⚛️ nuclear theory

Skewness dependence of the pion and kaon generalized parton distributions

Using the covariant Nambu-Jona-Lasinio model with Schwinger proper-time regularization, this study investigates the skewness and momentum transfer dependence of pion and kaon generalized parton distributions, finding that valence-quark distributions are strongly suppressed by increasing momentum transfer and skewness (especially at lower scales) while gluon distributions show weak dependence on both, with results at specific scales aligning well with experimental data, global JAM analyses, and lattice-QCD calculations.

Original authors: Fernando Chandra, Parada T. P. Hutauruk, Terry Mart

Published 2026-08-21

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CC BY 4.0

Original authors: Fernando Chandra, Parada T. P. Hutauruk, Terry Mart

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

Deep inside the protons and neutrons that make up our visible universe, a chaotic yet structured world of quarks and gluons exists. These fundamental particles are bound together by the strong force, the most powerful interaction in nature, but they cannot be seen in isolation. To understand how they are arranged inside a particle like a pion or a kaon, scientists use a tool called a generalized parton distribution. Think of this not as a simple list of ingredients, but as a three-dimensional map that reveals both how fast a particle is moving and where it sits within the larger structure. While we have long known the average speed of these internal particles, a newer question has emerged: how does this map change when the particle is hit hard and knocked off course? This is the realm of "skewness," a measure of how much the particle's momentum shifts during a collision, and "momentum transfer," a measure of the force of that hit. Understanding these shifts is crucial because it connects the internal motion of quarks to the very mass and stability of the matter around us, offering a glimpse into the hidden architecture of the universe.

In a recent study, researchers set out to map these changes for pions and kaons, the lightest and most fleeting members of the meson family. Using a theoretical framework known as the Nambu-Jona-Lasinio model, which effectively simulates how quarks are confined within these particles, the team calculated how the internal distributions of quarks and gluons respond to different levels of impact. They did not rely on a single snapshot but explored a range of scenarios, varying the force of the collision and the angle of the momentum shift. The calculations were then refined to match the energy scales of modern experiments, allowing the team to compare their theoretical maps with global analyses and high-precision computer simulations performed by other groups, given that direct experimental data on these specific distributions remains scarce.

The results revealed a clear and consistent pattern in how these particles behave under pressure. When the force of the collision increases, the distribution of the valence quarks—the primary building blocks of the pion and kaon—becomes noticeably suppressed. In simpler terms, as the particle is struck harder, the likelihood of finding these main quarks in their usual high-speed states drops significantly. This effect is most pronounced at lower energy scales, where the internal structure appears to be more sensitive to the disturbance. Interestingly, the researchers found that this suppression depends heavily on the force of the hit but is surprisingly indifferent to the angle of the momentum shift. Whether the particle is knocked slightly off course or at a sharper angle, the change in the quark distribution remains largely the same, provided the force of the impact is constant.

The behavior of the gluons, the particles that carry the strong force and glue the quarks together, tells a slightly different story. While the gluon distributions also decrease as the collision force increases, they are remarkably stable. They show very little sensitivity to either the force of the hit or the angle of the momentum shift. This suggests that while the primary quarks rearrange themselves significantly under stress, the glue holding them together maintains a more consistent presence across different conditions. When the researchers compared their findings for the forward limit—where the particle is not knocked off course at all—with existing global analyses and available data, their predictions aligned closely. This agreement gives confidence that their model accurately captures the fundamental physics of these particles.

Beyond the distributions themselves, the study also examined the generalized form factors, which are mathematical quantities that summarize the overall shape and mechanical properties of the pion and kaon. The team found that these form factors follow a distinct hierarchy. The form factors for the pion sit between those of the up-quark in the kaon and the strange-quark in the kaon. Specifically, the pion's form factors are stiffer, or more resistant to change, than those of the up-quark in the kaon, but softer, or more flexible, than those of the strange-quark. This ordering holds true across different energy levels and provides a clear, testable prediction for future experiments. The study also compared the ratios of these form factors to recent lattice-QCD simulations, which use supercomputers to solve the equations of the strong force from first principles. The researchers found that their results matched the lattice data well, particularly for the ratio of the pion's form factor to the kaon's up-quark form factor, which remained greater than one as the collision force increased.

This work serves as a vital theoretical bridge for upcoming experiments at major facilities around the world, including the Electron-Ion Collider and upgrades at Jefferson Lab. By providing precise predictions for how pions and kaons behave under various conditions, the study offers a roadmap for experimentalists who are preparing to measure these elusive particles directly. The findings confirm that while the internal structure of these particles is dynamic and responsive to external forces, it follows predictable rules that can be modeled and understood. As scientists move toward more precise measurements of the three-dimensional structure of matter, these insights into the skewness dependence of generalized parton distributions will help refine our understanding of how the universe is built from the ground up.

Technical Summary: Skewness Dependence of the Pion and Kaon Generalized Parton Distributions

Problem and Motivation
Generalized Parton Distributions (GPDs) provide a unified framework for understanding the three-dimensional spatial structure and internal dynamics of hadrons, linking parton distribution functions (PDFs) with electromagnetic form factors. While pion and kaon GPDs at zero skewness ( $\xi = 0$ ) have been studied extensively, their behavior at non-zero skewness remains less constrained due to the scarcity of experimental data and the theoretical challenges in modeling non-perturbative QCD dynamics. Specifically, the dependence of these distributions on the momentum transfer ( $-t$ ) and skewness ( $\xi$ ) is crucial for accessing the energy-momentum tensor and the mechanical structure of these pseudo-Goldstone bosons. This paper addresses the gap in theoretical descriptions of pion and kaon GPDs at non-zero skewness and varying momentum transfers.

