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, Hπv(x,ξ,t). Our analysis focuses primarily on ξ=0ξ=0, although we also capitalise on the symmetry-preserving nature of the RL truncation by connecting Hπv(x,ξ=±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 Hπv(x,0,t), expressed as the Radon transform of a single amplitude. Therewith we obtain results for Hπv(x,0,t) and the associated impact-parameter dependent distribution, qπv(x,|b→⊥|), which provide a qualitatively sound picture of the pion's dressed-quark structure at a hadronic scale. We evolve the distributions to a scale ζ=2 GeVζ=2 GeV, so as to facilitate comparisons in future with results from experiment or other nonperturbative methods.