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 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.We thank A. Besse, I.C. Cloet, D. Muller, P. Fromholz, C. Keppel, P. Kroll, J.-Ph. Lansberg, C. Lorce, J. Segovia and S. Wallon for valuable discussions. C.M., L.C., H.M., C.D.R. and J.R.-Q. are grateful for the chance to participate in the workshop "Many Manifestations of Nonperturbative QCD under the Southern Cross", Ubatuba, Sao Paulo, where significant parts of this work were first presented and improvements discussed. C.D.R. acknowledges support from an International Fellow Award from the Helmholtz Association; and this research was otherwise supported by: Commissariat a l'Energie Atomique; JRA "Study of Strongly Interacting Matter" (Grant Agreement no. 283286, HadronPhysics3) under the EU Seventh Framework Programme; GDR 3034 PH-QCD; ANR-12-MONU-0008-01 "PARTONS"; University of Adelaide and Australian Research Council through grant no. FL0992247; Spanish ministry Research Project FPA2011-23781; U.S. Department of Energy, Office of Science, Office of Nuclear Physics, contract no. DE-AC02-06CH11357; and Forschungszentrum Julich GmbH