⚛️ phenomenology

Charmonium-nucleon femtoscopy as a possible probe of the nucleon gravitational form factor

This paper proposes using charmonium-nucleon femtoscopy as a probe of the nucleon's gravitational form factors by constructing an effective potential from lattice-QCD data and HAL QCD constraints to evaluate correlation functions sensitive to the nucleon's $D$ -form factor.

Original authors: Ren Ejima, Daisuke Fujii, Mamiya Kawaguchi

Published 2026-07-02

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CC BY 4.0

Original authors: Ren Ejima, Daisuke Fujii, Mamiya Kawaguchi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the proton (a particle inside an atom's nucleus) not as a solid marble, but as a tiny, vibrating balloon filled with invisible forces. Scientists have long wanted to map out exactly how the energy and pressure are distributed inside this balloon. They call these maps "Gravitational Form Factors." One specific part of this map, called the "D-term," is like the blueprint for the internal pressure and stress—essentially, how the balloon holds itself together without popping.

For a long time, this "D-term" has been the "last unknown global property" of the proton. We know it exists, but we haven't been able to measure it directly. Usually, scientists try to figure it out by firing high-energy beams at protons and analyzing the scattered debris (a bit like trying to understand a car's engine by smashing it and looking at the broken parts).

The New Idea: Listening to the Echo
This paper proposes a different way to "listen" to the proton's internal pressure. Instead of smashing it, the authors suggest looking at how a proton interacts with a charmonium particle.

Think of charmonium (specifically the $J/\psi$ and $\psi(2S)$ particles) as a tiny, heavy, and very compact "probe" or "diver."

The proton is the ocean.
The charmonium is the diver.
The interaction is how the water pushes back on the diver.

The authors use a theoretical framework (based on Quantum Chromodynamics, or QCD) to build a "map" of the ocean (the proton) based on its pressure and energy. They then drop their "diver" (the charmonium) into this map to see how it behaves.

The Experiment: Two Different Divers
The researchers tested two different types of divers to see which one could best feel the pressure of the ocean:

The $J/\psi$ Diver: This is a standard, compact diver. The authors found that when this diver moves through the proton's "ocean," the resulting interaction (measured as a correlation function) is somewhat "deaf" to the specific details of the internal pressure (the D-term). It's like a small boat floating on a calm sea; it doesn't feel the subtle shifts in water pressure very well. The paper shows that changing the internal pressure settings doesn't change the boat's path much.
The $\psi(2S)$ Diver: This is a larger, more sensitive diver (a higher energy state). The authors found that this diver is much more sensitive to the internal pressure. When they adjusted the "D-term" (the pressure map) in their calculations, the path of this larger diver changed noticeably. It's like a large submarine that feels every shift in the water's density.

The Conclusion
The paper concludes that while the standard $J/\psi$ interaction isn't a great tool for measuring the proton's internal pressure, the larger $\psi(2S)$ interaction might be a very sensitive probe.

By studying how these particles correlate (how they move together) in high-energy collisions (like those at the Large Hadron Collider), scientists might be able to "see" the proton's internal pressure map in a new way. This wouldn't replace current methods but would offer a complementary perspective, like using a different kind of sonar to map the ocean floor.

Important Caveat
The authors are careful to note that this is currently a theoretical investigation. They haven't extracted the final, perfect map of the proton yet. They are essentially saying, "If we can pin down some of the unknown numbers in our model, this method could become a powerful new tool to measure the proton's internal stress." They are not claiming to have solved the mystery of the proton's structure today, but rather identifying a promising new path to do so.

Technical Summary: Charmonium-Nucleon Femtoscopy as a Probe of the Nucleon Gravitational Form Factor

Problem Statement
The internal mechanical structure of the nucleon, specifically the spatial distributions of energy, pressure, and shear forces, is encoded in the Gravitational Form Factors (GFFs) of the Energy-Momentum Tensor (EMT). While the mass and spin contributions are well-studied, the $D$ -term (the forward limit of the $D$ -form factor) remains experimentally unestablished and is often termed the "last unknown global property" of the proton. Current extractions of the $D$ -term and pressure distributions rely on Deeply Virtual Compton Scattering (DVCS) and near-threshold quarkonium photoproduction, where GFFs are inferred through Generalized Parton Distributions (GPDs). This indirect extraction involves theoretical model assumptions that may obscure underlying mechanical structures. Consequently, there is a need for complementary observables to probe nucleon mechanical properties from a different perspective.

