The vacuum preserving Lie algebra of a classical W-algebra

We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the `classical vacuum preserving algebra') containing the M\"obius $sl(2)$ subalgebra to any classical \W-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-primary fields. In the case of the \W_\S^\G-algebra constructed through the Drinfeld-Sokolov reduction based on an arbitrary $sl(2)$ subalgebra $\S$ of a simple Lie algebra \G, we exhibit a natural isomorphism between this finite Lie algebra and \G whereby the M\"obius $sl(2)$ is identified with $\S$.Comment: 11 pages, BONN-HE-93-25, DIAS-STP-93-13. Some typos had been removed, no change in formula

Non-invariant two-loop counterterms for the three-gauge-boson vertices

Some practical applications of algebraic renormalization are discussed. In particular we consider the two-loop QCD corrections to the three-gauge-boson vertices involving photons, Z and W bosons. For this purpose also the corresponding two-point functions have to be discussed. A recently developed procedure is used to analyze the breaking terms of the functional identities and explicit formulae for the universal counterterms are provided. Special attention is devoted to the treatment of infra-red divergences.Comment: Some minor corrections and improved discussion in the one-loop sectio