The leading-twist parton distribution functions of the pion and kaon are calculated for the first time using a rainbow-ladder truncation of QCD's Dyson-Schwinger equations (DSEs) that self-consistently sums all planar diagrams. The non-perturbative gluon dressing of the quarks is thereby correctly accounted for, which in practice means solving the inhomogeneous Bethe-Salpeter equation (BSE) for the quark operator that defines the spin-independent quark distribution functions. An immediate consequence of using this dressed vertex is that gluons carry 35% of the pion's and 30% of the kaon's light-cone momentum, with the remaining momentum carried by the quarks. The scale associated with these DSE results is µ 0 = 0.78 GeV. The gluon effects generated by the inhomogeneous BSE are inherently non-perturbative and cannot be mimicked by the perturbative QCD evolution equations. A key consequence of this gluon dressing is that the valence quarks have reduced support at low-to-intermediate x, where the gluons dominate, and increased support at large x. As a result, our DSE calculation of the pion's valence quark distribution is in excellent agreement with the Conway et al. pion-induced Drell-Yan data, but nevertheless exhibits the q π (x) (1 − x) 2 behavior as x → 1 predicted by perturbative QCD.

We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multi-dimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. Highlights from this study include predictions that the mean-squared impact parameter for the quarks in the pion and kaon are: b 2 T π u = 0.11 fm 2 , b 2 T K s = 0.08 fm 2 , and b 2 T K u = 0.13 fm 2 , and therefore the s quark in the kaon is much closer to the center of transverse momentum than the u quark. From the electromagnetic and gravitational form factors we find that the light-cone energy radii are about 60% smaller than the light-cone charge radii for each quark sector in the pion and kaon. A quantitative measure of the importance of the leading Fock state is obtained via comparison with a full DSE calculation (containing an infinite tower of Fock states) for the pion form factor.

We present results for the nucleon's leading-twist spin-independent valence parton distribution functions obtained from a theoretical framework based on the Dyson-Schwinger equations (DSEs) of QCD that previously gave an excellent description of nucleon electromagnetic form factors. We employ the rainbow-ladder truncation of the DSEs and utilize nucleon bound state amplitudes from the Poincaré-covariant Faddeev equation, where the dominant scalar and axial-vector quark-quark correlations are included. This DSE framework is used to numerically evaluate the first 20 moments of the valence u and d quark distribution functions, from which the x-dependence of the distributions is found to be well constrained. We find good agreement with empirical parameterizations of experimental data and make the prediction that the d/u ratio in the x → 1 limit, invariant under scale evolution, takes the value d/u → 0.087 ± 0.010. We find that this ratio is rather sensitive to the strength of axial-vector diquark correlations. However, contrary to a naive expectation, our result for the d/u ratio in the x → 1 limit does not vanish when only scalar diquark correlations are present, although it is an order of magnitude smaller than our d/u result that also includes axial-vector diquarks. The valence quark distribution results are set in a broader context via a simple pion cloud model estimate of sea-quark light-cone momenta and gluon light-cone momentum.