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A solvable systematic truncation scheme for the Dyson-Schwinger equations of Euclidean QCD in Landau gauge is presented. It implements the Slavnov-Taylor identities for the three-gluon and ghost-gluon vertices, whereas irreducible four-gluon couplings as well as the gluon-ghost and ghost-ghost scattering kernels are neglected. The infrared behavior of gluon and ghost propagators is obtained analytically: The gluon propagator vanishes for small spacelike momenta whereas the ghost propagator diverges more strongly than a massless particle pole. The numerical solutions are compared with recent lattice data for these propagators. The running coupling of the renormalization scheme approaches a fixed point, αc ≃ 9.5, in the infrared. A theoretical understanding of confinement of quarks and gluons into colorless hadrons could be obtained by proving the failure of the cluster decomposition property for color-nonsinglet gauge-covariant operators. One long established idea in this direction is based on the occurrence of infrared divergences to suppress the emission of colored states from color-singlet states [1]. Such a description of confinement in terms of perturbation theory necessarily has to fail.Thus, to study the infrared behavior of QCD amplitudes non-perturbative methods are required, and, since divergences are anticipated, a formulation in the continuum is desirable. Both of these are provided by studies of truncated systems of Dyson-Schwinger equations (DSEs), the equations of motion of QCD Green's functions. Typically, for their truncation, additional sources of information like the Slavnov-Taylor identities, entailed by gauge invariance, are used to express vertex functions in terms of the elementary 2-point functions, i.e., the quark, ghost and gluon propagators. Those propagators can then be obtained as selfconsistent solutions to non-linear integral equations representing a closed set of truncated DSEs. Some systematic control over the truncating assumptions can be obtained by successively including higher n-point functions in selfconsistent calculations, and by assessing their influence on lower n-point functions in this way. At present, even at the level of propagators no complete solution to truncated DSEs of QCD exists. In particular, even in absence of quarks, solutions for the gluon propagator in Landau gauge rely on neglecting ghost contributions [2][3][4][5]. While this particular problem is avoided in ghost free gauges such as the axial gauge, in studies of the gluon DSE in this gauge [6], the possible occurrence of an independent second term in the tensor structure of the gluon propagator has so far been disregarded [7]. In fact, if the complete tensor structure of the gluon propagator in axial gauge is taken into account, one arrives at equations of no less complexity than the ghost-gluon system in the Landau gauge [8].In addition to the prospect of some insight into confinement from studying the infrared behavior of QCD Green's functions, DSEs have proved to be a highly successful tool in devel...

A truncation scheme for the Dyson Schwinger equations of QCD in Landau gauge is presented which implements the Slavnov Taylor identities for the 3-point vertex functions. Neglecting contributions from 4-point correlations such as the 4-gluon vertex function and irreducible scattering kernels, a closed system of equations for the propagators is obtained. For the pure gauge theory without quarks this system of equations for the propagators of gluons and ghosts is solved in an approximation which allows for an analytic discussion of its solutions in the infrared: The gluon propagator is shown to vanish for small spacelike momenta whereas the ghost propagator is found to be infrared enhanced. The running coupling of the non-perturbative subtraction scheme approaches an infrared stable fixed point at a critical value of the coupling, : c & 9.5. The gluon propagator is shown to have no Lehmann representation. The results for the propagators obtained here compare favorably with recent lattice calculations.1998 Academic Press

The coupled system of renormalized Dyson-Schwinger equations for the quark, gluon and ghost propagators of Landau gauge QCD is solved within truncation schemes. These employ bare as well as non-perturbative ansätze for the vertices such that the running coupling as well as the quark mass function are independent of the renormalization point. The one-loop anomalous dimensions of all propagators are reproduced. Dynamical chiral symmetry breaking is found, the dynamically generated quark mass agrees well with phenomenological values and corresponding results from lattice calculations. The effects of unquenching the system are small. In particular the infrared behavior of the ghost and gluon dressing functions found in previous studies is almost unchanged as long as the number of light flavors is smaller than four.

We review the spectrum and electromagnetic properties of baryons described as relativistic three-quark bound states within QCD. The composite nature of baryons results in a rich excitation spectrum, whilst leading to highly non-trivial structural properties explored by the coupling to external (electromagnetic and other) currents. Both present many unsolved problems despite decades of experimental and theoretical research. We discuss the progress in these fields from a theoretical perspective, focusing on nonperturbative QCD as encoded in the functional approach via Dyson-Schwinger and Bethe-Salpeter equations. We give a systematic overview as to how results are obtained in this framework and explain technical connections to lattice QCD. We also discuss the mutual relations to the quark model, which still serves as a reference to distinguish 'expected' from 'unexpected' physics. We confront recent results on the spectrum of non-strange and strange baryons, their form factors and the issues of two-photon processes and Compton scattering determined in the DysonSchwinger framework with those of lattice QCD and the available experimental data. The general aim is to identify the underlying physical mechanisms behind the plethora of observable phenomena in terms of the underlying quark and gluon degrees of freedom.

We explore the analytic structure of the gluon and quark propagators of Landau gauge QCD from numerical solutions of the coupled system of renormalized Dyson-Schwinger equations and from fits to lattice data. We find sizable negative norm contributions in the transverse gluon propagator indicating the absence of the transverse gluon from the physical spectrum. A simple analytic structure for the gluon propagator is proposed. For the quark propagator we find evidence for a mass-like singularity on the real timelike momentum axis, with a mass of 350 to 500 MeV. Within the employed Green's functions approach we identify a crucial term in the quark-gluon vertex that leads to a positive definite Schwinger function for the quark propagator.

We highlight the progress, current status, and open challenges of QCD-driven physics, in theory and in experiment. We discuss how the strong interaction is intimately connected to a broad sweep of physical problems, in settings ranging from astrophysics and cosmology to strongly coupled, complex systems in particle and condensed-matter physics, as well as to searches for physics beyond the Standard Model. We also discuss how success in describing the strong interaction impacts other fields, and, in turn, how such subjects can impact studies of the strong interaction. In the course of the work we offer a perspective on the many research streams which flow into and out of QCD, as well as a vision for future developments.

The self-consistent chiral soliton of the Nambu-Jona-Lasinio model including the ω, ρ and a 1 (axial-) vector meson fields besides the chiral angle is investigated. The resulting energy spectrum of the one particle Dirac Hamiltonian is strongly distorted leading to a polarized Dirac sea which carries the complete baryon number. This supports Witten's conjecture that baryons can be described as topological solitons. The exploration of the isoscalar mean squared radius of the nucleon exhibits that the repulsive character of the isoscalar vector field ω as well as the attractive features of the (axial-) vector mesons ρ and a 1 are reproduced in the Nambu-Jona-Lasinio model. The axial charge of the nucleon g A comes out far too small. This can be understood as an artifact of the proper time regularization prescription. † Supported by the Deutsche Forschungsgemeinschaft (DFG) under contract Re 856/2-1. 1