A Class of Bicovariant Differential Calculi on Hopf Algebras

We introduce a large class of bicovariant differential calculi on any quantum group $A$, associated to $Ad$-invariant elements. For example, the deformed trace element on $SL_q(2)$ recovers Woronowicz' $4D_\pm$ calculus. More generally, we obtain a sequence of differential calculi on each quantum group $A(R)$, based on the theory of the corresponding braided groups $B(R)$. Here $R$ is any regular solution of the QYBE.Comment: 16 page

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

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 $Q$.Comment: 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 versio