Nucleon Parton Distribution Functions from Boosted Correlations in the Coulomb gauge

This study presents the first exploratory lattice QCD calculation of nucleon unpolarized, helicity, and transversity parton distribution functions using boosted correlators in the Coulomb gauge, demonstrating that the method yields results compatible with global analyses for valence-quark distributions while highlighting the need to address excited-state contamination in full-quark-channel extractions.

The Gist

Imagine the proton (the core of every atom in your body) not as a solid marble, but as a bustling, chaotic city inside a tiny, invisible bubble. Inside this city, there are tiny messengers called quarks zooming around at incredible speeds. Scientists want to know exactly how these messengers are distributed: how many are there, how fast are they going, and how are they spinning? This map of the city is called a Parton Distribution Function (PDF).

For decades, trying to draw this map has been like trying to photograph a speeding race car with a camera that only takes pictures of stationary objects. It's incredibly hard.

This paper introduces a new, clever camera technique to take a clearer picture of the proton's interior. Here is the story of how they did it, explained simply.

1. The Old Problem: The "Wilson Line" Rope

Previously, scientists used a method called LaMET (Large-Momentum Effective Theory) to study these quarks. To do this, they had to connect two quarks with a theoretical "rope" (called a Wilson line) to measure how they interact.

• The Analogy: Imagine trying to measure the wind speed by tying a long, heavy rope between two points. The rope itself gets tangled, creates drag, and is very hard to keep straight. In the math world, this "rope" creates huge, messy errors (called linear divergences) that make the calculations very difficult and slow.

2. The New Solution: The "Coulomb Gauge" Shortcut

The authors of this paper tried a different approach. They decided to stop using the heavy "rope" entirely. Instead, they used a specific mathematical setting called the Coulomb gauge.

• The Analogy: Instead of dragging a heavy rope through the city, they decided to look at the traffic patterns from a specific, fixed angle where the roads are naturally straight and clear. By removing the rope, they removed the tangles and the drag.
• The Result: The calculations became much cleaner, faster, and more efficient. It's like switching from a muddy, winding dirt path to a smooth, straight highway.

3. The Experiment: Boosting the Proton

To see the quarks clearly, you can't just look at a proton sitting still; they are too blurry. You have to "boost" it to high speeds.

• The Setup: The scientists used a supercomputer to simulate a proton moving at nearly the speed of light (up to 3.04 GeV of momentum).
• The Trick: They didn't just boost it straight ahead; they boosted it diagonally (off-axis). This allowed them to get a sharper, more detailed view of the city from a unique angle, revealing details that were previously hidden.

4. What They Found: The Map is Getting Clearer

They looked at three different "maps" of the proton's traffic:

1. Unpolarized: Just the general traffic flow (how many quarks are there?).
2. Helicity: How the traffic is spinning in the direction of the road.
3. Transversity: How the traffic is spinning sideways.

The Good News:
When they looked at the "real" part of their data (the main traffic flow), their new map matched perfectly with the best maps made by experimentalists using giant particle colliders (like the LHC). The new method works! It proves that you don't need the messy "rope" to get an accurate picture.

The Challenge:
However, they noticed a glitch in the "imaginary" part of the data (a more subtle, complex layer of the map). The results were a bit wobbly and didn't match up as well as they hoped.

• The Analogy: It's like looking at a city through a window. The main buildings (real part) are clear, but the reflections in the glass (imaginary part) are a bit distorted by the vibrations of the window frame (excited-state contamination). The scientists realized that the "noise" from the proton's internal vibrations was messing up this specific part of the measurement.

5. Why This Matters

This paper is a major milestone because:

• It's a Proof of Concept: It shows that the "no-rope" method works for protons, not just for simpler particles like pions.
• It's Faster and Cheaper: By removing the heavy "rope," future calculations can be done more efficiently, saving massive amounts of supercomputer time.
• It's a Roadmap: While there are still some kinks to work out (fixing the "wobbly" reflections), this study provides a clear blueprint for how to build even better maps of the proton in the future.

In a Nutshell:
The scientists found a way to take a high-speed photo of the proton's interior without dragging a heavy, messy rope through the scene. The resulting picture is sharp and matches reality, proving that this new, cleaner method is a powerful tool for understanding the fundamental building blocks of our universe.

Technical Summary

1. Problem Statement

Determining the internal partonic structure of hadrons (specifically the proton) via Parton Distribution Functions (PDFs) is a central challenge in hadron physics. While experimental data provides constraints in certain kinematic regions, lattice QCD offers a first-principles approach to calculate these non-perturbative quantities.

• Current Limitations: The dominant approach, Large-Momentum Effective Theory (LaMET), typically uses Gauge-Invariant (GI) quasi-PDF operators connected by Wilson lines. These operators suffer from:
  • Linear UV Divergences: Requiring complex non-perturbative renormalization (e.g., hybrid schemes, leading-renormalon resummation).
  • Signal-to-Noise Ratio (SNR) Issues: The SNR decreases exponentially with the length of the Wilson line, limiting precision at large distances.
  • Directional Constraints: Wilson lines restrict simulations to on-axis directions, limiting flexibility in accessing large off-axis momenta.
• The Gap: A novel approach using Coulomb Gauge (CG) fixed correlators without Wilson lines has been proposed to circumvent these issues, but its application to nucleon PDFs (unpolarized, helicity, and transversity) had not been comprehensively benchmarked prior to this work.

2. Methodology

The authors performed a lattice QCD calculation using the LaMET framework but implemented the Coulomb Gauge formulation.

