Generalized parton distributions of valence, sea, and gluon components of the proton

Using basis light-front quantization (BLFQ) with a confining-potential-free QCD Hamiltonian, this study computes generalized parton distributions (GPDs) for valence quarks, sea quarks, and gluons in the proton, including quark GPDs at nonzero skewness for the first time within this framework, yielding results that qualitatively align with but are smaller than the GUMP1.0 global extraction.

The Gist

Imagine the proton not as a solid, tiny marble, but as a bustling, chaotic city inside a microscopic bubble. For decades, scientists thought this city was simple: just three main residents (quarks) holding hands. But we now know it's a complex metropolis filled with a permanent population (valence quarks), a transient crowd of visitors (sea quarks), and a constant flow of energy and force carriers (gluons) zipping around.

This paper is like a high-tech 3D architectural blueprint of that city, created by a team of physicists using a powerful new simulation tool called BLFQ (Basis Light-Front Quantization).

Here is the story of their discovery, broken down into simple concepts:

1. The Challenge: Seeing the Invisible

To understand how a proton works, scientists need to know not just what is inside it, but where everything is and how it's moving.

• The Problem: You can't take a photo of a proton with a camera because it's too small and the particles move too fast.
• The Solution: They use "Generalized Parton Distributions" (GPDs). Think of GPDs as a super-advanced GPS map that tells you the location, speed, and spin of every particle inside the proton at the same time.

2. The Tool: The "Light-Front" Camera

The authors used a specific method called Light-Front Quantization.

• The Analogy: Imagine trying to understand a speeding train. If you take a photo from the side, it's a blur. But if you take a photo from a helicopter flying alongside the train at the exact same speed, the train looks frozen, and you can see every passenger clearly.
• The Application: This method "freezes" the proton in time from a specific angle, allowing the team to calculate the wave functions (the probability maps) of the particles inside.

3. The Simulation: Building the City

The team didn't just look at the three main quarks. They built a simulation that included:

• The Core: The three original quarks.
• The Glue: Gluons (the force that holds the quarks together).
• The Visitors: "Sea" quarks and antiquarks that pop in and out of existence like bubbles in boiling water.

They solved a massive mathematical puzzle (the Hamiltonian) to see how these components interact without needing to force them to stay together with an artificial "glue" rule. Nature does the gluing naturally through their equations.

4. The Discovery: Two Different Neighborhoods

The paper maps out two distinct "neighborhoods" inside the proton's GPS map:

• The DGLAP Region (The Main Street): This is where the main traffic flows. Here, the particles are moving in the same direction as the proton. The map shows that the "main street" is dominated by the three core quarks and the gluons.
• The ERBL Region (The Side Alley): This is a trickier area where particles can be moving in opposite directions or where a particle-antiparticle pair is created and destroyed.
  • The Surprise: The team was able to map this "side alley" for the first time using their method. They found that the "sea" of temporary particles is very active here, creating a complex, fluctuating pattern.

5. The Comparison: The "Low-Res" vs. "High-Res" Photo

The simulation starts at a "low resolution" (a low energy scale), which is like looking at the proton city with a slightly blurry lens.

• The Evolution: To compare their work with real-world experiments (like those at the Large Hadron Collider), they used "QCD Evolution" to sharpen the lens. This is like taking a low-res photo and using AI to upscale it to 4K, adding the details that appear at higher energies.
• The Result: When they zoomed in, their "upscaled" map looked very similar to the best maps scientists have made by combining data from many different experiments (called the GUMP 1.0 model).
  • The Catch: Their map was a bit "smaller" (less intense) than the experimental one, but it captured the same general shape and features. This is a huge success because they built this map from first principles (the laws of physics) rather than just fitting a curve to data.

6. The Payoff: Predicting the Future

Finally, they used their map to predict the outcome of a specific experiment called Deeply Virtual Compton Scattering (DVCS).

• The Analogy: If you throw a stone (a photon) at a pond (the proton), the ripples (the scattering) tell you about the shape of the pond.
• The Outcome: Their predictions matched the ripples observed in real experiments very well. This proves their "architectural blueprint" is accurate.

Summary

In short, this paper is a milestone in understanding the proton's interior.

• They built a 3D, dynamic map of the proton using only the fundamental laws of physics.
• They successfully mapped the hidden "sea" of particles that usually gets ignored.
• Their map matches real-world data, confirming that their theoretical approach is a powerful way to see inside the building blocks of our universe.

It's like finally getting a clear, high-definition tour of a city that was previously only visible as a blurry smudge.

Technical Summary

1. Problem Statement

Generalized Parton Distributions (GPDs) are fundamental quantities in Quantum Chromodynamics (QCD) that unify Parton Distribution Functions (PDFs) and electromagnetic form factors. They provide a three-dimensional tomographic picture of the nucleon, encoding information about the spatial distribution, spin, and orbital angular momentum of partons (quarks and gluons).

Despite extensive experimental efforts (e.g., at JLab, HERA, and future EICs), a rigorous first-principles theoretical calculation of GPDs that simultaneously includes valence quarks, sea quarks, and gluons remains a significant challenge. Existing approaches often rely on phenomenological models or are limited to specific Fock sectors (e.g., only valence quarks). There is a particular need for ab initio calculations that can describe the transition between the DGLAP region (parton emission/absorption) and the ERBL region (parton pair creation/annihilation) at non-zero skewness ($\xi$), which is crucial for understanding sea quark and gluon dynamics.

2. Methodology

The authors employ the Basis Light-Front Quantization (BLFQ) framework to solve the light-front QCD (LFQCD) Hamiltonian eigenvalue problem.

