Infrared behavior of non-Abelian gauge theories. II

We continue and extend our earlier work on infrared singularities of non-Abelian gauge theories (exemplified by quantum chromodynamics, or QCD), emphasizing the nature of these singularities in space-time rather than in momentum space. The leading logarithms of non-Abelian gauge theories sum to an eikonal (semiclassical) form for processes involving massive particles. This form is like the corresponding Abelian (QED) form, except that the coupling constant $g$ is replaced by the invariant coupling $\overline{g}\mathrm{}\left(t\right)\mathrm{}$ which depends on the momentum transfer $t$ to the gluon. It is not known whether the nonleading logarithms exactly cancel order by order, but there are strong cancellations which are governed by Ward identities. Besides four-dimensional QCD, two analogous theories which show confinement are studied: Two-dimensional QCD is developed in the light-cone gauge without recourse to the large-$N$ approximation of 't Hooft and is shown to have infrared singularities like those of a string theory with longitudinal modes. Three-dimensional QED is briefly discussed, and it is argued that confinement is revealed in the anomalous infrared behavior of the fermion propagator which is an entire function (of momentum) with no pole. Some speculations to this effect are made for four-dimensional QCD.