From scattering amplitudes to cross sections in QCD

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I describe how to calculate cross sections for hard-scattering processes in high energy collisions at next to leading order in QCD. I consider infrared-safe quantities and I assume that the scattering amplitudes are known in analytic form up to next-to-leading order. The main topic is the description of the algorithm for the analytic cancellation of the soft and collinear singularities in the loop and bremsstrahlung contributions. The method is systematic and general. It allows the construction of an analytic expression for finite next-to-leading order hard scattering cross sections suitable for numerical evaluation.