Abstract:
Dyson-Schwinger equations furnish a Poincare' 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. Key to the DSE's efficacious application is their expression of the materially important momentum-dependent dressing of parton propagators at infrared length-scales, which is responsible for the magnitude of constituent-quark masses and the length-scale characterising confinement in bound states. A unified quantitative description of light- and heavy-quark systems is achieved by capitalising on these features.

Abstract:
An overview of the theory and phenomenology of hadrons and QCD is provided from a Dyson-Schwinger equation viewpoint. Following a discussion of the definition and realisation of light-quark confinement, the nonperturbative nature of the running mass in QCD and inferences from the gap equation relating to the radius of convergence for expansions of observables in the current-quark mass are described. Some exact results for pseudoscalar mesons are also highlighted, with details relating to the U_A(1) problem, and calculated masses of the lightest J=0,1 states are discussed. Studies of nucleon properties are recapitulated upon and illustrated: through a comparison of the ln-weighted ratios of Pauli and Dirac form factors for the neutron and proton; and a perspective on the contribution of quark orbital angular momentum to the spin of a nucleon at rest. Comments on prospects for the future of the study of quarks in hadrons and nuclei round out the contribution.

Abstract:
Dynamical chiral symmetry breaking is a nonperturbative phenomenon that may be studied using QCD's gap equation. Model-independent results can be obtained with a nonperturbative and symmetry preserving truncation. The gap equation yields the massive dressed-quark propagator, which has a spectral representation when considered as a function of the current-quark mass. This enables an explication of the connection between the infrared limit of the QCD Dirac operator's spectrum and the quark condensate appearing in the operator product expansion.

Abstract:
We propose a method to probe Landau and Beliaev processes in dilute trapped atomic condensates with a multiple state structure using ultraslow light experimental configurations. Under certain conditions, damping rates from these collisional processes are directly proportional to the dephasing rates, making it possible to determine damping rates through measurement of the dephasing. In the ultraslow light systems we consider, Landau decay rates are enhanced at low momenta, which allows one to distinguish between Landau-dominated and Beliaev-dominated regimes at the same temperature. Furthermore, the enhancement of Landau rates potentially provides a way to measure low temperatures ($T \ll T_c$) in dilute condensates more accurately than current methods permit.

Abstract:
An analytic expression is derived for a force on a weak point impurity arising from the scattering of quantum fluctuations in a slow-moving, weakly interacting, three-dimensional Bose-Einstein condensate at zero temperature. In an infinitely extended geometry, this force is shown to exist at any arbitrarily small flow velocity below Landau's critical velocity. Furthermore, this force is shown to be directly proportional to the flow speed.

Abstract:
Dynamical chiral symmetry breaking (DCSB) is one of the keystones of low-energy hadronic phenomena. Dyson-Schwinger equations provide a model-independent quark-level understanding and correlate that with the behaviour of the pion's Bethe-Salpeter amplitude. This amplitude is a core element in the calculation of pion observables and combined with the dressed-quark Schwinger function required by DCSB it yields a valence-quark distribution function for the pion that behaves as (1-x)^2 for x~1, in accordance with perturbative analyses. This behaviour can be verified at contemporary experimental facilities.

Abstract:
We recapitulate on aspects of Dyson-Schwinger equation studies relevant to pseudoscalar mesons: lattice confirmation of the DSE prediction that propagators are nonperturbatively dressed in the infrared; and exact results, e.g., in the chiral limit the leptonic decay constant vanishes for every pseudoscalar meson except the pion.

Abstract:
The form factor for the anomalous process $\gamma \pi ^* \to \pi \pi$, $F^{3\pi}(s,t,u)$, is calculated as a phenomenological application of the QCD Dyson-Schwinger equations. The chiral-limit value dictated by the electromagnetic, anomalous chiral Ward identity, $F^{3\pi}(0,0,0)= e N_c/(12\pi^2 f_\pi^3)$, is reproduced, {\it independent} of the details of the modelling of the gluon and quark 2-point Schwinger functions. Using a parametrisation of the dressed $u$-$d$ quark 2-point Schwinger function that provides a good description of pion observables, such as $\pi$-$\pi$ scattering-lengths and the electromagnetic pion form factor, $F^{3\pi}(s,t,u)$ is calculated on a kinematic range that proposed experiments plan to explore. Our result confirms the general trend of other calculations; i.e., a monotonic increase with $s$ at fixed $t$ and $u$, but is uniformly larger and exhibits a more rapid rise with $s$.

Abstract:
Independent of assumptions about the form of the quark-quark scattering kernel, K, we derive the explicit relation between the flavour-nonsinglet pseudoscalar meson Bethe-Salpeter amplitude, Gamma_H, and the dressed-quark propagator in the chiral limit. In addition to a term proportional to gamma_5, Gamma_H necessarily contains qualitatively and quantitatively important terms proportional to gamma_5 gamma.P and gamma_5 gamma.k k.P, where P is the total momentum of the bound state. The axial-vector vertex contains a bound state pole described by Gamma_H, whose residue is the leptonic decay constant for the bound state. The pseudoscalar vertex also contains such a bound state pole and, in the chiral limit, the residue of this pole is related to the vacuum quark condensate. The axial-vector Ward-Takahashi identity relates these pole residues; with the Gell-Mann--Oakes-Renner relation a corollary of this identity. The dominant ultraviolet asymptotic behaviour of the scalar functions in the meson Bethe-Salpeter amplitude is fully determined by the behaviour of the chiral limit quark mass function, and is characteristic of the QCD renormalisation group. The rainbow-ladder Ansatz for K, with a simple model for the dressed-quark-quark interaction, is used to illustrate and elucidate these general results. The model preserves the one-loop renormalisation group structure of QCD. The numerical studies also provide a means of exploring procedures for solving the Bethe-Salpeter equation without a three-dimensional reduction.

Abstract:
Dyson-Schwinger equations furnish a Poincare' 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 behaviour 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.