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Photon and gluon gravitational form factors of proton in QCD

Using QCD sum rules, this study calculates the photon and gluon gravitational form factors of the proton, revealing that the gluon mass radius is smaller than the photon's, which indicates a more localized gluonic energy distribution within the proton.

Original authors: Z. Asmaee, K. Azizi

Published 2026-08-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Z. Asmaee, K. Azizi

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

To understand what holds a proton together, scientists have long looked at its parts: the quarks and the gluons that zip around inside it. Quarks are the fundamental building blocks of matter, while gluons are the particles that carry the strong force, acting like invisible glue to bind the quarks into a stable package. For decades, experiments have mapped out how these pieces share the proton's momentum and spin. However, a complete picture of the proton's internal structure requires more than just knowing how fast things move or how they spin; it demands an understanding of how energy and pressure are distributed within this tiny sphere. This is where the concept of "gravitational form factors" comes in. Even though a proton is far too small to be weighed on a scale or to generate a noticeable gravitational pull, physicists use the mathematics of gravity as a powerful tool to probe the proton's mechanical heart. By asking how the proton would respond to a gravitational field, researchers can map out where its mass is concentrated, how its internal pressure is balanced, and how the different forces inside it contribute to its overall stability.

In a recent study, researchers Z. Asmaee and K. Azizi turned their attention to two specific components of this internal landscape: the energy carried by gluons and the energy carried by the electromagnetic field, or photons, within the proton. While the role of quarks in the proton's structure is well-documented, the specific contributions of gluons and photons to its gravitational properties have been much harder to pin down. Gluons are notoriously difficult to study directly because they do not interact with light, and the electromagnetic contribution is often considered a minor player compared to the massive strong force. To investigate these elusive components, the team used a theoretical method known as QCD sum rules. This approach does not rely on direct observation but instead builds a bridge between the fundamental equations of quantum chromodynamics—the theory describing how quarks and gluons interact—and the measurable properties of the proton. By analyzing the mathematical relationships between these equations and the proton's behavior, the researchers could extract specific numbers that describe how mass and energy are distributed in the gluon and photon sectors.

The study focused on calculating four distinct gravitational form factors for both the gluon and photon sectors. These form factors act like a set of coordinates that tell us how the proton's mass, spin, and internal pressure are distributed in space. The researchers found that the gluon sector, which is responsible for the vast majority of the proton's mass, behaves quite differently from the photon sector. Their calculations showed that the gluons are concentrated in a tighter, more compact region near the center of the proton. In contrast, the energy associated with the photon field spreads out over a wider area. Specifically, the team determined that the "mass radius" of the gluon sector is approximately 1.07 femtometers, while the mass radius for the photon sector is larger, at about 1.33 femtometers. A femtometer is a unit of length equal to one quadrillionth of a meter, a scale so small that it is difficult to visualize; for context, it is roughly the size of a single proton. This difference in size suggests that the energy generated by the strong force is packed more densely than the energy associated with the proton's electromagnetic field.

Beyond just the size of these distributions, the researchers also looked at how the proton's internal pressure and mechanical stability are maintained by these different components. They found that the gluon contribution to the proton's momentum and angular momentum aligns well with previous theoretical predictions, reinforcing our understanding that gluons are the primary carriers of the proton's spin and motion. However, the study also highlighted that the "D-term," a specific form factor that describes the internal forces and pressure distribution, shows more variation depending on the method used to calculate it. This suggests that while the general picture of the proton's structure is becoming clearer, the precise details of how internal forces balance each other remain sensitive to the theoretical approach. The photon contribution, while much smaller in magnitude than the gluon contribution, was successfully isolated and calculated within a unified framework, addressing a gap where systematic determinations of complete photon gravitational form factors had previously remained largely unexplored. This provides a new baseline for understanding the electromagnetic role in the proton's gravitational structure.

The results of this work offer a fresh perspective on the proton's internal architecture, confirming that the strong force creates a more localized energy distribution than the electromagnetic force. By separating the gluon and photon contributions, the study provides a clearer view of how different fundamental forces shape the matter that makes up our world. These findings serve as a valuable reference for future experiments, such as those planned at the Electron-Ion Collider, which aim to probe the proton's structure with even greater precision. They also offer a benchmark for other theoretical methods, including lattice QCD simulations, helping to refine our collective understanding of how the universe's most fundamental particles hold themselves together. The study does not claim to have solved every mystery of the proton's interior, but it successfully maps out the distinct territories of the gluon and photon sectors, revealing a proton that is not just a uniform ball of energy, but a complex system where different forces carve out their own unique spatial footprints.

Technical Summary: Photon and Gluon Gravitational Form Factors of the Proton in QCD

Problem Statement
The internal gravitational structure of the proton is encoded in the gravitational form factors (GFFs) derived from the matrix elements of the QCD energy-momentum tensor (EMT). While significant progress has been made in understanding the total proton GFFs and the quark sector, the individual contributions of the gluon and photon sectors remain less explored. Accessing the gluonic structure is particularly challenging due to the absence of direct electromagnetic probes for gluons, and the photon contribution to the EMT has historically received limited systematic attention in composite hadrons. A unified, nonperturbative description of the separate gluon and photon contributions to the proton's energy-momentum structure is currently incomplete. This work addresses the need to calculate the photon and gluon GFFs independently to understand how these specific dynamical sectors contribute to the proton's mass, angular momentum, and mechanical properties.

