in the collection of data, design and commissioning of the experimental equipment, or the processing and analysis of the data. The following authors contributed especially to the main analysis and writing of this Article: D.R., J.P.C., K.S. and R.Z.

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Correspondence to K. Slifer.

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Nature Physics thanks Eva-Maria Kabuß, Salvatore Fazio and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Measured spin structure function systematic uncertainties.

The results are given in %. Systematic uncertainties arise from the target dilution factor f, from beam and target polarizations Pb and Pt, radiative corrections ΔσRC and Δσtail, the g1/g2 inputs into eq. (8), and dΩdE’, the error on the unpolarized cross section model. Systematic contributions from the out of plane polarization angle θOoP and the adjustment of the structure functions to constant Q2 were evaluated and found to contribute < 1% to the total systematic error. Reported values are typical of values around the Δ(1,232) resonance. The columns indicate the systematic associated with each individual Q2 kinematic setting, as indicated in the first row. †Longitudinal kinematic setting.

Source data

Source Data Fig. 1

Main data results of the Article.

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Ruth, D., Zielinski, R., Gu, C. et al. Proton spin structure and generalized polarizabilities in the strong quantum chromodynamics regime. Nat. Phys. 18, 1441–1446 (2022).

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