Measuring $\Lambda_{\overline{MS}}$ at LEP

Abstract

We show that standard next-to-leading order (NLO) perturbative QCD analyses used to extract $\alpha_{s}$ from LEP data do not serve to disentangle the completely unknown renormalization scheme (RS) invariant next-NLO (NNLO) and higher-order uncalculated corrections from those dependent on the renormalization scale in a predictable manner. The resulting quoted values of $\alpha_{s}(M_{Z})$ $(\tilde{\Lambda}_{\overline{MS}})$ with attendant scheme dependence uncertainties therefore reflect the ad hoc way in which they were extracted, rather than the actual values of these parameters. Choosing the scale so that the NLO coefficient vanishes (effective charge scheme) exposes the relative size of these unknown RS invariant higher order terms. They are seen to be sizeable for some of the LEP observables, indicating that they must be estimated if $\tilde{\Lambda}_{\overline{MS}}$ is to be determined reliably. This can be accomplished either from NNLO calculations, at present only available for the hadronic width of the $Z^{0}$ ($R_{Z}$), or non-perturbatively by writing the RS-invariant uncertainty in $\tilde{\Lambda}$ in terms of the running ${dR(Q)}/{d \ln Q}$ of the observable R(Q) with energy using an effective charge formalism. The NNLO calculations for $R_{Z}$ and LEP data supplemented by lower energy PETRA data lead to $\tilde{\Lambda}^{(5)}_{\overline{MS}}=287 \pm 100$ MeV. We also