Gluon Distributions in the Pion

TL;DR

This work presents a relativistic, nonperturbative description of the pion’s gluonic structure by embedding an active gluon within a light-front framework that also treats the spectator as a spin-1 system. Transverse dynamics are governed by a holographic light-front equation, while longitudinal confinement is captured by the ’t Hooft equation, yielding a cohesive three-dimensional bound-state picture and a mass spectrum consistent with data. Using the resulting light-front wavefunctions, the authors compute gluon PDFs, GPDs in the DGLAP region, TMDs, and the gluon contribution to the pion’s gravitational form factor, performing NNLO DGLAP evolution and comparing to lattice QCD and global fits. A minimal double-distribution extension with a D-term is explored to illustrate approximate polynomiality and ERBL-region implications. The results demonstrate good agreement with lattice results for the gravitational form factor and with global fits for the gluon PDF, providing a robust, QCD-based description of the pion’s three-dimensional gluonic structure and offering predictions for future experiments at facilities like the EIC and JLab.

Abstract

We formulate a light-front model for the pion that explicitly incorporates the gluonic degree of freedom. In this framework, high-energy scattering off the pion is described by an active gluon, while the remaining constituents are treated as a spectator system with an effective mass. The mass spectrum and light-front wave functions (LFWFs) of the pion are determined by solving two Schrödinger-like equations derived from quantum chromodynamics: the light-front holographic equation in the chiral limit and the 't~Hooft equation. Using the resulting LFWFs, we compute the gluon distributions within pion, including the parton distribution functions in comparison with the available global fits, generalized parton distributions, and transverse momentum-dependent distributions. The present analysis is restricted to the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) domain of the pion's gluon GPD, without explicit construction of the Efremov-Radyushkin-Brodsky-Lepage (ERBL) region. Consequently, the polynomiality condition for full Mellin moments is not satisfied by construction. Furthermore, we demonstrate that the obtained LFWFs provide a good description of the gravitational form factors of pion when compared with the recent lattice QCD results.

Figures (15)

• Figure 1: Our Regge trajectories for the pion family using the physical gluon mass $m_{1}=0$, the spectator mass $m_{2}=0.092$ GeV, $g=0.109$ GeV and $\kappa=0.523$ GeV.
• Figure 2: The longitudinal wave function $X(x)$ of gluon in the pion.
• Figure 3: The gluon GPD of the pion in the DGLAP region ($\xi < x < 1$) as a function of (a) $x$ and $-t$ [in GeV$^2$] at fixed skewness, $\xi=0.15$, and (b) $x$ and $\xi$ at fixed $-t=0.5$ GeV$^2$.
• Figure 4: The pion’s gluon PDF at (a) the model scale ($\mu_0^2 = 0.176$ GeV$^2$) and (b) the scale $\mu^2 = 5~\mathrm{GeV}^2$ evolved via DGLAP, varying the model scale within $0.144 < \mu_0^2 < 0.211~\mathrm{GeV}^2$. The evolved results are compared with global QCD fits: JAM Barry:2021osv (orange band), xFitter Novikov:2020snp (cyan band), Fant$\hat{\rm o}$mas Kotz:2025lio (gray band), a phenomenological model by the MAP Collaboration Pasquini:2023aaf (red band), and a Hamiltonian-based BLFQ framework Lan:2024ais (magenta band).
• Figure 5: Gluon PDF of the pion at the scale $5~\mathrm{GeV}^2$ evolved via DGLAP with different IR-freezing prescriptions: (a) varying $\alpha_s^{\mathrm{max}}$ in the range $\frac{\pi}{4} < \alpha_s^{\mathrm{max}} < 1.5$; (b) varying the freeze scale $\mu_f^2$ in $0.08 < \mu_f^2 < 0.36~\mathrm{GeV}^2$. Results are compared with global QCD fits JAM Barry:2021osv (orange), xFitter Novikov:2020snp (cyan), and Fant$\hat{\mathrm{o}}$mas Kotz:2025lio (gray), as well as the MAP model Pasquini:2023aaf (red) and BLFQ framework Lan:2024ais (magenta).