Quasi Parton Distribution Functions in Covariant Quark Models

This paper establishes general proofs for the convergence and sum rules of unpolarized quark and antiquark quasi parton distribution functions (QPDFs) within a broad class of gauge-free covariant quark models, specifically illustrating these findings with the Covariant Parton Model to derive analytical results for small-$x_v$ behavior and energy-momentum tensor form factors.

The Gist

Imagine the inside of a proton (a tiny particle inside an atom) as a bustling city filled with smaller particles called quarks. Physicists want to take a "snapshot" of how these quarks are moving and distributed. To do this, they use a map called a Parton Distribution Function (PDF). Think of a PDF as a perfect, high-resolution map of the city's traffic, showing exactly where every car is and how fast it's going.

However, there's a problem: creating this perfect map is incredibly difficult in the real world (specifically, in the mathematical framework of Quantum Chromodynamics or QCD). It's like trying to photograph a speeding car with a camera that only works in a specific type of light that doesn't exist in our current labs.

The New Tool: "Quasi" Maps (QPDFs)

To get around this, physicists invented a new tool called Quasi Parton Distribution Functions (QPDFs).

• The Analogy: Imagine you can't take a photo of the city while it's moving fast. Instead, you take a photo of the city while it's moving slowly, and then you use a special mathematical "zoom lens" to speed it up in your mind until it looks like the fast-moving city.
• How it works: QPDFs are like taking a picture of the quarks while the proton is moving at a very high speed (but not quite the speed of light). As the proton gets faster and faster, approaching the speed of light, this "Quasi" map slowly transforms and becomes identical to the perfect "PDF" map.

The Experiment: Testing the Lens with a Model

The authors of this paper wanted to understand how well this "zoom lens" works. They didn't just look at the real, messy universe; they built a simulation (a model) to test it.

They used a specific simulation called the Covariant Parton Model (CPM).

• The Metaphor: Think of the real world as a chaotic city with traffic jams, accidents, and complex rules (interactions between particles). The CPM is like a simplified, toy version of that city where the cars (quarks) don't crash into each other; they just drive in straight lines. This makes it much easier to see how the math works without getting lost in the chaos.

Key Findings from the Paper

1. The "Leaking" Phenomenon
In the perfect map (PDF), quarks and anti-quarks (the opposite of quarks) live in separate neighborhoods. But in the "Quasi" map (when the proton isn't moving at light speed yet), these neighborhoods start to bleed into each other.

• The Metaphor: Imagine a crowd of people wearing red shirts (quarks) and blue shirts (anti-quarks). When the crowd is standing still, the groups are mixed. But as the crowd starts running, the red shirts stay on the left and blue on the right. However, at medium speeds, some red shirts might accidentally run into the blue zone, and vice versa. The paper shows exactly how much they "leak" into each other's territory depending on how fast the proton is moving.

2. Two Different Camera Angles (Gamma 0 vs. Gamma 3)
The researchers tested two different ways to take the "Quasi" picture, which they call $\Gamma = \gamma_0$ and $\Gamma = \gamma_3$.

• The Result: They found that one angle ($\gamma_3$) is generally better. It converges (becomes the perfect map) faster and more smoothly, especially when looking at the "edges" of the city (where the quark numbers are very small or very large). The other angle ($\gamma_0$) sometimes creates weird wiggles or sign reversals (where the map says "negative traffic" in a place where there should be none) before it settles down.

3. The "Wandzura-Wilczek" Approximation
The paper notes that their simulation (CPM) essentially acts like a specific, simplified rule in physics called the "Wandzura-Wilczek approximation."

• The Metaphor: This is like saying, "If we ignore all the complicated arguments the quarks have with each other, we can predict their behavior with surprising accuracy." The paper shows that even with this simplification, the model correctly predicts how the "Quasi" maps turn into the "Real" maps.

4. Comparing to Real Lattice Calculations
The authors compared their simple toy model results to actual, complex computer simulations done by other scientists (called "Lattice QCD").

• The Finding: The toy model and the complex computer simulation agreed reasonably well in the middle of the map. However, they differed at the edges. The authors suggest this difference might be due to the fact that their toy model assumes quarks are "on-shell" (like perfect, free-moving cars), while the real world involves "off-shell" effects (cars that are accelerating, braking, or interacting). This difference helps physicists understand what parts of the complex computer simulations are due to the physics of the quarks themselves versus the limitations of the computer methods.

Summary

In simple terms, this paper is a stress test for a new mathematical tool. The authors used a simplified, easy-to-understand model of the proton to prove that:

1. The "Quasi" maps do indeed turn into the perfect "Real" maps when the proton moves fast enough.
2. There is a specific way to take these pictures ($\gamma_3$) that is cleaner and less prone to errors than the other way.
3. Even a simplified model can teach us valuable lessons about how complex computer simulations (Lattice QCD) behave, helping scientists understand where the "noise" in their data comes from.

The paper does not claim to cure diseases or build new technology; it is purely about refining the theoretical "maps" physicists use to understand the fundamental building blocks of the universe.

Technical Summary

Technical Summary: Quasi Parton Distribution Functions in Covariant Quark Models

Problem Statement
Parton Distribution Functions (PDFs) are fundamental to describing the internal structure of hadrons in deep-inelastic scattering. However, their standard definition involves non-local QCD operators with lightlike separations, rendering them inaccessible to direct computation in Euclidean lattice QCD. The introduction of Quasi Parton Distribution Functions (QPDFs) offers a pathway to compute these quantities on the lattice by evaluating matrix elements of hadrons moving with a finite velocity $v$ using spacelike separations. While QPDFs converge to physical PDFs in the limit $v \to 1$, this convergence is complicated by power corrections (suppressed by $M^2/P_z^2$) and distinct renormalization properties. A comprehensive understanding of these convergence mechanisms and the behavior of QPDFs at finite velocity is necessary to interpret lattice results and understand off-shell effects. This paper addresses the need to study QPDFs within theoretical frameworks that respect Lorentz symmetry but lack explicit gauge degrees of freedom, allowing for analytical insights into convergence and sum rules that are difficult to isolate in full QCD.

