We present a program for the numerical evaluation of form factors entering the calculation of one-loop amplitudes with up to six external legs. The program is written in Fortran95 and performs the reduction to a certain set of basis integrals numerically, using a formalism where inverse Gram determinants can be avoided. It can be used to calculate one-loop amplitudes with massless internal particles in a fast and numerically stable way.
We present a new Fortran code to calculate the scalar one-loop four-point integral with complex internal masses, based on the method of t Hooft and Veltman. The code is applicable when the external momenta fulfill a certain physical condition. In particular it holds if one of the external momenta or a sum of them is timelike or lightlike and therefore covers all physical processes at colliders. All the special cases related to massless external particles are treated separately. Some technical issues related to numerical evaluation and Landau singularities are discussed.
We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of $n$- and $(n-1)$-point scalar integrals that are finite in the limit of vanishing Gram determinant. These non-trivial combinations of dilogarithms, logarithms and constants are systematically obtained by either differentiating with respect to the external parameters - essentially yielding scalar integrals with Feynman parameters in the numerator - or by developing the scalar integral in $D=6-2e$ or higher dimensions. As an explicit example, we develop the tensor integrals and associated scalar integral combinations for processes where the internal particles are massless and where up to five (four massless and one massive) external particles are involved. For more general processes, we present the equations needed for deriving the relevant combinations of scalar integrals.
In this paper, we describe a numerical approach to evaluate Feynman loop integrals. In this approach the key technique is a combination of a numerical integration method and a numerical extrapolation method. Since the computation is carried out in a fully numerical way, our approach is applicable to one-, two- and multi-loop diagrams. Without any analytic treatment it can compute diagrams with not only real masses but also complex masses for the internal particles. As concrete examples we present numerical results of a scalar one-loop box integral with complex masses and two-loop planar and non-planar box integrals with masses. We discuss the quality of our numerical computation by comparisons with other methods and also propose a self consistency check.
We suggest a new approach for the automatic and fully numerical evaluation of one-loop scattering amplitudes in perturbative quantum field theory. We use suitably formulated dispersion relations to perform the calculation as a convolution of tree-level amplitudes. This allows to take advantage of the iterative numerical algorithms for the evaluation of leading order matrix elements.
We briefly sketch the methods for a numerically stable evaluation of tensor one-loop integrals that have been used in the calculation of the complete electroweak one-loop corrections to $PepPemto4 $fermions. In particular, the improvement of the new methods over the conventional Passarino--Veltman reduction is illustrated for some 4-point integrals in the delicate limits of small Gram (and other kinematical) determinants.
T.Binoth
,J.-Ph.Guillet
,G.Heinrich
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(2008)
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"Golem95: a numerical program to calculate one-loop tensor integrals with up to six external legs"
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Gudrun Heinrich
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