[Paper Review] Gluon Parton Distribution of the Pion and Nucleon from Lattice QCD

Zhouyou Fan, Huey-Wen Lin|arXiv (Cornell University)|Oct 27, 2021

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This study presents the first lattice QCD calculation of the $x$-dependent gluon parton distribution functions (PDFs) for the pion and nucleon using the pseudo-PDF approach on $2+1+1$ flavor HISQ ensembles with three lattice spacings and three pion masses. The results show good consistency with global phenomenological fits in the large-$x$ region, but significant deviations at small $x$, highlighting the need for improved statistics and systematic control in future work.

ABSTRACT

We present the $x$-dependent nucleon and pion gluon distribution from lattice QCD using the pseudo-PDF approach, on lattice ensembles with $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. We use clover fermions for the valence action and momentum smearing to achieve pion boost momentum up to 2.56~GeV on three lattice spacings $a\approx 0.9, 0.12$ and 0.15~fm and three pion masses $M_π\approx 220$, 310 and 690~MeV. We compare our pion and preliminary nucleon gluon results with the determination by global fits.

Motivation & Objective

• To compute the $x$-dependent gluon parton distribution functions (PDFs) for the pion and nucleon using lattice QCD, an ab initio nonperturbative method.
• To investigate the dependence of gluon PDFs on pion mass and lattice spacing, assessing systematic effects in the current statistical regime.
• To compare lattice QCD results with phenomenological global fits (e.g., CT18, NNPDF3.1, JAM, xFitter) to validate lattice predictions and identify discrepancies.
• To evaluate the reliability of functional form fits (1-, 2-, and 3-parameter) in reconstructing the gluon PDF from lattice data.
• To identify and address key systematic uncertainties, including finite-$\nu$ effects and quark PDF mixing, for future improvements.

Proposed method

• Employ the pseudo-PDF approach to extract the lightcone gluon PDF from Euclidean correlation functions computed on $2+1+1$ flavor HISQ lattices.
• Use clover valence fermions and momentum smearing to access pion boost momenta up to 2.56 GeV and reduce statistical noise.
• Apply two-state fits to two-point correlators and two-simultaneous-state fits to three-point correlators to extract ground-state matrix elements and suppress excited-state contamination.
• Construct the reduced pseudo-Ioffe-time distribution (pITD) from lattice correlators and perform functional fits using 1-, 2-, and 3-parameter forms to reconstruct the gluon PDF.
• Apply scheme conversion to map lattice results to the $\overline{\text{MS}}$ scheme at $\mu^2 = 4\,\text{GeV}^2$ for comparison with global fits.
• Use HYP smearing (five steps) on gluon loops to reduce statistical uncertainties in the correlator measurements.

Experimental results

Research questions

• RQ1 How do the lattice QCD results for the pion and nucleon gluon PDFs compare with global phenomenological fits across different $x$ regions?
• RQ2 What is the impact of varying the functional form (1-, 2-, 3-parameter) on the reconstructed gluon PDFs, and which form provides the most reliable and stable fit?
• RQ3 How do the lattice results depend on pion mass and lattice spacing, and is the dependence within current statistical uncertainties?
• RQ4 What are the dominant systematic uncertainties affecting the lattice extraction of gluon PDFs, particularly at small $x$?
• RQ5 To what extent do the current lattice data resolve the large-$x$ behavior of the gluon PDF, and how do they compare with global fits?

Key findings

• The lattice results for the nucleon and pion gluon PDFs are consistent with global fits (CT18 NNLO, NNPDF3.1 NNLO, JAM, xFitter) in the large-$x$ region ($x > 0.5$), with agreement within one standard deviation.
• Significant deviations are observed between lattice results and global fits in the small-$x$ region ($x < 0.3$), likely due to limited $\nu$-range and increasing errors at large $\nu$.
• The 1-parameter fit to lattice data is found to be unreliable, with large discrepancies in the $x < 0.4$ region; the 2- and 3-parameter fits show convergence, justifying the use of the 2-parameter form for final results.
• The extracted gluon PDFs from different ensembles (varying lattice spacing and pion mass) are consistent within one-sigma error, indicating weak dependence on these parameters under current statistics.
• The pion gluon PDF from lattice QCD is consistent with JAM and xFitter NLO PDFs within one sigma for $x > 0.2$, supporting the reliability of the lattice approach.
• The results suggest that future improvements in statistics and systematic control—especially for finite-$\nu$ effects and quark PDF mixing—will be essential for precise small-$x$ determinations.