⚛️ phenomenology

Gluon GTMDs in the exclusive electroproduction of heavy-quark pairs

This paper investigates the exclusive electroproduction of heavy quark-antiquark pairs off unpolarized nucleons by calculating structure functions in terms of gluon Generalized Transverse Momentum Dependent distributions (GTMDs) and Generalized Parton Distributions (GPDs) at leading order, introducing a new Lorentz-basis decomposition of the gluon-gluon correlation matrix to clarify helicity contributions relevant for future Electron Ion Collider experiments.

Original authors: Mattia Bellotti, Daniël Boer, Cristian Pisano

Published 2026-07-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mattia Bellotti, Daniël Boer, Cristian Pisano

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 atom not as a tiny solar system, but as a bustling, chaotic city inside a microscopic bubble. For decades, scientists have been trying to map the streets of this city, specifically the neighborhood where the "glue" lives. This glue, made of particles called gluons, is what holds the heavyweights of the city—protons and neutrons—together. But these gluons aren't just sitting still; they are zooming around, spinning, and carrying the city's momentum. To understand how the city works, physicists use special maps called "Generalized Parton Distributions" (GPDs). Think of GPDs as a 2D street map that tells you where the gluons are and how fast they are moving in a straight line.

However, a flat map isn't enough to capture the full, wild energy of the city. The gluons also have a "sideways" motion, a kind of jittery dance that the flat map misses. To see this, scientists need a 3D hologram, a more complex tool called "Generalized Transverse Momentum Dependent parton distributions" (GTMDs). If GPDs are a street map, GTMDs are a 3D simulation that shows not just where the gluons are, but exactly how they are spinning and swaying as they zip through the proton. Understanding this dance is crucial because it might hold the secret to where the proton's spin comes from—a mystery that has puzzled scientists for years. The big question is: Can we actually see this 3D dance in real experiments, or is it just a mathematical fantasy?

This paper, written by Mattia Bellotti, Daniël Boer, and Cristian Pisano, takes a giant step toward answering that question by looking at a very specific, high-energy event: smashing an electron into a proton to create a pair of heavy particles (like a charm or bottom quark) while keeping the proton intact. The authors act like master architects, designing a new set of blueprints to describe this collision. They calculated exactly what the "scattering pattern" (the way the particles fly out) should look like if the gluons are behaving according to the complex 3D GTMD rules.

The team performed two different calculations for the same event. First, they used the detailed 3D GTMD maps to predict the outcome. Then, they used the simpler, flat 2D GPD maps to see what would happen if they ignored the sideways dance. They found that when you zoom out and look at the big picture, the two calculations match perfectly. This is a huge relief for the scientific community; it proves that their new, complex 3D math is consistent with the old, trusted 2D math. But the real magic is in the details. The authors discovered that the collision creates a unique "fingerprint" in the form of wiggles and waves in the angles at which the heavy particles fly out. These wiggles are like musical notes that only the 3D GTMDs can produce.

Crucially, the paper shows that you don't need to shoot a spinning, polarized proton at the target to see these effects. Even if the proton is just a "normal," unpolarized blob, the heavy particles it spits out will still carry the secret signature of the gluons' internal spin and orbital motion. It's as if the proton is a drum that, even when hit gently, vibrates in a way that reveals the tension of the drumhead. The authors have provided a complete list of all the possible "vibrations" (angular modulations) that could be seen in future experiments at the Electron Ion Collider. They haven't measured these vibrations yet—that's for the experimentalists to do—but they have provided the exact recipe and the musical score to listen for. By extending the known formulas and introducing a cleaner way to organize the math, this paper gives scientists the tools they need to finally decode the 3D dance of the gluons inside the proton.

Technical Summary: Gluon GTMDs in the Exclusive Electroproduction of Heavy-Quark Pairs

Problem and Motivation
Hard exclusive processes, such as Deeply Virtual Compton Scattering (DVCS), have been extensively utilized to study Generalized Parton Distributions (GPDs), primarily for quarks. While exclusive dijet production and heavy quarkonium production have been suggested as probes for gluon GPDs, particularly in the small- $x$ region, analyses indicate that the relevant distributions are often transverse momentum dependent (TMD) rather than purely collinear. This necessitates a description in terms of more general quantities: Wigner distributions or Generalized Transverse Momentum Dependent parton distributions (GTMDs). GTMDs are crucial for investigating orbital angular momentum and spin-orbit correlations within the nucleon. Although previous studies have shown sensitivity to transverse momentum integrals of GTMDs through various processes, there is a need to extend this framework to open heavy quark pair production (e.g., $D$ -meson or $B$ -meson pairs). This paper addresses the exclusive electroproduction of heavy quark-antiquark pairs off nucleons, aiming to express the differential cross section in terms of gluon GTMDs and to verify consistency with the GPD description in the collinear limit.

