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Towards Systematic Control of the Nucleon Gluon PDF with the Pseudo-PDF Method

This paper presents updated lattice QCD calculations of the unpolarized nucleon gluon parton distribution function using the pseudo-PDF method on a finer $2+1+1$ flavor HISQ ensemble with the Lanczos method to control systematics, revealing a strongly suppressed gluon distribution at large $x$ and demonstrating significant progress toward high-precision, systematically controlled extractions for global PDF analyses.

Original authors: Alex NieMiera, Kai Ebira, William Good, Huey-Wen Lin

Published 2026-10-23

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CC BY 4.0

Original authors: Alex NieMiera, Kai Ebira, William Good, Huey-Wen Lin

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

Inside the heart of every atom lies the proton, a tiny, bustling city of matter that gives our visible universe its weight. For decades, physicists have understood that protons are not solid spheres but complex assemblies of even smaller particles called quarks and gluons. Quarks are the fundamental building blocks, while gluons are the invisible threads that bind them together, acting as the carriers of the strong force. While we know quarks make up the proton's identity, the gluons are the hidden majority, carrying roughly half of the proton's total momentum and playing a crucial role in generating the mass of everything we see. However, mapping exactly how these gluons are distributed inside the proton has remained one of the most difficult puzzles in modern physics. Because gluons do not interact directly with light, they cannot be photographed or measured easily in particle collisions, leaving scientists with only indirect clues about their behavior.

To solve this, a team of researchers at Michigan State University has turned to a powerful new approach using supercomputers to simulate the laws of nature from the ground up. They focused on creating a detailed map of the gluon distribution, specifically looking at how much momentum a gluon carries relative to the whole proton. This is not just a theoretical exercise; knowing the precise shape of this distribution is essential for understanding high-energy collisions at facilities like the Large Hadron Collider, where scientists search for new particles and test the fundamental laws of the universe. Without a clear picture of the gluons, predictions for these experiments remain uncertain, and our understanding of how the universe is built stays incomplete.

The researchers tackled this challenge using a technique called lattice quantum chromodynamics, which treats space and time as a grid of tiny points, much like the pixels on a screen, but at a scale so small it is almost unimaginable. On this digital grid, they simulated the behavior of protons made of quarks and gluons, calculating how the gluons move and interact. A major hurdle in these simulations is that the signal from gluons is incredibly faint and easily drowned out by noise, similar to trying to hear a whisper in a crowded stadium. To overcome this, the team used a finer grid than ever before, allowing them to see details at much smaller distances, and they boosted the protons to very high speeds within the simulation to make the gluon patterns clearer. They also employed a sophisticated mathematical method to filter out unwanted background noise, ensuring that the data they extracted came purely from the stable, ground-state proton rather than fleeting, excited states.

After running these massive calculations, the team analyzed the results to reconstruct the gluon distribution. They found that the gluons are not spread out evenly; instead, their behavior changes dramatically depending on how much momentum they carry. The most striking discovery is that at very high momentum fractions, the probability of finding a gluon drops off sharply. In other words, it is extremely rare to find a gluon carrying a large share of the proton's total momentum. This finding aligns with some previous experimental guesses but contradicts others, offering a new, more precise constraint on how nature organizes these particles. The researchers also confirmed that their results are robust, showing no significant dependence on the specific mass of the particles used in the simulation, which suggests their findings are a true reflection of physical reality rather than an artifact of their computer model.

This work represents a significant step forward in bringing the abstract world of quantum chromodynamics into sharp focus. By combining finer grids, higher speeds, and advanced noise-reduction techniques, the team has provided a clearer view of the invisible glue that holds our universe together. While the picture is not yet complete, and more data is needed to refine the details, these results offer a solid foundation for future experiments. They demonstrate that with the right tools and methods, scientists can systematically control the uncertainties that have long plagued these measurements, paving the way for a deeper understanding of the fundamental structure of matter.

Technical Summary: Towards Systematic Control of the Nucleon Gluon PDF with the Pseudo-PDF Method

Problem Statement
The unpolarized gluon parton distribution function (PDF), $g(x, Q^2)$ , is critical for understanding nucleon structure, mass, and spin, as well as for precision phenomenology at hadron colliders (e.g., Higgs production). However, determining this quantity precisely remains challenging. In experimental deep-inelastic scattering, gluons do not couple directly to virtual photons; their distribution is inferred indirectly via $Q^2$ evolution and complementary processes, leading to significant uncertainties, particularly at large momentum fractions ( $x$ ). While lattice Quantum Chromodynamics (QCD) offers a first-principles approach via Large-Momentum Effective Theory (LaMET) and the pseudo-PDF method, extracting gluon PDFs is hindered by poor signal-to-noise ratios in gluonic matrix elements. These elements are disconnected from valence-quark lines and suffer from excited-state contamination, discretization effects, and higher-twist contributions, which complicate the isolation of the ground-state matrix element and the subsequent reconstruction of the PDF.

