The Infra-Red Behaviour of the Running Coupling Constant in Yang-Mills Theories

Abstract

In Quantum Electrodynamics (QED) the infra-red (IR) structure of the theory is well understood. In Quantum Chromodynamics (QCD) it is not, though considerable effort has been spent in recent years in studying it. One reason for this interest is the hope that (because of the apparently stronger IR singularities in QCD over those in QED) confinement can be understood through the IR properties of QCD. Indeed, it has been argued1) that an “effective potential” in momentum space between quarks is proportional to g2(q2)/q2 where g(q2) is singular in the IR (that is, as q2 → 0) then the “effective potential” produces confinement. In particular, if g2(q2) ~ 1/q2 as q2 → 0, this can be crudely translated into a potential growing linearly with distance for large distances. There are clearly difficulties and uncertainties with these ideas. First the identification of g2(q2)/q2 as an “effective potential” has been made only in a leading log study of the IR properties of QCD, and an investigation of non-leading logs casts doubt on it2). Second, g2(q2) is not really gauge invariant, and it is clearly uncomfortable to attribute a physical result, such as confinement, to a non-gauge invariant quantity.