Methodology
The authors calculate the differential cross section for the process $e N \to e Q \bar{Q} N$ (where $Q$ is a heavy quark) in the framework of perturbative Quantum Chromodynamics (pQCD) at leading order (LO) in $\alpha_s$ and first order in a collinear expansion.

Formalism: The study utilizes the gluon-gluon correlator defined within QCD. The authors introduce a decomposition of this correlator into a Lorentz basis of symmetric traceless tensors constructed from the partonic transverse momentum $k_T$ and the momentum transfer $\Delta_T$ .
GTMD Parametrization: The correlator is parametrized in terms of sixteen leading-twist GTMDs ( $F^g_i, G^g_i, H^g_i$ ). A specific notation is adopted where these functions relate to nucleon helicity states at the amplitude level (helicity flip and non-flip components) rather than the polarization states of the incoming nucleon. This distinction is critical for understanding which matrix elements contribute to scattering off unpolarized nucleons.
Calculation: The hard scattering subprocess $\gamma^* g \to Q \bar{Q} g$ is calculated using Feynman diagrams. The amplitude is squared, and the result is convoluted with the gluon correlators. The calculation is performed for unpolarized incoming nucleons but includes all possible azimuthal modulations of the final state.
GPD Limit: To ensure consistency, the authors also derive the structure functions in terms of GPDs. This is achieved either by directly using the GPD parametrization of the correlator or by integrating the GTMD expressions over $k_T$ and applying known relations between GTMDs and GPDs (e.g., Eq. 12 and 17).

Key Contributions and Results

GTMD Decomposition: The paper provides a convenient decomposition of the gluon-gluon correlation matrix using a Lorentz basis of symmetric traceless tensors. This formulation explicitly separates contributions based on helicity difference and helicity flip at the amplitude level.
Structure Functions: The authors derive explicit expressions for the structure functions of the differential cross section in terms of integrals of gluon GTMDs (Compton form factors). These expressions cover all possible angular modulations, including terms dependent on $\cos(2(\phi_\Delta - \phi_\perp))$ , $\cos(4(\phi_\Delta - \phi_\perp))$ , and various combinations involving the lepton scattering plane angle $\phi_\ell$ .
GPD Consistency: The study confirms that the GTMD-based calculation matches the GPD-based calculation when the GTMDs are considered to the first order in a collinear expansion. This validates the theoretical framework and clarifies which specific terms in the cross section constrain which GTMDs.
Extension of Literature: The results extend existing GPD expressions found in the literature (e.g., Refs. [1, 11, 12]) by including all azimuthal modulations and explicitly handling the helicity structure for unpolarized beams.
Sivers Function Sensitivity: The analysis reinforces the observation that exclusive scattering with unpolarized nucleons can access helicity-flip matrix elements. Specifically, it highlights sensitivity to the GTMD that reduces to the Sivers TMD in the forward limit, effectively probing the square of the Sivers function contribution in the amplitude squared.
Dipole vs. WW Link: The process considered involves a product of two gluon GTMDs with a dipole link structure ( $[+, -]$ ), distinguishing it from processes like exclusive double $\eta_c$ production which involve a product of four GTMDs with a Weizsäcker-Williams link structure ( $[+, +]$ ). The dipole correlator is noted for its connection to Wilson loop correlators in the small- $x$ limit.

Significance and Claims
The paper claims that the presented results are relevant for future experimental studies of exclusive heavy quark pair production at the Electron Ion Collider (EIC). By providing a complete set of angular modulations and their corresponding structure functions in terms of GTMDs, the work offers a comprehensive toolkit for extracting information about gluon orbital angular momentum and spin-orbit correlations. The authors emphasize that while they restrict their analysis to unpolarized incoming nucleons, the explicit treatment of helicity states at the amplitude level allows for a transparent identification of which helicity-flip and helicity-difference contributions are accessible. The work does not propose new experimental setups but rather provides the theoretical expressions necessary to interpret data from such processes, extending the current understanding of gluon dynamics beyond the collinear approximation.

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