Systematic Study of the Self-Renormalized Nucleon Gluon PDF in Large-Momentum Effective Theory

TL;DR

This work delivers a systematic lattice-QCD study of the nucleon gluon PDF using self-renormalized LaMET on $N_f=2+1+1$ MILC HISQ ensembles, addressing UV divergences and renormalon effects by combining lattice matrix elements with perturbative input. By varying gauge-link smearing, lattice spacing, pion mass, and nucleon boost momentum up to $P_z \approx 2.2~\mathrm{GeV}$, the authors demonstrate robust control over smearing and discretization, with only mild dependence on pion mass and lattice spacing, while momentum dependence remains a major source of uncertainty. The renormalized matrix elements yield quasi-PDFs that are modestly negative before matching and largely corrected by NLO perturbative matching to light-cone PDFs, though some negativity persists at large $x$, signaling remaining systematics. The results chart a path toward precision lattice determinations of the gluon PDF, highlighting the need for higher boosts and resummation strategies to reach the accuracy required for LHC and future Electron-Ion Collider phenomenology.

Abstract

We present a systematic study of the nucleon gluon parton distribution function (PDF) using the self-renormalized large-momentum effective theory (LaMET) approach in lattice QCD. This work extends previous gluon-PDF extractions by performing a detailed analysis of key systematic effects, including gauge-link smearing, lattice spacing, pion mass, and nucleon boost momentum. The self-renormalization framework mitigates ultraviolet divergences associated with Wilson-line self-energy and renormalon contributions by combining lattice matrix elements with perturbative short-distance information, thereby preserving the correct infrared structure. Calculations are performed on $N_f=2+1+1$ HISQ ensembles generated by the MILC Collaboration at three lattice spacings and two pion masses, with boosted nucleon states reaching momenta up to 2.2~GeV. We determine renormalization factors from zero-momentum matrix elements and apply hybrid renormalization to suppress discretization artifacts. After extrapolating large-separation behavior and performing Fourier transforms, we reconstruct quasi-PDFs and match them to lightcone PDFs using next-to-leading order Wilson coefficients. Our results demonstrate that smearing and lattice-spacing effects are under control, and pion-mass and lattice-spacing dependence is mild relative to the current $O(10^6)$ statistics; however, momentum dependence remains a significant source of uncertainty. Future work including even larger boost momenta will be essential to reduce systematics in lattice determinations of the gluon PDF and to advance toward precision QCD phenomenology at the LHC and the future Electron-Ion Collider.

Figures (9)

• Figure 1: The renormalized matrix elements for analyses (top) W$123$, (middle) W$235$, and (bottom) W$333$ compared to the perturbative one-loop $\overline{\text{MS}}$ result illustrated by the black curve.
• Figure 2: The renormalized gluon matrix elements at $M_\pi \approx 690$ MeV are compared across the three smearing schemes at three boosted momenta $P_z$ and lattice spacings $a \approx 0.09$ fm (top), $0.12$ fm (middle), and $0.15$ fm (bottom).
• Figure 3: Renormalized matrix elements compared across each of the three lattice spacings $a \approx 0.09$, $0.12$, $0.15$ three smearing analyses at $M_\pi \approx 690$ (left column) and $310$ (right column) MeV for each of the three analyses W$123$ (top row), W$235$ (middle row), and W$333$ (bottom row).
• Figure 4: Selected large-$\nu$ extrapolations at select momentum and ensemble for (top row) the W123 analysis, (middle row) the W235 analysis, and (bottom row) the W333 analysis. We use data from $z \gtrsim 0.6$ fm across all examples and the black dashed lines illustrate the range of $z$ used in the fit
• Figure 5: A selection of quasi-PDFs and their corresponding lightcone PDFs after matching. From top to bottom, the lattice parameters correspond to $a \approx 0.15,\; 0.09,\; 0.12$ fm, $M_\pi \approx 690$, $310$ MeV, $P_z = 2.05,\; 2.18,\; 1.71$ GeV, and smearing schemes W$123$, W$235$, and W$333$, respectively.