We have e?cient programs like FeynArts 108, GRACE 109,110 or QGRAF 111 to deal with multi-leg one loop feynman diagrams. But due to the presence of IR divergences the evaluation for more than four external legs is still far from automated. This loop contribution usually obtained analytically on a case by case basis and extensive computer algebra is necessary to extract the IR and UV singularities from these one loop graphs that appear as poles in the dimensional regularization. This is the bottleneck for producing NLO corrections for many multi particle processes. For one-loop amplitude the most straightforward procedure and historically the ?rst one relies on the use of recursion relations to reduce the tensor integrals occurring in the one-loop amplitude to a set of known basis integrals 112–132. A formalism for numerical evaluation of multi-leg amplitudes for the massless case hase been proposed in 129, but this formalism produces spurious inverse Gram determinants. A method is proposed in 130 about to deal with them. A formalism given in 126 avoids inverse Gram determinants in the reduction of pentagon integrals but deals with massive particles only. In 131, another algorithm is presented, using spinor helicity methods. Based on the formalism of 131, an evaluation of one-loop integrals in massless gauge theories for up to 12 external legs has been given recently in 132. One loop calculations was revolutionized by merging the idea of four dimensional unitarity cuts 133,134 with the understanding of the universal algebraic form of any one loop integral in four dimension contained in the OPP method 135–137. Unitarity-based methods and integrand-level reduction techniques provided the theoretical background for development of e?cient computational algorithms for NLO calculations in perturbation theory, which have been implemented in various automated codes.

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