Sketching the pion's valence-quark generalised parton distribution

In order to learn effectively from measurements of generalised parton distributions (GPDs), it is desirable to compute them using a framework that can potentially connect empirical information with basic features of the Standard Model. We sketch an approach to such computations, based upon a rainbow-ladder (RL) truncation of QCD's Dyson-Schwinger equations and exemplified via the pion's valence dressed-quark GPD, $H_\pi^{\rm v}(x,\xi,t)$. Our analysis focuses primarily on $\xi=0$, although we also capitalise on the symmetry-preserving nature of the RL truncation by connecting $H_\pi^{\rm v}(x,\xi=\pm 1,t)$ with the pion's valence-quark parton distribution amplitude. We explain that the impulse-approximation used hitherto to define the pion's valence dressed-quark GPD is generally invalid owing to omission of contributions from the gluons which bind dressed-quarks into the pion. A simple correction enables us to identify a practicable improvement to the approximation for $H_\pi^{\rm v}(x,0,t)$, expressed as the Radon transform of a single amplitude. Therewith we obtain results for $H_\pi^{\rm v}(x,0,t)$ and the associated impact-parameter dependent distribution, $q_\pi^{\rm v}(x,|\vec{b}_\perp|)$, which provide a qualitatively sound picture of the pion's dressed-quark structure at an hadronic scale. We evolve the distributions to a scale $\zeta=2\,$GeV, so as to facilitate comparisons in future with results from experiment or other nonperturbative methods.
Comment: 7 pages, 4 figures