Methodology
The authors employ a two-pronged approach:

1. General Quark Model Framework: The first part of the work establishes a general formalism for unpolarized QPDFs in any quark model characterized by the absence of gauge fields (e.g., no Wilson lines). The models are constrained only by Lorentz symmetry and the requirement that amplitudes are either finite or renormalized/regularized to preserve Lorentz invariance. The authors express QPDFs in terms of Lorentz-invariant amplitudes $A_i(P \cdot k, k^2)$ derived from the quark correlator. They utilize the nucleon velocity $v$ as the primary variable rather than momentum $P_z$ to simplify expressions and demonstrate the convergence to PDFs as $v \to 1$.
2. Covariant Parton Model (CPM): The second part applies this general framework to a specific model, the Covariant Parton Model (CPM). The CPM is a systematic extension of Feynman's parton model where partons are on-shell. The authors derive analytical expressions for QPDFs, their sum rules, and their behavior at small and large momentum fractions ($x_v$). They utilize NLO LHAPDF parametrizations as input to generate numerical results for various quark flavors ($u, d, \bar{u}, \bar{d}, s, \bar{s}$) and compare the convergence of two operator choices: $\Gamma = \gamma^0$ and $\Gamma = \gamma^3$.

Key Contributions and Results

• General Proofs and Sum Rules: The authors provide general proofs demonstrating that in any Lorentz-invariant quark model, unpolarized QPDFs converge to the corresponding PDFs as $v \to 1$. They derive independent proofs for the flavor number sum rules and the momentum sum rules (second Mellin moments) for both $\Gamma = \gamma^0$ and $\Gamma = \gamma^3$. Notably, they show that the momentum sum rule for $\Gamma = \gamma^3$ involves a form factor $\bar{c}_q(0)$ related to the non-conservation of individual quark contributions to the energy-momentum tensor, whereas the $\Gamma = \gamma^0$ case does not exhibit this specific twist-4 contamination in the sum rule.
• Radyushkin Formula: The paper re-proves the Radyushkin formula, which relates QPDFs to Transverse Momentum Dependent PDFs (TMDs), demonstrating that this relation holds in general quark models as a consequence of Lorentz invariance.
• Choice of Operator ($\gamma^0$ vs. $\gamma^3$): A significant finding is the comparison between the two operator choices. While coordinate-space arguments suggest $\Gamma = \gamma^0$ is preferable due to the absence of power-suppressed terms, the momentum-space analysis reveals that for $\Gamma = \gamma^3$, power corrections present in the leading term are mutually compensated by the power-suppressed term proportional to the amplitude $J_q$. Consequently, the authors suggest that $\Gamma = \gamma^3$ may exhibit faster convergence to the PDF limit in simple quark models, a hypothesis supported by their numerical results.
• "Leaking" Phenomenon: The study identifies a "leaking" phenomenon where, at finite velocities ($v < 1$), the distinction between quark and antiquark contributions is lost. Specifically, the covariant distribution for quarks ($G_q$) contributes to the antiquark QPDF region and vice versa. This effect is more pronounced for $\Gamma = \gamma^0$, where contributions can have opposite signs leading to cancellations and even sign reversals in the QPDF at low velocities. For $\Gamma = \gamma^3$, the leaking contributions have the same sign, leading to a more uniform convergence.
• Analytical Behavior: The authors derive analytical results for the small-$x_v$ and large-$x_v$ behavior of QPDFs in the CPM. They show that the CPM results effectively constitute a Wandzura-Wilczek (WW) approximation, automatically entailing complete target-mass corrections.
• Energy-Momentum Tensor Form Factor: A specific prediction of the CPM (and the WW approximation) is derived: $\bar{c}_q(0) = -\frac{1}{4} A_q(0)$. The authors compare this to phenomenological and lattice QCD results, finding agreement within 20–40%, noting that deviations arise from the trace anomaly in QCD which is absent in the free-parton CPM.
• Numerical Convergence: Numerical results using realistic PDF inputs show that QPDFs converge to PDFs within a few percent accuracy at $v \approx 0.9$. The convergence is observed to be faster for $\Gamma = \gamma^3$ in the small-$x_v$ region compared to $\Gamma = \gamma^0$.
• Lattice Comparison: The paper compares CPM results with recent lattice QCD calculations for the isovector QPDF. While the model captures the general shape and convergence, discrepancies are observed. The authors attribute these differences to the model's on-shell assumption (missing off-shell dynamics) and potential systematic uncertainties in the lattice calculation, such as the truncation of the Fourier transform at small $x_v$.

Significance
The paper claims that its primary significance lies in providing a rigorous, Lorentz-invariant framework to study QPDFs in models without gauge fields, thereby isolating kinematic effects from dynamical QCD interactions. By demonstrating that the CPM results effectively represent a Wandzura-Wilczek approximation, the work offers a benchmark for understanding target-mass corrections and the convergence of QPDFs. The finding that $\Gamma = \gamma^3$ may offer faster convergence in certain regimes challenges previous assumptions based solely on coordinate-space power counting. Furthermore, the comparison with lattice QCD highlights the role of off-shellness effects, suggesting that deviations between model and lattice results could guide the development of more realistic models that relax the on-shell condition. The work serves as a theoretical laboratory to interpret lattice QCD studies of hadron structure, particularly regarding the systematic uncertainties associated with finite momentum and the extraction of PDFs from QPDFs.