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 rainbow-ladder (RL) truncation of QCD's Dyson-Schwinger equations and exemplified via the pion's valence dressed-quark GPD, $H_{\ddot{I}}{€}^{\mathrm{v}}(x,\hat{I}¾,t)$. Our analysis focuses primarily on $\hat{I}¾=0$, although we also capitalise on the symmetry-preserving nature of the RL truncation by connecting $H_{\ddot{I}}{€}^{\mathrm{v}}(x,\hat{I}¾=\pm 1,t)$ 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 $H_{\ddot{I}}{€}^{\mathrm{v}}(x,0,t)$, expressed as the Radon transform of a single amplitude. Therewith we obtain results for $H_{\ddot{I}}{€}^{\mathrm{v}}(x,0,t)$ and the associated impact-parameter dependent distribution, $q_{\ddot{I}}{€}^{\mathrm{v}}(x,|{\vec{b}}_{\perp }|)$, which provide a qualitatively sound picture of the pion's dressed-quark structure at an hadronic scale. We evolve the distributions to a scale $\hat{I}\P =2$GeV, so as to facilitate comparisons in future with results from experiment or other nonperturbative methods.

7 pages, 4 figures