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Probing the core of the strong nuclear interaction


The strong nuclear interaction between nucleons (protons and neutrons) is the effective force that holds the atomic nucleus together. This force stems from fundamental interactions between quarks and gluons (the constituents of nucleons) that are described by the equations of quantum chromodynamics. However, as these equations cannot be solved directly, nuclear interactions are described using simplified models, which are well constrained at typical inter-nucleon distances1,2,3,4,5 but not at shorter distances. This limits our ability to describe high-density nuclear matter such as that in the cores of neutron stars6. Here we use high-energy electron scattering measurements that isolate nucleon pairs in short-distance, high-momentum configurations7,8,9, accessing a kinematical regime that has not been previously explored by experiments, corresponding to relative momenta between the pair above 400 megaelectronvolts per c (c, speed of light in vacuum). As the relative momentum between two nucleons increases and their separation thereby decreases, we observe a transition from a spin-dependent tensor force to a predominantly spin-independent scalar force. These results demonstrate the usefulness of using such measurements to study the nuclear interaction at short distances and also support the use of point-like nucleon models with two- and three-body effective interactions to describe nuclear systems up to densities several times higher than the central density of the nucleus.

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Fig. 1: Using electron scattering measurements to test the nuclear interaction.
Fig. 2: Dependence of the two- to one-proton knockout reaction yield ratio on the missing momentum.
Fig. 3: Missing-momentum and energy dependence of one- and two-proton knockout reaction yields.

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The raw data from this experiment are archived in Jefferson Lab’s mass storage silo.


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We acknowledge the efforts of the staff of the Accelerator and Physics Divisions at Jefferson Lab that made this experiment possible. The analysis presented here was carried out as part of the Jefferson Lab Hall B data-mining project supported by the US Department of Energy (DOE). The research was also supported by the National Science Foundation, the Israel Science Foundation, the Pazi Foundation, the Chilean Comisión Nacional de Investigación Científica y Tecnológica, the French Centre National de la Recherche Scientifique and Commissariat a l'Energie Atomique, the French–American Cultural Exchange, the Italian Istituto Nazionale di Fisica Nucleare, the National Research Foundation of Korea, and the UK Science and Technology Facilities Council. Jefferson Science Associates operates the Thomas Jefferson National Accelerator Facility for the DOE, Office of Science, Office of Nuclear Physics under contract DE-AC05-06OR23177.

Author information





The CEBAF Large Acceptance Spectrometer was designed and constructed by the CLAS Collaboration and Jefferson Lab. Data acquisition, processing and calibration, Monte Carlo simulations of the detector and data analyses were performed by a large number of CLAS Collaboration members, who also discussed and approved the scientific results. The analysis presented here was performed primarily by A.S. and J.R.P. R. Weiss and N.B. provided theoretical input and helped implement parts of the GCF event generator. M.S. and A.L. provided theoretical input and helped implement the light-cone formalism. A. Denniston and E.P.S. helped implement parts of the GCF event generator and performed the model systematic uncertainty studies. A.H. calculated the CLAS acceptance maps. O.H., E. Piasetzky, and L.B.W. guided and supervised the analysis.

Corresponding author

Correspondence to O. Hen.

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The authors declare no competing interests.

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

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Extended data figures and tables

Extended Data Fig. 1 SRC pair breakup.

Diagrammatic representation and four-momentum kinematics of the two-nucleon knockout A(eeNp) reaction within the SRC model. The dashed red lines represent off-shell particles and solid black lines represent detected particles. The A−2 system is undetected.

Extended Data Fig. 2 Kinematical distributions and A(eepp)/A(eep) ratios for A = 12–208 nuclei.

ac, Comparison of the number of A(eep) event reactions versus the (eep) missing momentum pmiss (a), Q2 (b) and xB (c). df, Comparison of the number of A(eepp) event reactions versus the (eep) missing momentum pmiss (d), Q2 (e) and xB (f). The total number of counts in aluminium (cyan), iron (orange), and lead (purple) was scaled to match that of carbon (dark blue). The shaded bands indicate the 1σ statistical uncertainty of the data.

Extended Data Fig. 3 Momentum-transfer and missing-momentum angular correlations.

ad, Distribution of the relative angle between the momentum transfer q and the (eep) missing momentum for A(eep) (a, b) and A(eepp) (c, d). The panels at left compare the 12C data and GCF calculations using different NN interaction models (a, c); and the panels at right compare the data for carbon (blue), aluminium (cyan), iron (orange), and lead (purple) nuclei (b, d). The total number of counts in the aluminium, iron and lead data has been scaled to match that of carbon. In a and c, the width of the band and the data error bars show the model systematic uncertainties and data statistical uncertainties, respectively, each at the 1σ confidence level. The shaded bands in b and d indicate the 1σ statistical uncertainty of the data.

Extended Data Fig. 4 Electron and proton kinematics.

Momentum and angle distributions of scattered electrons and protons for 12C(eep) (a, c, e, h) and 12C(eepp) (b, d, f, g, i, j) events. Coloured bands show the various GCF calculations. The width of the shaded band and the data error bars show the model systematic uncertainties and data statistical uncertainties, respectively, each at the 1σ confidence level.

Extended Data Fig. 5 Kinematic correlations of 12C(eep) events.

a, Q2 versus the missing-momentum distribution of 12C(eep) data. Owing to the event selection criteria, as pmiss approaches 1 GeV/c, the minimum Q2 of the data approaches 3 GeV/c. b, Emiss versus pmiss of the 12C(eep) data. The red line indicates the expected Emisspmiss correlation for the breakup of a stationary pair.

Extended Data Fig. 6 Universal functions for pp and np pairs and the momentum dependence of their ratio.

The relative momentum distributions for different NN interaction models studied in this work, for pn (a) and pp (b). c, The momentum dependence of the fraction of protons belonging to pp SRC pairs in 12C.

Extended Data Fig. 7 Light-cone calculations of the nuclear spectral function and momentum fractions.

aj, As in Fig. 3a–j. k, l, The reconstructed initial light-cone momentum fraction carried by the struck nucleon for (eep) (k) and (eepp) (l) events. m, The total pair light-cone momentum fraction for (eepp) events. The data points are identical to those in Fig. 3a–j. The bands are different and show the results of the GCF calculations using light-cone formalism and various NN interaction models. The width of the shaded band and the data error bars show the model systematic uncertainties and data statistical uncertainties, respectively, each at the 1σ confidence level.

Extended Data Fig. 8 Scale dependence and non-local interactions.

ac, As in Fig. 3a, b (a, b) and Fig. 2a (c), but also including the non-local N3LO (600 MeV/c) interaction. The width of the shaded band and the data error bars show the model systematic uncertainties and data statistical uncertainties, respectively, each at the 1σ confidence level. See Methods for details.

Extended Data Table 1 Extracted contact ratios \({{\boldsymbol{C}}}_{{\boldsymbol{p}}{\boldsymbol{p}}}^{{\boldsymbol{s}}={\bf{0}}}/{{\boldsymbol{C}}}_{{\boldsymbol{n}}{\boldsymbol{p}}}^{{\boldsymbol{s}}={\bf{1}}}\) for different nuclei

Supplementary information

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This file contains Supplementary Materials, including Supplementary Table 1, Supplementary Figures 1 and 2, and additional references.

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Schmidt, A., Pybus, J., Weiss, R. et al. Probing the core of the strong nuclear interaction. Nature 578, 540–544 (2020).

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