We review the current status of nonperturbative studies of gauge field theory using the Dyson-Schwinger equation formalism and its application to hadronic physics. We begin with an introduction to the formalism and a discussion of renormalisation in this approach. We then review the current status of studies of Abelian gauge theories [e.g., strong coupling quantum electrodynamics] before turning our attention to the non-Abelian gauge theory of the strong interaction, quantum chromodynamics. We discuss confinement, dynamical chiral symmetry breaking and the application and contribution of these techniques to our understanding of the strong interactions. KEYWORDS confinement of quarks and gluons; dynamical chiral symmetry breaking; Dyson-Schwinger equations; hadrons; quantum electrodynamics; quantum chromodynamics.

We present an unquenched calculation of the quark propagator in Landau gauge with 2+1 flavors of dynamical quarks. We use configurations generated with an improved staggered ("Asqtad") action by the MILC collaboration. This quark action has been seen to have excellent rotational symmetry and scaling properties in the quenched quark propagator. Quenched and dynamical calculations are performed on a 20 3 × 64 lattice with a nominal lattice spacing of a = 0.125 fm. The matched quenched and dynamical lattices allow us to investigate the relatively subtle sea quark effects, and even in the quenched case the physical volume of these lattices gives access to lower momenta than our previous study. We calculate the quark mass function and renormalization function for a variety of valence and sea quark masses.

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By combining the constraints of charge symmetry with new chiral extrapolation techniques and recent low mass lattice QCD simulations of the individual quark contributions to the magnetic moments of the nucleon octet, we obtain a precise determination of the strange magnetic moment of the proton. The result, namely G s M = −0.046 ± 0.019 µN , is consistent with the latest experimental measurements but an order of magnitude more precise. This poses a tremendous challenge for future experiments.

The complete tensor structure of the quark-gluon vertex in Landau gauge is determined at two kinematical points ('asymmetric' and 'symmetric') from lattice QCD in the quenched approximation. The simulations are carried out at β = 6.0, using a meanfield improved Sheikholeslami-Wohlert fermion action, with two quark masses ∼ 60 and 115 MeV. We find substantial deviations from the abelian form at the asymmetric point. The mass dependence is found to be negligible. At the symmetric point, the form factor related to the chromomagnetic moment is determined and found to contribute significantly to the infrared interaction strength.

We study the scaling behavior of the quark propagator on two lattices with similar physical volume in Landau gauge with 2+1 flavors of dynamical quarks in order to test whether we are close to the continuum limit for these lattices. We use configurations generated with an improved staggered ("Asqtad") action by the MILC collaboration. The calculations are performed on 28 3 × 96 lattices with lattice spacing a = 0.09 fm and on 20 3 × 64 lattices with lattice spacing a = 0.12 fm. We calculate the quark mass function, M (q 2 ), and the wave-function renormalization function, Z(q 2 ), for a variety of bare quark masses. Comparing the behavior of these functions on the two sets of lattices we find that both Z(q 2 ) and M (q 2 ) show little sensitivity to the ultraviolet cutoff.

The Laplacian gauge is a nonperturbative gauge fixing that reduces to Landau gauge in the asymptotic limit. Like Landau gauge, it respects Lorentz invariance, but it is free of Gribov copies; the gauge fixing is unambiguous. In this paper we study the infrared behavior of the lattice gluon propagator in Laplacian gauge by using a variety of lattices with spacings from a = 0.125 to 0.35 fm, to explore finite volume and discretization effects. Three different implementations of the Laplacian gauge are defined and compared. The Laplacian gauge propagator has already been claimed to be insensitive to finite volume effects and this is tested on lattices with large volumes.

Using lattice quantum chromodynamics (QCD) we perform an unquenched calculation of the gluon propagator in Landau gauge. We use configurations generated with the AsqTad quark action by the MILC collaboration for the dynamical quarks and compare the gluon propagator of quenched QCD (i.e., the pure Yang-Mills gluon propagator) with that of 2+1 flavor QCD. The effects of the dynamical quarks are clearly visible and lead to a significant reduction of the nonperturbative infrared enhancement relative to the quenched case.