First Nucleon Gluon PDF from Large Momentum Effective Theory

TL;DR

This work delivers the first nucleon gluon PDF obtained with Large-Momentum Effective Theory (LaMET) using a hybrid-renormalization scheme. It derives the one-loop Wilson coefficients and matching kernels that relate the hybrid-renormalized quasi-PDF to the light-front PDF and applies them to high-statistics lattice data on MILC HISQ $N_f=2{+}1{+}1$ ensembles at $M_\pi \approx 310$ and $690$ MeV with two smearings, achieving a consistent renormalization across hadrons. The resulting gluon PDF, obtained after large-$\nu$ extrapolation and Fourier transformation, agrees with global fits like CT18 within current uncertainties, demonstrating a viable path to ab initio gluon structure from LaMET. The study also identifies key systematics, such as smearing and heavy pion masses, and outlines a roadmap toward continuum extrapolation and physical pion masses to solidify the gluon PDF from lattice QCD.

Abstract

We report the first nucleon gluon parton distribution function (PDF) using Large-Momentum Effective Theory (LaMET). We focus on the gluon operator which was demonstrated to have the best signal-to-noise in the previous attempt [1] in computing gluon PDFs using LaMET. We compute the corresponding Wilson coefficients needed for the hybrid-renormalized matrix elements and the matching kernel to convert the quasi-PDF to the lightcone one at the one-loop level. We demonstrate that with the proper Wilson coefficients in place, the counterterms for the renormalization are independent of the hadron and mass within statistical error. Using the resulting renormalization, we then compute the nucleon PDF using a HISQ ensemble generated by the MILC collaboration with $N_f=2+1+1$, $a \approx 0.12$ fm, with valence pion masses of 310 and 690 MeV and two gauge link smearing techniques. Despite the physics effects of the heavier than physical pion masses and gauge link smearing, this calculation provides excellent proof of principle and compares reasonably with selected global fit results.

Figures (9)

• Figure 1: Fit results for the $\delta m + m_0$ at $\mu = 2$ GeV for varying central point $z$ for each hadron for the two smearings, Wilson-3 (left) and HYP5 (right). The error bands give statistical fluctuations only. The points represent the choice of $\delta m + m_0$ that is used in the subsequent hybrid renormalization.
• Figure 2: Lattice (points) hybrid renormalized nucleon matrix elements with $z_s = 0.36$ fm for the Wilson-3 (left column) and HYP5 (right column) smearing for $M_\pi \approx 690$ (top row) and $310$ (bottom row) GeV compared to the matrix elements reconstructed from the CT18 gluon PDF Hou:2019efy, shown by the colored bands which have the same color scheme as the lattice data for differing momenta.
• Figure 3: Large-$\nu$ extrapolation of the hybrid renormalized $P_z = 1.71$ GeV matrix elements with $M_\pi \approx 690$ MeV and Wilson3 smearing, using the data from $z \in [7a, 16a]$, the extent of which is shown by the two dashed, vertical lines.
• Figure 4: Quasi-PDF (green) from the Fourier transform of the hybrid renormalized $P_z = 1.71$ GeV matrix elements with $M_\pi \approx 310$ MeV and HYP5 smearing and the light-front (LF-)PDF (purple) after matching the quasi-PDF (qPDF) to the light cone. The gray bands represent the region where LaMET matching begins to break down.
• Figure 5: (Left) the MSULat nucleon gluon PDFs at $P_z = 1.71$ for the different pion masses and smearings all compared. (Right) The MSULat PDF from large momentum effective theory at $P_z = 1.71$ GeV at $M_\pi \approx 690$ (blue) and $310$ (red) MeV with HYP5 smearing compared to the CT18 Hou:2019efy (orange), JAM24 Anderson:2024evk (cyan), and CJ22 with multiplicative higher twist corrections Cerutti:2025yji (green) gluon PDFs. In both plots, the gray bands represent the region where LaMET matching begins to break down.