Methodology
The authors investigate the skewness dependence of pion and kaon GPDs within the covariant $SU(3)$ Nambu–Jona-Lasinio (NJL) model. Key methodological features include:

Regularization and Confinement: The model employs the Schwinger proper-time regularization scheme to regulate ultraviolet divergences in quark loop integrals, thereby implementing an effective description of quark confinement.
Model Parameters: The calculation utilizes parameters determined from the NJL gap equation and Bethe-Salpeter equation (BSE) solutions, including dressed quark masses ( $M_q$ ), meson masses ( $m_M$ ), and meson-quark coupling constants ( $g_{Mqq}$ ). The inputs assume $SU(2)$ isospin symmetry for up and down quarks.
GPD Calculation: The vector GPDs are calculated using the leading-twist-2 approximation, dominated by two Feynman diagrams involving quark propagators. The expressions are derived analytically for arbitrary quark masses, applicable to both pions ( $\pi$ ) and kaons ( $K$ ).
Evolution: The distributions are calculated at an initial scale $\mu_0^2 = 0.18 \text{ GeV}^2$ and subsequently evolved to higher renormalization scales $\mu^2 = 4 \text{ GeV}^2$ and $\mu^2 = 27 \text{ GeV}^2$ using Next-to-Leading-Order (NLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations.
Kinematic Range: The study covers momentum transfers $-t = 0, 0.11, 0.5, 1.0, 1.5 \text{ GeV}^2$ and skewness values $\xi = 0, 0.05, 0.15, 0.25$ .

Key Contributions and Results
The paper presents numerical results for valence-quark and gluon distributions, as well as generalized form factors (GFFs), for both pions and kaons.

Valence-Quark Distributions:
- $-t$ Dependence: The valence-quark distributions for both pions and kaons exhibit a strong dependence on momentum transfer. As $-t$ increases, the distributions are increasingly suppressed, particularly in the region $0 \le x \lesssim 0.8$ for kaons and $0 \le x \lesssim 0.7$ for pions. This suppression is more pronounced at the lower scale ( $\mu^2 = 4 \text{ GeV}^2$ ).
- $\xi$ Dependence: The dependence on skewness $\xi$ is found to be weak. For pions, the distribution decreases with increasing $\xi$ in the region $0 \le x \le 0.6$ but increases for $x \gtrsim 0.6$ . For kaons, a similar weak dependence is observed, with a decrease for $x \lesssim 0.2$ and an increase for $x > 0.2$ (for up-valence quarks).
- Comparison with Data: In the forward limit ( $\xi=0, -t=0$ ) at $\mu^2 = 27 \text{ GeV}^2$ , the predicted pion valence-quark distribution agrees well with available experimental data (Conway et al.) and the global JAM analysis.
Gluon Distributions:
- The pion and kaon gluon distributions, obtained via NLO DGLAP evolution, show a weak dependence on both $-t$ and $\xi$ .
- Similar to valence quarks, gluon distributions decrease slightly with increasing $-t$ , but the magnitude of change is small compared to the forward limit.
- The predicted pion gluon distribution at $\mu^2 = 4 \text{ GeV}^2$ is consistent with the JAM analysis.
Generalized Form Factors (GFFs):
- The Mellin moments of the GPDs are used to extract generalized form factors $A_{n, 2i}(t)$ .
- The calculated pion GFFs at $\mu^2 = 4 \text{ GeV}^2$ show good agreement with lattice-QCD results and experimental data (e.g., JLab).
- A distinct hierarchy is observed in the evolved GFFs at both $\mu^2 = 4$ and $27 \text{ GeV}^2$ : $A_{n0}^{uK}(t) < A_{n0}^{u\pi}(t) < A_{n0}^{sK}(t)$ for $n=1, 2, 3, 4$ . This indicates that the strange-quark GFFs in the kaon are the "stiffest" (slowest falloff with $-t$ ), followed by the pion, with the up-quark kaon GFFs being the "softest."
- Ratios of form factors, such as $A_{20}^{u\pi}(t)/A_{20}^{uK}(t)$ , are consistent with lattice-QCD simulations, remaining greater than unity but showing a larger magnitude at higher $-t$ compared to lattice results.

Significance and Claims
The authors claim that their study provides a consistent covariant description of pion and kaon GPDs at non-zero skewness, effectively incorporating quark confinement through the NJL model. The work extends previous zero-skewness studies to a broader kinematic range relevant for future experiments.

The paper asserts that its results serve as useful theoretical input for the extraction of pion and kaon GFFs in upcoming experiments at facilities such as the Electron-Ion Collider (EIC), the Electron-ion Collider in China (EicC), AMBER/COMPASS++ at CERN, J-PARC, and the 22-GeV upgrade of Jefferson Lab (JLab). Additionally, the findings offer benchmarks for future lattice-QCD simulations. The authors remain modest regarding the comparison of higher-order moments with lattice data, noting that discrepancies (e.g., in $A_{50}$ ) may be subject to large uncertainties in current lattice calculations.

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