• Lattice Setup:

  • Ensemble: $2+1$ flavor Highly Improved Staggered Quark (HISQ) gauge configurations from the HotQCD collaboration.
  • Parameters: Lattice spacing $a = 0.06$ fm, Volume $48^3 \times 64$, Valence pion mass $m_\pi = 300$ MeV.
  • Valence Fermions: Tadpole-improved clover Wilson fermions with HYP smearing.
  • Momentum: Utilized off-axis momentum directions ($\vec{n} = (1,1,0)$) to leverage 3D rotational symmetry, achieving nucleon momenta of $P_z = 2.43$ GeV and $3.04$ GeV.
  • Statistics: 553 gauge configurations with All-Mode Averaging (AMA) to improve SNR.

• Operator Definition:

  • Constructed non-local quark correlators in Coulomb gauge ($\vec{\nabla} \cdot \vec{A} = 0$) without Wilson lines.
  • Calculated matrix elements for unpolarized, helicity, and transversity channels using specific Dirac structures ($\tilde{\Gamma}$).

• Renormalization Strategy (Key Innovation):

  • Real Part: Used a Hybrid Renormalization scheme. Due to the absence of Wilson lines, linear divergences are absent. The authors adopted a single-ratio scheme (omitting the $z=0$ normalization) to avoid propagating discretization artifacts from the local point to the entire distribution.
  • Imaginary Part: Proposed a Static RI'-MOM (SRI'-MOM) scheme. The standard hybrid scheme was deemed suboptimal for the imaginary part because the zero-momentum denominator is purely real. The SRI'-MOM scheme uses spatial correlation functions to define the renormalization constant $Z_{SRI'}$, which is then matched to the $\overline{\text{MS}}$ scheme.

• Extraction and Matching:

  • Ground State Extraction: Employed the Feynman-Hellmann (FH) method combined with Bayesian joint fits of two-point and three-point correlators to suppress excited-state contamination.
  • Matching: Performed perturbative matching from the CG quasi-PDF to the light-cone PDF at Next-to-Leading Order (NLO) and Next-to-Leading Logarithmic (NLL) accuracy.
  • Extrapolation: Applied an asymptotic extrapolation strategy to the large-$z$ region to mitigate Fourier transform artifacts.

3. Key Contributions

1. First Comprehensive Nucleon Benchmark: This is the first study to apply the CG method to calculate all three nucleon spin structures (unpolarized, helicity, transversity) simultaneously.
2. Novel Renormalization Scheme: Introduction and validation of the SRI'-MOM scheme for the imaginary part of quasi-PDF matrix elements in the Coulomb gauge, addressing limitations of the standard hybrid approach.
3. High-Momentum Reach: Achieved nucleon momenta up to 3.04 GeV using off-axis directions, which is among the highest boosts achieved for nucleon PDFs at comparable pion masses, helping to suppress power corrections.
4. Systematic Analysis: Provided a detailed investigation of systematic uncertainties, including:
   • Excited-state contamination (noting the imaginary part is significantly more sensitive).
   • Discretization effects at $z=0$.
   • Dependence on the choice of Dirac structure ($\tilde{\Gamma}$).
   • Comparison between different renormalization schemes.

4. Results

• Valence Quark PDFs (Real Part):

  • The extracted valence distributions show good convergence with increasing nucleon momentum ($2.43 \to 3.04$ GeV).
  • Results at $P_z = 3.04$ GeV are compatible with recent global analyses (e.g., NNPDF4.0, JAMpol25, JAM3D-22) across all spin structures in the moderate-$x$ region.
  • The perturbative matching shows stable behavior between NLO and NLL orders.

• Full Quark Channel PDFs (Imaginary Part):

  • The full distributions (valence + sea) extracted from the imaginary part exhibit discrepancies between the two momenta ($2.43$ vs $3.04$ GeV).
  • However, the results at the higher momentum ($3.04$ GeV) are consistent with phenomenology.
  • The discrepancy is attributed to stronger excited-state contamination in the imaginary matrix elements, a known issue in lattice QCD that is more severe for these components.

• Comparison with Gauge-Invariant (GI) Methods:

  • When compared to previous GI-based calculations (using the same or different ensembles), the CG results show broad consistency in the moderate-$x$ region.
  • Differences near the endpoint regions ($x \to 1$) are observed, likely due to different power corrections inherent to the operator definitions.

5. Significance

• Validation of the CG Approach: The work demonstrates that the Coulomb gauge method is a viable, efficient, and accurate alternative to the Wilson-line-based GI approach for calculating nucleon PDFs.
• Efficiency and Simplicity: By eliminating Wilson lines, the method avoids linear divergences and renormalons, simplifying the renormalization procedure and potentially improving the signal-to-noise ratio at large distances.
• Benchmark for Future Work: This study serves as a critical benchmark for the broader application of the CG method, including future calculations of Transverse Momentum Dependent distributions (TMDs) and generalized parton distributions (GPDs).
• Roadmap for Precision: The paper identifies that while the method is promising, future improvements require:
  • Simulations at multiple lattice spacings for controlled continuum extrapolation.
  • Higher statistics at large source-sink separations to better control excited-state contamination (especially for the imaginary part).
  • Refined strategies for treating excited states (e.g., Lanczos algorithm, GEVP).

In conclusion, this paper establishes the Coulomb gauge method as a robust tool for precision lattice QCD calculations of nucleon structure, offering a complementary cross-check to traditional methods and paving the way for more precise determinations of the proton's internal structure.