Methodology
This work presents an updated lattice QCD calculation of the nucleon gluon PDF using the pseudo-PDF methodology. The study utilizes an $N_f = 2+1+1$ ensemble of highly improved staggered quarks (HISQ) generated by the MILC Collaboration, featuring a fine lattice spacing of $a \approx 0.06$ fm and a volume of $48^3 \times 144$ . The calculation is performed at two valence pion masses ( $M_\pi \approx 310$ and $690$ MeV) using Wilson-clover fermions in the valence sector.

Key methodological advancements include:

Lattice Setup and Smearing: The authors employ Gaussian momentum smearing to enhance the signal for boosted nucleon states up to momenta of $P_z \approx 4.4$ GeV. Wilson flow (at $\tau = 3a^2$ ) is applied to gauge links to suppress ultraviolet fluctuations in the gluon-loop contributions.
Excited-State Control: To address excited-state contamination, the authors implement the Lanczos method for the first time in this context, comparing it against their previous two-state fit approach. The Lanczos method constructs a Krylov space from the correlators to extract ground-state matrix elements without the need for arbitrary fit window selection, thereby reducing model dependence.
Renormalization and Matching: The bare matrix elements are renormalized multiplicatively to construct the reduced pseudo-Ioffe-time distribution (RpITD). The matching between the pseudo-ITD and the light-cone ITD is performed using a parameterization that explicitly accounts for:
- Leading-twist contributions via a functional form for the gluon PDF ( $x^\alpha(1-x)^\beta$ ).
- Higher-twist effects and discretization errors through explicit parameterizations ( $z^2 B_1(\nu)$ and $z^{-1} P_1(\nu)$ ), where the latter controls $O(a/z)$ lattice artifacts.
Fitting Strategy: A correlated $\chi^2$ minimization is performed on the RpITD data to extract six fit parameters ( $\alpha, \beta, b_1, b_2, p_1, p_2$ ). Bootstrap resampling is used to estimate uncertainties and correlations.

Key Contributions

Finer Lattice Spacing: The use of $a \approx 0.06$ fm allows access to multiple short-distance separations within the perturbative regime ( $z \lesssim 0.3\text{--}0.4$ fm) and higher nucleon boost momenta than previous gluon pseudo-PDF studies.
Systematic Control via Lanczos: The application of the Lanczos method provides a robust alternative to multi-state fits, demonstrating good agreement with the two-state fit approach while exhibiting reduced sensitivity to analysis choices such as fit windows.
Explicit Systematic Parameterization: The study incorporates explicit terms for higher-twist and discretization effects directly into the fitting framework, moving beyond simple leading-twist assumptions to better control systematic uncertainties.

Results

Ground-State Extraction: The Lanczos and two-state fit methods yield consistent ground-state energies and matrix elements, agreeing within $1\sigma$ for most kinematic points. The Lanczos method successfully isolates the ground state even at high boosts where signal-to-noise ratios degrade rapidly.
Pion Mass Dependence: The extracted gluon PDFs show no statistically significant dependence on the valence pion mass between $M_\pi \approx 310$ and $690$ MeV.
PDF Shape: The resulting gluon distributions indicate a strongly suppressed gluon distribution at large $x$ ( $x \gtrsim 0.5$ ). This finding is consistent with a subset of phenomenological global fits (specifically JAM24 and CJ22) but shows tension ( $>1\sigma$ ) with the CT18 extraction in the range $0.3 \lesssim x \lesssim 0.6$ .
Comparison to Phenomenology: At small $x$ ( $x \le 0.2$ ), the lattice results lie at the $1\sigma$ boundary of phenomenological fits, though the authors note that the RpITD data have limited sensitivity in this region. The lattice results generally agree with global fits across various $x$ ranges, though the broad range of compatible parameterizations suggests flexibility in the current constraints.

Significance and Claims
The authors claim that these results represent "important steps toward systematically controlled, high-precision extractions of the gluon PDF." By combining finer lattice spacing, the Lanczos method for excited-state control, and explicit parameterization of systematic effects, the study demonstrates progress toward providing lattice-QCD constraints that can be integrated into future global PDF analyses. The authors modestly note that while the current results are consistent with some phenomenological extractions, the broad range of compatible parameterizations implies that further improvements in precision (particularly at large $\nu$ ) and the inclusion of multiple lattice spacings are necessary to uniquely separate discretization effects and fully constrain the PDF parameters. The work suggests that these RpITD results could serve as useful constraints in joint global fits, analogous to recent analyses for pion PDFs.

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