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

Abstract

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. E Hexagon relations from helicity decomposition 61 2

Gluon-inducedW-boson pair production at the LHC

“…[25]. 6 The tadpole integral does not appear in this list, as it can be viewed as a degenerate 2-point integral.…”

Towards LHC phenomenology at the loop level: A new method for one-loop amplitudes

“…To avoid the latter, we use integrals with Feynman parameters in the numerator as reduction endpoints. Explicit representations for all form factors for r ≤ N ≤ 5 can be found in [5]. The form factors are expressed in terms of the following basis integrals…”

“…IR divergencies are only present for N = 3 and d = 4 − 2ǫ. For the IR divergent integrals explicit analytical formulas can be found in [5]. The numerical evaluation of the remaining IR finite integrals is problematic due to kinematical singularities, which occur if the quadratic form x · S · x changes sign.…”

“…Due to its phenomenological relevance, a lot of activity has been going on in this direction during the last years [1,2,3,4,5,6,7,8,9,10,11,12].…”

“…In this talk, a method [5] for the evaluation of one-loop diagrams occurring in multi-particle computaions at NLO is presented. This approach allows for a good analytical control over the expressions, but also provides stable numerical representations in phase space regions which typically pose numerical problems.…”