⚛️ phenomenology

Pion generalized parton distributions at zero skewness

This paper systematically calculates the zero-skewness generalized parton distributions of the pion within a covariant Nambu–Jona-Lasinio model using proper time regularization, demonstrating excellent agreement with experimental data, lattice QCD simulations, and global QCD analyses while providing specific values for the pion's scalar, vector, and tensor charge radii.

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

Published 2026-07-09

📖 5 min read🧠 Deep dive

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

Imagine the pion (a tiny particle that acts like a messenger in the atomic nucleus) not as a solid marble, but as a bustling, invisible city made of even smaller particles called quarks. For a long time, scientists have tried to map this city, but the rules of the universe (Quantum Chromodynamics, or QCD) make it incredibly hard to calculate exactly how these quarks move and interact.

This paper is like a team of cartographers (Fernando Chandra, Parada Hutauruk, and Terry Mart) using a specific set of tools to draw a detailed 3D map of this "quark city" at a specific moment in time.

Here is the breakdown of their journey, explained simply:

1. The Map-Making Tool: The NJL Model

To draw the map, the scientists used a theoretical framework called the Nambu–Jona-Lasinio (NJL) model. Think of this as a sophisticated simulation game engine.

The Problem: In their simulation, the math sometimes explodes into infinity (like a video game glitching out).
The Fix: They used a technique called Proper Time Regularization. Imagine this as putting a "speed limit" and a "boundary fence" around their simulation.
- The speed limit stops the math from going to infinity (UV cutoff).
- The fence keeps the quarks from escaping the pion and becoming free particles (IR cutoff). In the real world, quarks are never found alone; they are always stuck inside particles. This fence mimics that "confinement" rule.

2. The "3D Photo": Generalized Parton Distributions (GPDs)

Usually, scientists take a flat, 2D snapshot of the pion, showing how much momentum the quarks have (like a speedometer reading). This is called a Parton Distribution Function (PDF).

However, this paper focuses on GPDs. Think of a GPD as a 3D hologram or a CT scan of the pion. It doesn't just tell you how fast the quarks are going; it tells you:

How fast they are going (longitudinal momentum).
Where they are located inside the pion (transverse position).
How the pion changes shape when you poke it (momentum transfer).

The authors calculated these "holograms" for three different types of quark behaviors:

Vector: The standard movement of the quarks.
Tensor: How the quarks spin or flip.
Scalar: A specific type of interaction related to the mass and structure of the pion.

3. Taking the "Flat Photo" (PDFs)

To check if their 3D hologram was accurate, they flattened it out to create a standard 2D map (the PDF).

The Result: When they compared their flat map to real-world data from experiments and other supercomputer simulations (Lattice QCD), it was a perfect match.
The Analogy: It's like they built a complex 3D model of a car engine, flattened it into a blueprint, and found that the blueprint matched the actual engine specs perfectly. This gave them confidence that their 3D model was correct.

4. Measuring the "Shape" (Form Factors)

Next, they used their 3D map to measure the "size" and "shape" of the pion in different ways. They calculated three specific "form factors" (which are like measuring the radius of a balloon):

Vector Radius: How big the pion looks when you look at its electric charge.
Tensor Radius: How big it looks when you look at its spin.
Scalar Radius: How big it looks when you look at its mass structure.

The "Dressed" vs. "Bare" Surprise:
Initially, their raw calculations (the "bare" pion) were too big compared to real data. It was like measuring a balloon without accounting for the air pressure inside.

They then applied a "dressing" effect (accounting for how the quarks interact with the surrounding energy field).
The Result: Once "dressed," their measurements matched the real-world data and supercomputer simulations almost perfectly.

5. The Final Measurements (Charge Radii)

Finally, they calculated the actual size (radius) of the pion based on these three different views. They found:

Tensor Radius: 0.83 femtometers (the largest).
Vector Radius: 0.63 femtometers.
Scalar Radius: 0.56 femtometers (the smallest).

They noted that the "spin" view (Tensor) makes the pion look the biggest, while the "mass" view (Scalar) makes it look the smallest. This ordering matches what other scientists have found using massive supercomputers.

Summary

In simple terms, this paper says:
"We built a 3D simulation of a pion using a specific set of rules that keep the quarks trapped inside. We checked our work by flattening the 3D data into a 2D map, and it matched real-world data perfectly. We then used this 3D map to measure the pion's size from three different angles. When we accounted for the 'dressing' of the quarks, our size measurements matched the best supercomputer simulations available. We now have a reliable, detailed 3D picture of how the pion is built."

The authors note that while we can't take a direct photo of this 3D structure yet, future giant microscopes (like the Electron-Ion Collider) might one day be able to verify these specific maps.

