Nucleons, including protons and neutrons, are the fundamental building blocks of nuclei, which together with electrons form the visible matter in our universe. For over a century, physicists have been exploring the internal structure of nucleons, devoting to develop a comprehensive picture of their inner structures and dynamics. It has long been understood that nucleons are not fundamental point-like particles; the electric charge is distributed over a finite spetial extension inside nulecons, resulting in a nonzero “charge radius”. This is largely attributed to experiments using electrons as probes to "bombard" the proton and observe its responses. However, a potentially even more fundamental question has puzzled the scientific community for years: What is the “mass radius” of a nucleon, and how are mass and internal stresses distributed inside the nucleon?

Mass is an intrinsic property of matter determining the magnitude of gravitational force that an object experiences in a gravitational field. The mass of a proton, which is mainly generated by the strong interaction, is about 1.7×10-24 grams (or approximately 938 MeV in natural units commonly used by particle physicists). Gravitational interactions are extremely weak, making it impossible to directly measure the spatial distribution of mass for microscopic particles like the proton in the same way as for electromagnetic interactions. Fortunately, in the 1960s, it was discovered that one can use nucleon matrix elements of the energy-momentum tensor associated with the strong interaction to define a set of physical quantities known as nucleon gravitational form factors. Gravitational form factors encode the distributions of mass, angular momentum, and even internal stresses within composite particles. They are closely related to our understanding of the "origin of mass" from the fundamental theory of strong interaction—quantum chromodynamics (QCD). Experimental studies of the origin of mass in matter constitute an important part of the physics programs at the Electron-Ion Collider in the U.S. and the Electron-Ion Collider in China, which has been under discussion. Recently, the J/ψ-007 Collaboration at the Jefferson Lab published a paper in Nature [Nature 615 (2023) 7954], reporting the extraction of the proton gravitational form factor using near-threshold J/ψ photoproduction data. However, the results rely on model assumptions, and the theoretical uncertainties are not fully under control.

Previous theoretical studies of gravitational form factors have mainly relied on two approaches. The first is phenomenological modeling, such as the chiral quark soliton model, the holographic QCD model, and so on. Although these models provide valuable insights, they are based on certain assumptions and lack a rigorous connection to QCD, making it difficult to reliably estimate the corresponding theoretical uncertainties. The second approach is first-principle calculations based on QCD itself—lattice QCD calculations performed on supercomputers. However, due to computational resource limitations, most lattice QCD calculations to date have been carried out in a “virtual world” where the quark masses are heavier than those in our real world; this effect is usually characterized by the deviation of the pion mass (the lightest hadron, about 140 MeV in Nature) from its physical value. Thus, there is an urgent need for a theoretical method that is both model-independent and applicable under real-world conditions, enabling realiable determinations of gravitational form factors for matter particles.

In response to this long-standing theoretical bottleneck, the research team has made a breakthrough. They employed dispersion relations, a powerful mathematical tool, to build a bridge between experimental low-energy hadron scattering data and the theoretical results for gravitational form factors. First, by analyzing low-energy two-pion scattering data and incorporating fundamental principles of quantum field theory (specifically, analyticity, which corresponds to causality, and unitarity, which is a consequence of probability conservation), they determined the gravitational form factors of the pion, which act as an essential input for the study of nucleon gravitational form factors. After achieving a precise description of the pion gravitational form factors, they related the nucleon gravitational structure to the cross-channel process of pion-nucleon scattering, utilizing high-precision results from Roy-Steiner equations. This approach guarantees that the theoretical framework respects both analyticity and unitarity and on the same time has an excellent agreement with low-energy experimental data. For the first time, this has enabled a model-independent and precise determination of the nucleon gravitational form factors, overcoming the shortcomings of previous results that were either model-dependent or not directly calculated in the real physical world.

In addition to familiar properties such as charge, mass, spin, and magnetic moment, nucleons and other microscopic particles also possess a property known as the D-term, which is encoded in the gravitational form factors. Unlike other fundamental properties, the D-term is not constrained by spacetime symmetries and is extremely difficult to probe directly via electromagnetic interactions, making it notoriously the nucleon's “last global unknown property”. In the past, values of the D-term have relied primarily on model calculations. In recent years, thanks to advances in computational techniques, some results have been obtained using lattice QCD at unphysical quark masses (for example, the calculation by the MIT group used a pion mass of 170 MeV), but these results do not yet cover the parameter space of the real physical world. This study fills that gap, determining the value of the D-term with an uncertainty of only 10%.

In recent years, experimental physicists have precisely determined the proton charge radius to be about 0.84 femtometers (1 femtometer equals 10-15 meters) using modern techniques such as muonic hydrogen spectroscopy. This value can be understood as the typical range over which charged quarks and antiquarks are distributed inside a proton. However, within the proton there are not only charged quarks, but also electrically neutral gluons that are responsible for "gluing" them together. In fact, it is the intricate non-perturbative interactions between gluons that make the low-energy properties of QCD a major scientific challenge. The up and down quarks that make up nucleons have very small masse, each of them having a mass much less than one percent of the nucleon mass. If the up and down quark masses are neglected, QCD at the level of classical field theory exhibits a scale invariance; however, quantum effects break this symmetry, making the trace of the energy-momentum tensor nonvanishing. This effect, known as the "trace anomaly", arises mainly from gluon contributions. The team's calculations show that the root-mean-square radius defined by the spatial distribution of the trace of QCD energy-momentum tensor, i.e., the nucleon scalar radius, is 0.97±0.03 femtometers, apparently larger than the proton charge radius. This provides a modern picture of the nucleon internal structure: rather than quarks and gluons being completely mixed together, the space in which virtual quark-antiquark pairs are distributed is enveloped by a more broadly distributed cloud of gluons. The team also obtained the nucleon mass radius to be  femtometers. Comparing various radii, one sees that the scalar radius of the nucleon is the largest among known radii characterizing various internal distributions related to the strong interaction. Therefore, the scalar radius can be regarded as the "color confinement radius" of the strong interaction.

The research team has further extended their work, in particular having calculated various spatial distributions corresponding to the nucleon gravitational form factors. These results were summarized in a review article, which has been recently submitted to the arXiv server (arXiv:2507.05375) and as an invited article to the journal Eur. Phys. J. ST.

This research was supported by the National Natural Science Foundation of China, the Chinese Academy of Sciences, the China Postdoctoral Science Foundation, and the Hunan Provincial Outstanding Youth Science Foundation.

Original article link: https://doi.org/10.1038/s41467-025-62278-9