Methodology
The authors investigate the charmonium-nucleon ( $J/\psi$ - $N$ and $\psi(2S)$ - $N$ ) system as a potential probe for nucleon GFFs using femtoscopy (two-particle correlation functions). The methodology proceeds through the following steps:

GFF Parametrization: The nucleon GFFs ( $A(t)$ , $J(t)$ , and $D(t)$ ) are parametrized using a dipole-like ansatz fitted to existing lattice-QCD data. This allows for the calculation of the energy ( $\epsilon(r)$ ) and pressure ( $p(r)$ ) distributions inside the nucleon via Fourier transformation in the Breit frame.
Effective Potential Construction: The charmonium-nucleon interaction is modeled using the QCD multipole expansion in the leading chromo-electric dipole approximation. In this framework, the effective potential $V(r)$ is expressed directly in terms of the nucleon's energy and pressure distributions:
$V(r) = -\frac{\alpha_{nS}}{4\pi^2} \frac{b}{g^2} \left(\frac{g^2}{g_s^2}\right) [\nu \epsilon(r) - 3p(r)]$
where $\alpha_{nS}$ is the chromo-electric polarizability of the charmonium state.
Parameter Fixing and Smearing: To ensure physical consistency, the model parameters (specifically $\alpha_{1S}$ , the ratio of coupling constants, and the parameter $\nu$ ) are fixed by requiring the resulting $J/\psi$ - $N$ potential and scattering phase shift to reproduce estimates derived from HAL QCD lattice data. Because the point-like approximation of the potential deviates from lattice results at short distances ( $r < 1$ fm), the authors introduce a smearing procedure to account for the finite size of the charmonium state ( $\sigma_{J/\psi} \approx 0.3$ fm).
Correlation Function Calculation: The $J/\psi$ - $N$ and $\psi(2S)$ - $N$ correlation functions, $C(k^*)$ , are calculated using the Koonin–Pratt formula, incorporating the relative wave functions derived from the Schrödinger equation with the constructed potentials. The sensitivity of these correlation functions to the $D$ -term ( $D_0$ ) is then examined by varying $D_0$ within the uncertainties of lattice-QCD results.

Key Contributions and Results

Validation of the Framework: The study successfully constructs a $J/\psi$ - $N$ potential that, after smearing, aligns with HAL QCD lattice data in the long-distance region and reproduces the scattering phase shift. This validates the use of the QCD multipole expansion framework for describing charmonium-nucleon interactions in terms of nucleon GFFs.
Sensitivity of $J/\psi$ - $N$ System: The analysis reveals that the $J/\psi$ - $N$ correlation function exhibits only weak dependence on the nucleon $D$ -term ( $D_0$ ). Varying $D_0$ within the range of $-2.69 \pm 1$ produces minimal changes in the correlation function, suggesting this system is not a highly sensitive probe for the $D$ -term under current constraints.
Enhanced Sensitivity in $\psi(2S)$ - $N$ System: The study identifies the $\psi(2S)$ - $N$ system as a more promising probe. Due to the larger chromo-electric polarizability ( $\alpha_{2S}$ ) of the $\psi(2S)$ state, the dependence of the potential on the pressure distribution (and thus the $D$ -term) is amplified. The results show that the $\psi(2S)$ - $N$ correlation function displays a visible sensitivity to variations in $D_0$ , particularly in the near-threshold region (small relative momentum $k^*$ ). The slope of the correlation function at $k^* \approx 0$ varies significantly with $D_0$ for specific values of $\alpha_{2S}$ .

Significance and Claims
The paper posits that charmonium-nucleon femtoscopy offers a complementary approach to studying nucleon GFFs, distinct from the GPD-based analyses of DVCS. The authors explicitly state that their goal is not to extract GFFs in a model-independent manner, but rather to demonstrate that two-particle correlation functions can be sensitive to the internal mechanical structure encoded in the $D$ -form factor.

The significance lies in the potential for femtoscopic observables to constrain GFFs without first reconstructing GPDs, thereby providing additional input to reduce ambiguities in three-dimensional nucleon structure models. However, the authors maintain a modest tone regarding immediate application: they emphasize that for femtoscopy to serve as a quantitative probe, the model parameters (particularly $\alpha_{2S}$ for the $\psi(2S)$ system) must be better constrained. The current study serves as a first step, establishing the theoretical link and identifying the $\psi(2S)$ - $N$ system as a candidate for future investigation once these parameters are refined.

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