An algebraic/numerical formalism for one-loop multi-leg amplitudes

We present a formalism for the calculation of multi-particle one-loop amplitudes, valid for an arbitrary number N of external legs, and for massive as well as massless particles. A new method for the tensor reduction is suggested which naturally isolates infrared divergences by construction. We prove that for N ⩾ 5, higher dimensional integrals can be avoided. We derive many useful relations which allow for algebraic simplifications of one-loop amplitudes. We introduce a form factor representation of tensor integrals which contains no inverse Gram determinants by choosing a convenient set of basis integrals. For the evaluation of these basis integrals we propose two methods: An evaluation based on the analytical representation, which is fast and accurate away from exceptional kinematical configurations, and a robust numerical one, based on multi-dimensional contour deformation. The formalism can be implemented straightforwardly into a computer program to calculate next-to-leading order corrections to multi-particle processes in a largely automated way.