Techniques for evaluating one-loop Feynman diagrams and their application
For a full understanding of QCD and a precise comparison of the theory with experiment, QCD observables must be calculated to next-to-leading order in the strong coupling constant. This thesis will discuss some of the techniques used for calculating the one-loop Feynman diagrams which are required for such calculations, and their associated tensor integrals. In particular, conventional methods introduce Gram determinants. This can lead to unnecessarily complicated expressions and numerical instabilities in the limit of vanishing Gram determinant. An alternative method is presented which removes these problems by gathering together scalar integrals in combinations which are finite as the Gram determinant vanishes. These combinations are related to the corresponding scalar integrals in higher dimensions. This method is applied to the evaluation 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. These matrix elements are required for the next-to-leading order corrections to four jet production in electron-positron annihilation, the production of a gauge boson and two jets in hadron-hadron collisions, and three jet production in lepton- nucleon scattering.