The Schwinger-Dyson equations and confinement in quantum chromo-dynamics
The Schwinger-Dyson equations for the gluon and quark propagators are investigated in the covariant gauge. The renormalization functions are approximated suitably and the value of the parameters are determined by requiring that the functions be numerically self-consistent solutions over appropriate ranges of momenta. In the case of the gluon the Schwinger-Dyson equation is truncated by neglecting the the two loop contributions and the triple gluon vertex is approximated by a form proposed by Mandelstam which has the same behaviour as the more complicated longitudinal vertex determined from the Slavnov-Taylor identity. The equation is then closed and the integrals are calculated by dimensional regularization and renormalized to remove a mass term. In the quark case the dominant part of the quark-gluon vertex is determined from the Ward-Takahashi identity to give, with the gluon, a closed equation. The angular integrals are then calculated by an appropriate choice of coordinate frame. The quark function is approximated by a power series in the non-perturbative regime and the usual perturbative result elsewhere. The radial integrals are then calculated with appropriate regularization and renormalization. It is found that the gluon propagator has approximately a singularity of the form 1/q(^4) which leads to a roughly linear confining potential. The effect of this enhanced singularity on the quark propagator is to suppress the propagation of quarks at low momenta.