Parton distribution and fragmentation functions with massive gluons

TL;DR

This work addresses the nonperturbative structure of hadrons by computing the pion PDF and fragmentation functions within two nonperturbative QCD frameworks: the Qin–Chang (QC) model and the Curci–Ferrari (CF) model with a dynamical gluon mass. PDFs and FFs are obtained from cut diagrams using dressed quark propagators and a pion Bethe–Salpeter amplitude, with the Drell–Levy–Yan relation connecting elementary and full fragmentation functions. The results show that the CF model yields PDFs and FFs in quantitative agreement with the QC framework and lattice benchmarks, supporting the CF approach as a viable tool for hadron phenomenology in the infrared. The findings motivate extending CF-based calculations beyond the chiral limit to other hadrons and to a physical pion BSA, potentially broadening the applicability of massive-gluon models in hadron structure studies.

Abstract

The correct description of the hadron's structure requires understanding how quarks and gluons form the observable hadrons and how they are distributed within them. Two key nonperturbative quantities encapsulate this information: parton distribution functions (PDFs) and fragmentation functions (FFs). The former define a probabilistic light-front momentum distribution of partons within a hadron, whereas the latter describe the hadronization process of high-energy partons. Computing these functions analytically poses significant challenges, as it demands models that accurately incorporate the nonperturbative infrared dynamics of Quantum Chromodynamics (QCD). In this work, we compute the pion PDF and its elementary and full FFs using the Curci-Ferrari (CF) model. This model enables the exploration of nonperturbative QCD effects by introducing a gluon-mass scale within the Landau gauge QCD Lagrangian. The two-point quark and gluon correlation functions derived from the CF model agree well with lattice QCD results and reproduce the pion decay constant in the chiral limit, consistent with chiral perturbation theory. The resulting pion PDF and FFs computed with the CF quark propagator and pion Bethe-Salpeter amplitude are in good qualitative and quantitative agreement with those obtained using the Qin-Chang model, a benchmark approach to nonperturbative QCD. These findings support the broader applicability of the Curci-Ferrari model in hadron phenomenology.

Figures (3)

• Figure 1: Cut-diagram representation of the parton distribution functon (a) and elementary fragmentation function (b). The shaded circle with outgoing/incoming double-solid lines denote the pion in the fragmentation process, while solid lines are quark propagators and solid dots represent the $\gamma^+$ between the two quarks with momentum $k$.
• Figure 2: Normalized PDF computed with Eq. \ref{['pdf_func']}, the QC and CF frameworks.
• Figure 4: Comparison of the (un)normalized elementary and full fragmentation functions, Eqs. \ref{['ff_func']} (dotted line), \ref{['frag_sum_rule']} (solid line) and \ref{['full_frag']} (dashed line). Left-hand side: predictions of the QC model \ref{['sec:qc_model']}; Right-hand side: predictions of the CF model \ref{['sec:curci_ferrari']}.