Nonperturbative approach to quantum chromodynamics

Abstract

In this article a nonperturbative approach to quantum chromodynamics in the confinement phase is developed. The principal idea is to assume confinement, rather than attempting to prove it, and then to examine the consequences of the resulting quantum field theory. We implement confinement by assuming that the gauge field propagator becomes infrared singular like $q^{-4\mathrm{}}$ at small momenta. The consequences of this confinement Ansatz are explored for the quark propagator, ghost propagator, and quark-gluon proper vertex. Slavnov-Taylor-Ward identities are used to fix the low-momentum behavior of the Green's functions, so no approximations are made in the Dyson equations. In both the covariant Landau gauge and the noncovariant axial gauge we obtain differential equations for the quark propagator which are solved. Vacuum ${\gamma }_5$ invariance is dynamically broken so the solutions satisfy partial conservation of axial-vector current. We also obtain the solution for the quark propagator in the case in which the flavor symmetry is explicitly broken by a quark mass term. Some implications of this approach, which is an expansion about the infrared behavior of the amplitudes, for the bound-state problem are briefly discussed.