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Unpolarized gluon PDF of the nucleon from lattice QCD at physical point in the continuum limit

This paper presents a state-of-the-art lattice QCD calculation of the nucleon's unpolarized gluon parton distribution function using large-momentum effective theory on 2+1 flavor CLQCD ensembles, which incorporates distillation techniques and hybrid renormalization to extrapolate results to the physical point and continuum limit.

Original authors: Chen Chen, Chunhua Zeng, Hongxin Dong, Liuming Liu, Xiaomin Shen, Peng Sun, Xiaonu Xiong, Yi-Bo Yang, Fei Yao, Jian-Hui Zhang, Shiyi Zhong

Published 2026-07-31

📖 4 min read🧠 Deep dive

CC0 1.0

Original authors: Chen Chen, Chunhua Zeng, Hongxin Dong, Liuming Liu, Xiaomin Shen, Peng Sun, Xiaonu Xiong, Yi-Bo Yang, Fei Yao, Jian-Hui Zhang, Shiyi Zhong

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

The Invisible Glue Inside the Atom

Imagine the nucleus of an atom as a bustling city. Inside, tiny particles called protons and neutrons zoom around, but they aren't just solid marbles; they are more like chaotic crowds of even smaller particles called quarks and gluons. Quarks are the citizens, but gluons are the invisible glue that holds them together, carrying the force that keeps the city from flying apart. This glue is governed by a set of rules called Quantum Chromodynamics (QCD), which is notoriously difficult to solve with a pencil and paper because the interactions are so intense.

To understand how this city works, physicists use something called "Parton Distribution Functions" (PDFs). Think of a PDF as a census report. It doesn't just tell you how many people live in the city; it tells you the probability of finding a specific type of citizen (like a gluon) carrying a specific amount of the city's total energy or momentum. While we have good maps for the quark citizens, the gluon census has been a mystery, especially for the "wealthy" gluons that carry a huge chunk of the momentum. Without a clear map of these gluons, our predictions for high-energy particle collisions—like those at the Large Hadron Collider—remain a bit fuzzy.

Mapping the Glue with a Digital Microscope

In this new study, a team of researchers has taken a giant leap forward in mapping these elusive gluons. They didn't use a physical microscope; instead, they built a massive digital one using a technique called Lattice QCD. Imagine the universe as a giant 3D grid, like a massive game of Minecraft, where space and time are broken down into tiny blocks. By simulating the laws of physics on this grid, they can watch how gluons behave inside a proton.

The team, led by Chen Chen and colleagues, ran their simulation on eight different versions of this digital grid. They didn't just use one size; they used five different "pixel sizes" (lattice spacings) ranging from 0.105 fm down to 0.0519 fm, and they adjusted the "weight" of the particles (pion masses) from 136 MeV to 317 MeV. This is crucial because, in the real world, these particles have a specific, physical weight. By running the simulation at many different settings and then mathematically smoothing the results together, they could effectively remove the "pixelation" and the artificial weight, revealing what the gluons look like in the real, physical world.

To get a clear picture, they had to overcome some serious noise. Imagine trying to hear a whisper in a rock concert; that's what calculating gluon signals is like. The team used clever tricks, such as "distillation" (filtering out the static) and "momentum smearing" (tuning the signal to the right frequency), to boost the clarity of their data. They pushed the simulated protons to move at incredibly high speeds, up to 3 GeV, to get a better view of the internal structure.

The Findings: A Clearer, but Still Fuzzy, Picture

After all this heavy lifting, the team produced a new map of the unpolarized gluon PDF. This map shows the probability of finding a gluon carrying a fraction $x$ of the proton's momentum. Their results, which have been extrapolated to the "infinite momentum limit" (a theoretical state where the proton is moving so fast it reveals its true structure), show that the gluon distribution is consistent with the best experimental guesses we have so far. Specifically, their data aligns well with global fit models like CT18, NNPDF, and JAM24 within a margin of error of two standard deviations ( $2\sigma$ ).

However, the map isn't perfect yet. The researchers found that at the very edges of the momentum spectrum—where gluons carry either almost no momentum ( $x < 0.2$ ) or almost all of it ( $x > 0.8$ )—the results become less reliable. This is expected because the mathematical tools they used start to break down in these extreme zones. The team notes that their final results still carry a significant amount of uncertainty, primarily because they had to simultaneously correct for the grid size, the speed of the proton, and the particle masses all at once.

While this isn't the final, perfect census of the gluon city, it is the most complete and precise lattice QCD calculation of the nucleon gluon PDF to date. It confirms that the theoretical tools are working and that the "glue" behaves roughly as we expect, even if the exact details at the extremes are still being sharpened. The authors suggest that future improvements, like better mathematical resummation techniques, could clear up the remaining fuzziness, bringing us closer to a perfect understanding of the invisible glue that holds our universe together.

