Sketching the pion’s valence-quark generalised parton distribution

Abstract

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 rainbowladder (RL) truncation of QCD’s Dyson–Schwinger equations and exemplified via the pion’s valence dressed-quark GPD, Hvπ (x, ξ, t). Our analysis focuses primarily on ξ = 0, although we also capitalise on the symmetry-preserving nature of the RL truncation by connecting Hvπ (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 Hvπ (x, 0, t), expressed as the Radon transform of a single amplitude. Therewith we obtain results for Hvπ (x, 0, t) and the associated impact-parameter dependent distribution, qv π (x, | b⊥|), which provide a qualitatively sound picture of the pion’s dressedquark structure at a hadronic scale. We evolve the distributions to a scale ζ =2 GeV, so as to facilitate comparisons in future with results from experiment or other nonperturbative methods.