The One-loop QCD Corrections for $\gamma ^* \to Q\overline{Q}q\overline{q}$

We calculate 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 ${\cal O}(\alpha_s^3)$ 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 use standard techniques for computing the interference of one-loop and tree level Feynman diagrams, but organise the results in terms of combinations of scalar loop integrals that are finite in the limit of vanishing Gram determinants and are therefore numerically stable