Three-jet production at LEP and the bottom quark mass

We consider the possibility of extracting the bottom quark mass from LEP data. The inclusive decay rate for $\text{zbb}+\cdots$ is obtained at order $\text{as}$ by summing up the one-loop two-parton decay rate to the tree-level three-parton rate. We calculate the decay width of the $Z$-boson into two and three jets containing the $b$-quark including complete quark mass effects. In particular, we give analytic results for a slight modification of the JADE clustering algorithm. We also study the angular distribution with respect to the angle formed between the gluon and the quark jets, which has a strong dependence on the quark mass. The impact of higher order QCD corrections on these observables is briefly discussed. Finally, we present numerical results for some popular jet-clustering algorithms and show that, indeed, these three-jet observables are very sensitive to the $b$-quark mass and well suited for its determination at LEP.