• Hamiltonian and Fock Space:
  • The study utilizes a light-front QCD Hamiltonian in the light-cone gauge ($A^+=0$) without an explicit confining potential.
  • The calculation includes three dominant Fock sectors:
    1. $|qqq\rangle$ (three valence quarks)
    2. $|qqqg\rangle$ (three quarks + one gluon)
    3. $|qqq q\bar{q}\rangle$ (three quarks + one quark-antiquark pair)
  • The inclusion of the $|qqq q\bar{q}\rangle$ sector is critical as it allows for the description of sea quarks and non-diagonal transitions required for the ERBL region.
• Basis and Truncation:
  • The basis consists of plane waves in the longitudinal direction (within a box of length $2L$) and 2D Harmonic Oscillator (2D-HO) states in the transverse plane.
  • Truncation parameters are set to $N_{max} = 7$ (transverse excitation) and $K = 16.5$ (longitudinal momentum resolution).
  • Model parameters (quark masses, coupling constants) are fitted to reproduce the proton mass and electromagnetic properties.
• GPD Calculation:
  • DGLAP Region ($|x| > \xi$): Calculated via diagonal overlaps of identical $N$-particle Fock states.
  • ERBL Region ($|x| < \xi$): Calculated via non-diagonal overlaps involving transitions between different Fock sectors (specifically $N \to N+2$), enabled by the inclusion of the $|qqq q\bar{q}\rangle$ sector.
  • The authors compute the chiral-even GPDs $H(x, \xi, t)$ and $E(x, \xi, t)$ for up, down, and strange quarks, as well as gluons.
• Evolution and Observables:
  • The calculated GPDs are defined at a low hadronic scale (non-perturbative regime).
  • QCD Evolution: The results are evolved to higher momentum scales ($\mu^2 = 1, 3, 10$ GeV$^2$) using the APFEL++ code to solve the coupled DGLAP and ERBL evolution equations.
  • Compton Form Factors (CFFs): The evolved GPDs are convoluted to compute CFFs, which are directly comparable to Deeply Virtual Compton Scattering (DVCS) experimental data.

3. Key Contributions

• First BLFQ Calculation at Non-Zero Skewness: This is the first application of the BLFQ framework to evaluate quark GPDs at non-zero skewness ($\xi \neq 0$) covering both the DGLAP and ERBL regions.
• Inclusion of Sea Quarks and Gluons: By incorporating the $|qqq q\bar{q}\rangle$ Fock sector, the authors provide a systematic description of sea-quark GPDs and their contribution to the ERBL region, alongside gluon GPDs.
• Unified First-Principles Approach: The work demonstrates a fully relativistic, non-perturbative solution of the QCD Hamiltonian that generates valence, sea, and gluon distributions simultaneously without relying on ad-hoc confinement potentials.
• Comparison with Global Fits: The study provides a direct comparison of BLFQ predictions with the GUMP 1.0 global extraction (based on experimental and lattice QCD data) and neural-network-based global analyses of DVCS observables.

4. Key Results

• Qualitative Features: The calculated GPDs exhibit qualitative features consistent with global extractions (e.g., GUMP 1.0) but are systematically smaller in magnitude.
• Region-Specific Behavior:
  • DGLAP Region: Dominated by the $|qqq\rangle$ and $|qqqg\rangle$ sectors. The up-quark distribution ($H_u$) is significantly larger than others.
  • ERBL Region: Dominated by the $|qqq q\bar{q}\rangle$ sector. The sea contributions show specific symmetry properties (e.g., $H_d$ and $H_s$ are nearly symmetric under $x \to -x$), while valence contributions dominate the sign of $H_u$.
  • Gluon GPDs: At the initial low scale, gluon distributions do not rise at small $x$ (unlike at experimental scales); instead, they peak around $x \approx 0.3$. The strong enhancement at small $x$ is generated dynamically through QCD evolution.
• Evolution Effects: As the energy scale increases, the GPDs are suppressed at large $x$ and enhanced at small $x$. A new peak emerges at small $x$ ($x \approx 0.075$) for most distributions, except $E_d$, where valence and sea contributions cancel.
• Compton Form Factors (CFFs):
  • The imaginary part of the CFF ($\text{Im } \mathcal{H}$) agrees well with global analysis data in the range $0.03 < \xi < 0.13$.
  • Deviations occur at very low $\xi$ (BLFQ predicts larger values) and high $\xi$ (BLFQ predicts smaller values).
  • The real part of the CFF is consistent with experimental results.

5. Significance

This work represents a major step forward in the first-principles understanding of nucleon structure. By successfully computing GPDs for valence, sea, and gluon components within a single, unified Hamiltonian framework, the authors bridge the gap between non-perturbative QCD dynamics and experimental observables.

• Validation of BLFQ: The results validate the BLFQ framework as a viable tool for calculating complex 3D nucleon structure observables, including the challenging ERBL region.
• Insight into Sea and Gluon Dynamics: The study elucidates how sea quarks and gluons contribute to the proton's spin and spatial structure, revealing that their behavior is distinct from valence quarks and highly sensitive to the Fock sector composition.
• Benchmark for Future Experiments: The predictions for CFFs and evolved GPDs provide a crucial benchmark for current and future facilities like the Electron-Ion Collider (EIC) and the 12 GeV upgrade at JLab, helping to guide the interpretation of DVCS and DVMP data.
• Theoretical Consistency: The agreement with global analyses (GUMP 1.0) and neural-network fits, despite the model being defined at a low scale, suggests that the underlying non-perturbative dynamics captured by BLFQ are robust and consistent with the broader QCD landscape.