Renormalization-prescription dependence of the quantum-chromodynamic coupling constant

Abstract

Massless quantum chromodynamics cannot be renormalized on-shell; various possible off-shell renormalization prescriptions yield different definitions of a scale-dependent coupling constant $g$. We show how to relate physical predictions computed in different renormalization schemes. In particular, we compute the dimensionally regularized two- and three-point functions at the symmetric point in momentum space through one-loop order, and deduce the relation between ${g}_{min}$ defined by minimal subtraction and ${g}_{\mathrm{mom}}$ defined by momentum-space subtraction. We find that ${g}_{\mathrm{mom}}$ is fairly insensitive to which vertex one chooses to define it, and only weakly gauge dependent. ${g}_{min}$ is shown to depend strongly on the dimensional-regularization procedure, and can therefore differ quite dramatically from ${g}_{\mathrm{mom}}$. The scale dependence of $g$ is conventionally parametrized by a scale-invariant mass $\ensuremath{\Lambda}$; the ratio of $\ensuremath{\Lambda}$'s defined by any two renormalization schemes is a pure number which we show is exactly deducible from our one-loop results.