Technical Summary: Pion Generalized Parton Distributions at Zero Skewness

Problem Statement
The internal structure of the pion, as a pseudo-Goldstone boson associated with spontaneous chiral symmetry breaking in Quantum Chromodynamics (QCD), remains difficult to compute from first principles. While QCD-inspired models offer a pathway to investigate hadron structure, there is a specific need to understand Generalized Parton Distributions (GPDs) at zero skewness ( $\xi = 0$ ). These distributions unify information regarding the pion's Parton Distribution Functions (PDFs) and Electromagnetic Elastic Form Factors (EMFFs), providing a three-dimensional tomographic description. However, direct experimental data for pion GPDs are currently unavailable. Consequently, theoretical frameworks must be developed to predict these distributions and derive observable quantities (PDFs and form factors) that can be compared against existing experimental data, lattice QCD simulations, and global QCD analyses.

Methodology
The authors employ the covariant Nambu–Jona-Lasinio (NJL) model to systematically study pion GPDs. Key methodological features include:

Regularization and Confinement: The study utilizes the Schwinger proper-time regularization scheme. This approach introduces both ultraviolet (UV) and infrared (IR) cutoffs. The UV cutoff ( $\Lambda_{UV}$ ) renders the theory finite, while the IR cutoff ( $\Lambda_{IR}$ ) removes the unphysical quark-production threshold, effectively simulating quark confinement.
Model Parameters: The model assumes $SU(2)$ isospin symmetry with a dynamical quark mass $M_q = 400$ MeV. The IR cutoff is set to $\Lambda_{IR} = 240$ MeV. The coupling constant $G_\pi$ and UV cutoff $\Lambda_{UV}$ are fixed by fitting the physical pion mass ( $m_\pi = 140$ MeV) and pion decay constant ( $f_\pi = 93$ MeV).
Calculations: The authors compute vector, tensor, and scalar pion GPDs at zero skewness.
- PDFs: Derived from the forward limit ( $\xi=0, t=0$ ) of the vector GPDs.
- Form Factors: Obtained via Mellin moments. The vector and tensor form factors are derived from the $n=0$ Mellin moments of the twist-2 GPDs, while the scalar form factor is derived from the twist-3 scalar GPD.
Evolution: To facilitate comparison with high-energy data, the calculated PDFs are evolved from the intrinsic model scale ( $\mu_0^2 = 0.18$ GeV $^2$ ) to higher scales ( $\mu^2 = 4$ and $27$ GeV $^2$ ) using DGLAP QCD evolution equations.
Dressing Effects: The authors distinguish between "bare" form factors (calculated directly from the model) and "dressed" form factors, where quark-photon vertex dressing is included to better match experimental and lattice data.

Key Contributions and Results

Pion GPDs: The study presents numerical results for vector, tensor, and scalar pion GPDs as functions of the longitudinal momentum fraction $x$ and momentum transfer $t$ . The results show that the shapes of these distributions evolve mildly with increasing momentum transfer.
Pion PDFs: The valence-quark pion PDFs derived from the GPDs, after evolution to $\mu^2 = 27$ GeV $^2$ , show excellent agreement with existing experimental data and the JAM global QCD analysis. Similarly, at $\mu^2 = 4$ GeV $^2$ , the results align well with JAM analysis and lattice QCD scales. The study notes that the asymptotic behavior ( $x \to 1$ ) follows a $(1-x)^1$ power counting rule, consistent with JAM but differing from some other models (e.g., DSE, NLChQM) that predict $(1-x)^2$ .
Generalized Form Factors:
- Vector: The bare vector form factor overestimates data, but the dressed vector form factor shows excellent agreement with both recent lattice QCD results and experimental measurements.
- Tensor: Similar to the vector case, the bare tensor form factor overestimates lattice data, while the dressed version agrees well with lattice calculations.
- Scalar: The scalar form factor, derived from twist-3 GPDs, presents a unique result. The bare scalar form factor shows reasonable agreement with lattice data at intermediate $Q^2$ , whereas the dressed scalar form factor underestimates the lattice results.
Charge Radii: The authors compute the charge radii for the scalar, vector, and tensor form factors. The dressed results yield:
- Vector radius ( $r_V^\pi$ ): 0.63 fm
- Tensor radius ( $r_T^\pi$ ): 0.83 fm
- Scalar radius ( $r_S^\pi$ ): 0.56 fm
  These values satisfy the ordering $r_T^\pi \geq r_V^\pi \geq r_S^\pi$ , which is consistent with recent lattice QCD findings and other theoretical models. The dressed vector radius is consistent with experimental measurements and the Particle Data Group average.

Significance and Claims
The paper claims that the covariant NJL model, augmented by the proper-time regularization scheme, provides a robust framework for describing the internal structure of the pion. The primary significance lies in the successful derivation of pion PDFs and form factors from GPDs that are consistent with a wide range of external data (experimental, JAM analysis, and lattice QCD).

The authors modestly emphasize that while direct experimental verification of pion GPDs remains challenging due to the lack of current data, the derived quantities (PDFs and form factors) serve as critical benchmarks. They assert that their results are relevant for future experimental programs, specifically citing the Electron-Ion Collider (EIC) at BNL, the Electron-ion collider in China (EicC), J-PARC, the upgraded JLAB 22 GeV, and COMPASS/AMBER++ at CERN, where such data may become accessible. The study validates the model's approach by reproducing known observables (PDFs and EMFFs) from the GPD formalism, thereby supporting the use of this framework for future three-dimensional tomographic studies of the pion.