Methodology
The authors employ the framework of Quantum Chromodynamics Sum Rules (QCDSR) to investigate the photon and gluon GFFs. The methodology involves the following steps:

Correlation Function Construction: A three-point correlation function is defined involving proton interpolating currents and the specific EMT operators for the gluon ( $T^g_{\mu\nu}$ ) and photon ( $T^\gamma_{\mu\nu}$ ) sectors.
Hadronic Representation: On the physical side, the correlation function is expressed using a complete set of intermediate hadronic states. The ground-state proton contribution is isolated using a double Borel transformation to suppress excited states and the continuum. The matrix elements are parametrized in terms of four independent GFFs: $A_j(t)$ (mass/energy), $J_j(t)$ (angular momentum), $D_j(t)$ (D-term/mechanical), and $\bar{c}_j(t)$ (non-conserved scalar), where $j$ denotes the gluon or photon sector.
QCD Representation: On the theoretical side, the correlation function is calculated using the Operator Product Expansion (OPE) in terms of quark and gluon degrees of freedom.
- Gluon Sector: The calculation includes two distinct classes of contributions: (i) the vacuum expectation value of the gluon field-strength tensors (gluon condensate) and (ii) the interaction of gluon fields with light-quark fields via the quark-gluon vertex, which induces corrections to the light-quark propagators.
- Photon Sector: Since the vacuum expectation value of the electromagnetic EMT vanishes, the non-vanishing contribution arises solely from the coupling of photon fields to light-quark fields through the quark-photon interaction vertex, leading to photon-induced corrections to the quark propagators.
- The OPE is carried out up to dimension-seven operators, including perturbative terms and nonperturbative condensate contributions (quark condensate, mixed condensate, and gluon condensate).
Numerical Analysis: The sum rules are matched by equating the coefficients of independent Lorentz structures in the hadronic and QCD representations. The analysis utilizes standard input parameters (quark masses, condensates, coupling constants) and determines working intervals for the Borel parameter ( $M^2$ ) and continuum threshold ( $s_0$ ) based on pole contribution and OPE convergence criteria.

Key Contributions

Unified Framework: The paper provides a simultaneous calculation of both gluon and photon GFFs within a single QCDSR framework, allowing for a direct comparison of their roles in the proton's gravitational structure.
Explicit Non-Conserved Term: By treating the gluon and photon sectors separately, the analysis consistently retains the non-conserved form factor $\bar{c}_j(t)$ , which is often omitted or constrained when considering the total EMT.
Propagator Corrections: The derivation explicitly incorporates interaction-induced corrections to the light-quark propagators arising from the insertion of the EMT operators, a crucial step for capturing the nonperturbative dynamics of the gluon and photon sectors.

Results

Form Factors: The momentum transfer ( $Q^2$ $Q^{2}$ ) dependence of the four GFFs ( $A, J, \bar{c}, D$ $A, J, \overset{c}{ˉ}, D$ ) for both gluon and photon sectors was extracted and fitted using a $p$ $p$ -pole parametrization.
- The gluon GFFs exhibit significantly larger magnitudes than the photon GFFs, reflecting the dominant role of the strong interaction in the proton's energy-momentum structure.
- The gluon results at zero momentum transfer ( $A_g(0) \approx 0.46$ , $J_g(0) \approx 0.26$ , $D_g(0) \approx -2.54$ ) are in good agreement with previous lattice QCD, holographic QCD, and phenomenological studies.
- The photon GFFs are found to be small ( $A_\gamma(0) \approx 0.03$ ), consistent with the subleading nature of the electromagnetic contribution to the proton's mass and momentum.
Radii: Using the derived GFFs, the mass radii ( $\langle r_m \rangle$ $⟨ r_{m} ⟩$ ) and scalar radii ( $\langle r_s \rangle$ $⟨ r_{s} ⟩$ ) were calculated:
- Gluon: $\langle r_m \rangle_g = 1.066 \pm 0.013$ fm, $\langle r_s \rangle_g = 1.307 \pm 0.035$ fm.
- Photon: $\langle r_m \rangle_\gamma = 1.327 \pm 0.003$ fm, $\langle r_s \rangle_\gamma = 1.542 \pm 0.017$ fm.
- The results indicate that the gluon mass radius is smaller than the photon mass radius, suggesting a more localized distribution of gluonic energy within the proton compared to the electromagnetic contribution.
Comparison: The gluon mass radius obtained is consistent with lattice QCD (tripole fit) and holographic QCD results, while the scalar radius shows a wider spread across different theoretical approaches, attributed to its sensitivity to the $D$ -term.

Significance
The authors claim that this work provides new nonperturbative information on the gravitational structure of the proton, specifically isolating the gluon and photon sectors. The findings offer complementary theoretical results for future lattice QCD calculations and phenomenological studies. By establishing a unified description of these sectors, the study aids in understanding how different fundamental degrees of freedom contribute to the proton's energy-momentum structure. The explicit determination of the photon GFFs fills a gap in the literature, while the gluon results reinforce the understanding of the gluonic contribution to proton spin and mass decomposition. The paper positions these results as a basis for further investigations into the QCD energy-momentum tensor and the internal mechanical properties of hadrons.

Technical Summary: Photon and Gluon Gravitational Form Factors of the Proton in QCD

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