Gluon PDF from Quark dressing in the Nucleon and Pion

Gluon dressing of the light quarks within hadrons is very strong and extremely important in that it dynamically generates most of the observable mass through the breaking of chiral symmetry. The quark and gluon parton densities, $q(x)$ and $g(x)$, are necessarily interrelated since any gluon emission and absorption process, especially dressing of a quark, contributes to $g(x)$ and modifies $q(x)$. Guided by long-established results for the parton-in-parton distributions from a strict 1-loop perturbative analysis of a quark target, we extend the non-perturbative QCD approach based on the Rainbow-Ladder truncation of the Dyson-Schwinger equations to describe the interrelated valence $q_{\rm v}(x)$ and the dressing-gluon $g(x)$ for a hadron at its intrinsic model scale. We employ the pion description from previous DSE work that accounted for the gluon-in-quark effect and introduce a simple model of the nucleon for exploratory purposes. We find typically \mbox{$\langle x \rangle_g \sim 0.20$} for both pion and nucleon at the model scale, and the valence quark helicity contributes 52\% of nucleon spin. We deduce both $q_{\rm v}(x)$ and $g(x)$ from 30 calculated Mellin moments, and after adopting existing data analysis results for $q_{\rm sea}(x)$, we find that NLO scale evolution produces $g(x)$ in good agreement with existing data analysis results for the pion at 1.3 GeV and the nucleon at 5 GeV$^2$. At the scale 2 GeV typical of lattice-QCD calculations, we obtain \mbox{$\langle x \rangle_g^{\rm N} = 0.42$} in good agreement with 0.38 from the average of recent lattice-QCD calculations.