This paper shows how to define and measure the force on quarks inside a proton using existing experimental data and numerical calculations. The authors use Quantum Chromodynamics (QCD), the theory of the strong interaction, and find that the force acting on quarks is attractive and roughly constant over a range of positions. They call this direct evidence for confinement—the long-known fact that quarks cannot be pulled free of protons and neutrons—though they do not claim a formal mathematical proof of confinement from QCD.

To make the force concrete, the researchers start from the QCD energy-momentum tensor (EMT), a quantity that encodes momentum and stress in the proton. The divergence of the EMT gives a force density, which in QCD takes the form of a color version of the Lorentz force familiar from electromagnetism. To connect that theory to data, they use experimentally measured form factors—functions that describe how the proton responds to probes in processes such as deeply virtual Compton scattering (DVCS) and deep exclusive meson production (DVMP)—and results from lattice QCD, a numerical method that calculates QCD quantities on a spacetime grid.

A key technical step is defining a quark position in a way consistent with relativity and quantum uncertainty. The authors evaluate densities in an infinite-momentum frame, which lets them speak about transverse positions inside a fast-moving proton. They obtain the force on a quark by dividing the force density (from the EMT scalar form factor) by the quark density (from the proton and neutron Dirac form factors). This gives a radial force in the transverse plane that can be plotted as a function of transverse distance r⊥.

Using current global fits that combine experimental data and lattice-QCD input, the paper reports specific numbers. One extraction based mainly on experimental analyses gives an average force of about −0.382(92) GeV/fm over r⊥ = 1.0–1.4 fm (the negative sign means the force is inward, i.e., attractive). A separate fit dominated by lattice-QCD input (the GYZ fit) finds −0.217+0.016−0.015 GeV/fm over r⊥ = 0.7–1.2 fm. The authors note these values are compatible with forces inferred from relativistic quark models and that, at intermediate distances, the results are consistent with a linear confining potential.