Convergence and Gauge Dependence Properties of the Resummed One-loop Quark-Quark Scattering Amplitude in Perturbative QCD

Abstract

The one-loop QCD effective charge ${\alpha }_s^{eff}$ for quark-quark scattering is derived by diagrammatic resummation of the one-loop amplitude using an arbitary covariant gauge. Except for the particular choice of gauge parameter $\xi =-3$, ${\alpha }_s^{eff}$ is found to {\it increase} with increasing physical scale, $Q$, as $\ln Q$ or ${\ln }^{2}Q$. For $\xi =-3$, ${\alpha }_s^{eff}$ decreases with increasing $Q$ and satisfies a renormalisation group equation. Also, except for the case $\xi =19/9$, convergence radii of geometric series are found to impose upper limits on $Q$.

Comments: 28 pages, 5 tables, 5 figures. v3 The one-loop amplitudes in Section 2 are recalculated using dimensional regularisation, and several errors in the on-shell calculation of Reference[1] are pointed out. v4 one figure removed one added. Three tables and new text in Section 5 added. Published version