Technical Summary: Unpolarized Gluon PDF of the Nucleon from Lattice QCD at the Physical Point in the Continuum Limit

Problem Statement
Parton distribution functions (PDFs) are fundamental to understanding the internal structure of hadrons, yet the determination of the gluon PDF has historically relied on phenomenological global fits to experimental data. These extractions are model-dependent and suffer from significant uncertainties, particularly in the large- $x$ region where experimental constraints are sparse. While Large Momentum Effective Theory (LaMET) has enabled first-principles lattice QCD calculations of quark PDFs, gluon PDF calculations face unique challenges, including notoriously noisy matrix elements and the difficulty of performing simultaneous extrapolations to the continuum limit, infinite momentum limit, and physical pion mass. Previous lattice studies of gluon PDFs have largely been limited to unphysical pion masses or lacked a full continuum extrapolation.

Methodology
This work presents a state-of-the-art calculation of the unpolarized gluon PDF using LaMET on $2+1$ flavor CLQCD ensembles. The study employs a comprehensive simulation setup involving eight ensembles with five distinct lattice spacings ( $a = 0.0519$ to $0.105$ fm) and pion masses ranging from 136 MeV to 317 MeV, covering nucleon momenta up to 3 GeV.

Key technical components include:

Operator and Smearing: The calculation utilizes a multiplicatively renormalizable gluon operator constructed from field strength tensors and Wilson lines. To enhance the signal-to-noise ratio of the gluonic matrix elements, the authors apply the distillation quark smearing method combined with momentum smearing techniques. Ten-step hypercubic (HYP) smearing is applied to the gluon operators to suppress linear divergence errors.
Renormalization: A hybrid renormalization scheme is employed to handle both linear and logarithmic ultraviolet divergences non-perturbatively. Short-distance divergences are removed by dividing by the zero-momentum bare matrix element, while long-distance divergences are addressed via a self-renormalization procedure. The renormalized matrix elements are matched to the continuum $\overline{\text{MS}}$ scheme using one-loop perturbative matching.
Extrapolation Strategy: The final light-cone PDF is obtained through a combined extrapolation to:
1. The continuum limit ( $a \to 0$ ), accounting for $O(a^2)$ discretization effects.
2. The infinite momentum limit ( $P_z \to \infty$ ), removing power corrections of order $1/P_z^2$ .
3. The physical pion mass ( $m_\pi \approx 135$ MeV).
  Systematic uncertainties associated with the extrapolation ansatz are estimated using the Akaike Information Criterion (AIC) to average over different fitting forms. Large- $\lambda$ ( $\lambda = zP_z$ ) extrapolation is used to complete the coordinate-space data required for the Fourier transform to momentum space.

Key Contributions and Results

First Physical Point Continuum Result: This work provides the first lattice QCD calculation of the unpolarized gluon PDF that simultaneously reaches the physical pion mass, the continuum limit, and the infinite momentum limit.
Signal Improvement: The application of distillation and momentum smearing successfully mitigates the noise typically associated with gluon matrix elements, allowing for the extraction of clean signals at high nucleon momenta.
Comparison with Global Fits: The final results for the normalized gluon PDF, $xg(x)/\langle x \rangle$ , are compared with global fit results from CT18, NNPDF, and JAM24. The lattice results are found to be consistent with these phenomenological extractions within $2\sigma$ .
Large- $x$ Behavior: The calculation confirms the suppression of the gluon PDF at large $x$ , with theoretical and experimental results converging toward zero in this region.
Uncertainty Analysis: The study identifies that the large uncertainty in the final results is primarily driven by the simultaneous extrapolation to the continuum, infinite momentum, and physical pion mass limits, particularly the treatment of the $1/P_z^2$ term.

Significance
The authors claim this work represents a significant step forward in the precision determination of gluon PDFs from first principles. By providing a complete lattice QCD calculation that addresses the continuum, infinite momentum, and physical mass limits, the study offers a rigorous, model-independent benchmark for the gluon structure of the nucleon. The consistency of the results with global fits validates the LaMET framework for gluon PDFs and establishes a foundation for future precision studies. The paper notes that while the current results have large uncertainties due to the complexity of the combined extrapolation, the methodology is robust and consistent with other recent lattice calculations (e.g., Ref. [43]), despite differences in lattice setup. Future improvements are suggested through renormalization-group resummation and leading-renormalon resummation to extend the effective range of LaMET and reduce scale dependence.

The Invisible Glue Inside the Atom

Mapping the Glue with a Digital Microscope

The Findings: A Clearer, but Still Fuzzy, Picture

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