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Determining the gluonic gravitational form factors of the proton


The proton is one of the main building blocks of all visible matter in the Universe1. Among its intrinsic properties are its electric charge, mass and spin2. These properties emerge from the complex dynamics of its fundamental constituents—quarks and gluons—described by the theory of quantum chromodynamics3,4,5. The electric charge and spin of protons, which are shared among the quarks, have been investigated previously using electron scattering2. An example is the highly precise measurement of the electric charge radius of the proton6. By contrast, little is known about the inner mass density of the proton, which is dominated by the energy carried by gluons. Gluons are hard to access using electron scattering because they do not carry an electromagnetic charge. Here we investigated the gravitational density of gluons using a small colour dipole, through the threshold photoproduction of the J/ψ particle. We determined the gluonic gravitational form factors of the proton7,8 from our measurement. We used a variety of models9,10,11 and determined, in all cases, a mass radius that is notably smaller than the electric charge radius. In some, but not all cases, depending on the model, the determined radius agrees well with first-principle predictions from lattice quantum chromodynamics12. This work paves the way for a deeper understanding of the salient role of gluons in providing gravitational mass to visible matter.

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Fig. 1: Using a bremsstrahlung photon beam and the decay of the J/ψ to an e+e pair to measure the production.
Fig. 2: The differential cross-sections versus t.
Fig. 3: The gluonic gravitational form factors.

Data availability

The raw data from the experiment are archived in the Jefferson Lab mass storage silo and at the Argonne National Laboratory. The analysed data are archived at the Argonne National Laboratory. The data are available in the Supplementary Information and a CSV file of the cross-section data is available in the Supplementary Data.


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This work was supported in part by the US Department of Energy Office of Science, Office of Nuclear Physics, under contract numbers DE-AC02-06CH11357 and DE-FG02-94ER40844, and contract number DE-AC05-06OR23177, under which Jefferson Science Associates operates the Thomas Jefferson National Accelerator Facility.

Author information

Authors and Affiliations



S. Joosten, M.K.J., Z.-E.M., M.P. and E.C. are joint spokespersons for the experiment. The data analysis was carried out by B.D., S.J., M.K.J., S.P., C.P. and Z.-E.M. All authors reviewed the manuscript. The entire J/ψ-007 collaboration participated in the data collection and in the online analysis of the experiment.

Corresponding author

Correspondence to Z.-E. Meziani.

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The authors declare no competing interests.

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Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Fit to differential cross-sections versus t.

The Differential cross-sections data are similar to Fig. 2 but the black solid curve is a dipole fit to the data according to ref. 9 while the grey band shows its uncertainty. The parameters are listed in Extended Data Table 2.

Extended Data Fig. 2 Mass radius and trace anomaly.

Left panel: The extracted radius as a function of the photon energy according to ref. 9 together with the GlueX result. Both our and the GlueX extractions used a dipole fit of the form factor. The charge radius from CODATA and the latest electron scattering6 (labeled PRad) are plotted as lines with error bands. The lattice result12 is plotted as a grey line with grey error band. Right panel: The extracted Ma/M according to Ji’s mass decomposition40 following32 along with a recent direct lattice calculation of the same quantity41.

Extended Data Table 1 Spectrometers Settings
Extended Data Table 2 Fitting parameters, mass radius, and trace anomaly

Supplementary information

Supplementary Information

This file contains Supplementary Figs. 1–12, Supplementary Tables 1–8 and references.

Supplementary Data

CSV file and legend with the two-dimensional cross-section results.

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Duran, B., Meziani, ZE., Joosten, S. et al. Determining the gluonic gravitational form factors of the proton. Nature 615, 813–816 (2023).

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