The one-loop QCD corrections for γ∗ → QQqq

We present the first calculation of the one-loop QCD corrections for the decay of an off-shell vector boson with vector couplings into two pairs of quarks of equal or unequal flavours keeping all orders in the number of colours. These matrix elements are relevant for the calculation of the next-to-leading order O(αs3) corrections to four jet production in electron-positron annihilation, the production of a gauge boson accompanied by two jets in hadron-hadron collisions and three jet production in deep inelastic scattering. We compute the interference of one-loop and tree level Feynman diagrams, and organise the matrix elements in terms of combinations of scalar loop integrals that are well behaved in the limit of vanishing Gram determinants. The results are therefore numerically stable and ready to be implemented in next-to-leading order Monte Carlo calculations of the e+e− → 4 jet, e±p → e± + 3 jet and pp → V + 2 jet processes.