Numerical evaluation of loop integrals

Abstract

We present a new method for the numerical evaluation of arbitrary loop integrals in dimensional regularization. We first derive Mellin-Barnes integral representations and apply an algorithmic technique, based on the Cauchy theorem, to extract the divergent parts in the ǫ → 0 limit. We then perform an ǫ-expansion and evaluate the integral coefficients of the expansion numerically. The method yields stable results in physical kinematic regions avoiding intricate analytic continuations. It can also be applied to evaluate both scalar and tensor integrals without employing reduction methods. We demonstrate our method with specific examples of infrared divergent integrals with many kinematic scales, such as two-loop and three-loop box integrals and tensor integrals of rank six for the one-loop hexagon topology.

Single scale tadpoles and O(GFmt2αs3) corrections to the ρ parameter

“…Although not all of the above numerical values could be checked with independent techniques, we were able to verify many of them, either with the sector decomposition method [27], or with Mellin-Barnes techniques [28,29] implemented in the MB package [30] (see also [31]). …”

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ε-expansion of generalized hypergeometric functions with one half-integer value of parameter

“…It is presently unknown what type of sums (beyond generalized harmonic sums) are expressible in terms of known special functions. 5 The aim of this paper is to prove the following theorems:…”

“…refs. [4,5]). One fruitful approach to the calculation of Feynman diagrams is based on their representation in terms of hypergeometric functions [6] or multiple series [7,8].…”

The rational parts of one-loop QCD amplitudes I: The general formalism

“…The current state of the art in NLO computation is 5-point for QCD processes and 6-point for electroweak processes [6,7,8]. The recent development in tackling the multi-leg amplitudes by semi-numerical/analytic methods shows promise for improving traditional capabilities [9,10,11,12,13,14,15,16]. All helicity configurations for the 6-gluon amplitude are evaluated for a single-space point [9].…”