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The pressure distribution inside the proton

Matters Arising to this article was published on 05 June 2019


The proton, one of the components of atomic nuclei, is composed of fundamental particles called quarks and gluons. Gluons are the carriers of the force that binds quarks together, and free quarks are never found in isolation—that is, they are confined within the composite particles in which they reside. The origin of quark confinement is one of the most important questions in modern particle and nuclear physics because confinement is at the core of what makes the proton a stable particle and thus provides stability to the Universe. The internal quark structure of the proton is revealed by deeply virtual Compton scattering1,2, a process in which electrons are scattered off quarks inside the protons, which  subsequently emit high-energy photons, which are detected in coincidence with the scattered electrons and recoil protons. Here we report a measurement of the pressure distribution experienced by the quarks in the proton. We find a strong repulsive pressure near the centre of the proton (up to 0.6 femtometres) and a binding pressure at greater distances. The average peak pressure near the centre is about 1035 pascals, which exceeds the pressure estimated for the most densely packed known objects in the Universe, neutron stars3. This work opens up a new area of research on the fundamental gravitational properties of protons, neutrons and nuclei, which can provide access to their physical radii, the internal shear forces acting on the quarks and their pressure distributions.

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Fig. 1: Radial pressure distribution in the proton.
Fig. 2: Fits to the beam-spin asymmetry results.
Fig. 3: Examples of fits to unpolarized cross-sections.
Fig. 4: Example of a fit to gravitational form factor d1(t).


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We thank M. Polyakov and P. Schweitzer for discussions on the subject matter of this work. We are indebted to G. D. Cates for reading the manuscript and suggesting text improvements. This material is based on work supported by the US Department of Energy, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.

Reviewer information

Nature thanks G. D. Cates and the other anonymous reviewer(s) for their contribution to the peer review of this work.

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Authors and Affiliations



V.D.B. wrote the manuscript initiated the analysis, and coordinated and oversaw the research and the validation of the results. L.E. worked with theory experts on the feasibility of the present analysis to develop a procedure connecting the data to the pressure measurements and worked on the analysis of the DVCS beam spin asymmetry. F.X.G. provided the detailed analysis of beam asymmetry and the cross-section measurements, performed the fits to extract the D-term and used the dispersion relation analysis to relate the results to the confinement form factor d1(t).

Corresponding author

Correspondence to V. D. Burkert.

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Extended data figures and tables

Extended Data Fig. 1 Compton form factor \(\pmb{\mathscr{H}}\) at −t = 0.15 GeV2.

The graphs show the imaginary (left) and real (right) parts of \({\mathscr{H}}\) versus the momentum transfer to the quark, ξ. The inner red curve shows the result of our global fit. The grey band shows the estimated uncertainties from the contributions of other CFFs, the outer red band shows the total uncertainties of the imaginary part of the amplitude, and the outer blue band (right) includes the uncertainties related to the D-term. All uncertainties represent one standard deviation.

Extended Data Fig. 2 Compton form factor \(\pmb{\mathscr{H}}\) at −t = 0.11 GeV2 and at −t = 0.20 GeV2.

The imaginary (left) and real (right) parts of \({\mathscr{H}}\) are shown as a function of ξ. The red curve and the grey, red and blue bands are as in Extended Data Fig. 1. All uncertainties represent one standard deviation.

Extended Data Fig. 3 Compton form factor \(\pmb{\mathscr{H}}\) at −t = 0.26 GeV2 and at −t = 0.34 GeV2.

The imaginary (left) and real (right) parts of \({\mathscr{H}}\) are shown as a function of ξ. The red curve and the grey, red and blue bands are as in Extended Data Fig. 1. All uncertainties represent one standard deviation.

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Burkert, V.D., Elouadrhiri, L. & Girod, F.X. The pressure distribution inside the proton. Nature 557, 396–399 (2018).

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