A non-perturbative study of the infra-red behaviour of QCD

Abstract

The non-perturbative behaviour of the non-Abelian gauge theory of strong interactions, namely QCD, is investigated using the Schwinger-Dyson equations. Using an approximation based on solving the Slavnov-Taylor identities, we derive a closed integral equation for the full gluon propagator. We numerically solve this equation, finding a consistent solution which is as singular as 1/p(^4) the momentum p(^2) → 0, whilst at large momenta the gluon propagates like a free particle. This infra-red behaviour can be seen as a signal for the confinement of quarks and gluons, implying, for example, that the Wilson loop operator behaves an 'axea law'. We then derive an equation for the full massless quark propagator. Using our solution for the gluon, we find the quark propagator to be suppressed at low momentum, to such an extent that the physical particle pole is removed, and free quarks cannot propagate. This is just what we might expect of a confining theory. The inclusion of quarks means we must study their dynamical effects via closed fermion loops in the gluon propagator equation. This couples the two equations together. We solve the two equations simultaneously, finding that the previous infra-red behaviour still holds. As we introduce more flavours of fermions, however, the infra-red enhancement of the gluon propagator is diminished, and this in turn means that the quark propagator is less suppressed. This exhibits the dynamical importance of quarks. These physically realistic results demonstrate the importance and validity of the Schwinger-Dyson equations as a valuable tool for investigating the non-perturbative features of gauge theories.