Dyson-Schwinger equations furnish a Poincaré covariant framework within which to study hadrons. A particular feature is the existence of a nonperturbative, symmetry preserving truncation that enables the proof of exact results. The gap equation reveals that dynamical chiral symmetry breaking is tied to the long-range behavior of the strong interaction, which is thereby constrained by observables, and the pion is precisely understood, and seen to exist simultaneously as a Goldstone mode and a bound state of strongly dressed quarks. The systematic error associated with the simplest truncation has been quantified, and it underpins a one-parameter model efficacious in describing an extensive body of mesonic phenomena. Incipient applications to baryons have brought successes and encountered challenges familiar from early studies of mesons, and promise a covariant field theory upon which to base an understanding of contemporary large momentum transfer data.

Features of the dressed-quark-gluon vertex and their role in the gap and Bethe-Salpeter equations are explored. It is argued that quenched lattice data indicate the existence of net attraction in the colour-octet projection of the quark-antiquark scattering kernel. This attraction affects the uniformity with which solutions of truncated equations converge pointwise to solutions of the complete gap and vertex equations. For current-quark masses less than the scale set by dynamical chiral symmetry breaking, the dependence of the dressed-quark-gluon vertex on the current-quark mass is weak. The study employs a vertex model whose diagrammatic content is explicitly enumerable. That enables the systematic construction of a vertex-consistent Bethe-Salpeter kernel and thereby an exploration of the consequences for the strong interaction spectrum of attraction in the colour-octet channel. With rising current-quark mass the rainbow-ladder truncation is shown to provide an increasingly accurate estimate of a bound state's mass. Moreover, the calculated splitting between vector and pseudoscalar meson masses vanishes as the current-quark mass increases, which argues for the mass of the pseudoscalar partner of the \Upsilon(1S) to be above 9.4 GeV. The absence of colour-antitriplet diquarks from the strong interaction spectrum is contingent upon the net amount of attraction in the octet projected quark-antiquark scattering kernel. There is a window within which diquarks appear. The amount of attraction suggested by lattice results is outside this domain.Comment: 22 pages, 12 figure

We survey contemporary studies of hadrons and strongly interacting quarks using QCD's DysonSchwinger equations, addressing: aspects of confinement and dynamical chiral symmetry breaking; the hadron spectrum; hadron elastic and transition form factors, from small-to large-Q 2 ; parton distribution functions; the physics of hadrons containing one or more heavy quarks; and properties of the quark gluon plasma.

An exact form is presented for the axial-vector Bethe-Salpeter equation, which is valid when the quark-gluon vertex is fully dressed. A Ward-Takahashi identity for the Bethe-Salpeter kernel is derived therefrom and solved for a class of dressed quark-gluon vertex models. The solution provides a symmetry-preserving closed system of gap and vertex equations. The analysis can be extended to the vector equation. This enables a comparison between the responses of pseudoscalarand scalar meson masses to nonperturbatively dressing the quark-gluon vertex. The result indicates that dynamical chiral symmetry breaking enhances spin-orbit splitting in the meson spectrum.