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

By Thomas Binoth, J. Guillet, Gudrun Heinrich, E. Pilon, C. Schubert. 2005

178 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/4d110963ce8fb9d4e3a3645ef9e38d50cfa733ce

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Citations in the corpus

NLO QCD corrections to off-shell top-antitop production with leptonic decays at hadron colliders (2012, background): “To regularize intermediate top-quark resonances in a gauge-invariant (7)Similar procedures based on numerical evaluations of specific one-loop integrals [75,79] or expansions in small determinants [80] have also been proposed by other authors.”
GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond (2014, methodology): “The algebraic expression of the integrands are automatically generated by means of the Golem technology [26,28–30].”
General-purpose event generators for LHC physics (2011, methodology): “In [306] the NLO corrections to ZZ+jet production have been calculated for the first time, relying on Golem [307] for the generation of the loop-amplitude expressions.”
Collider physics at the precision frontier (2020, background): “ iin eq. (28), tells us that the contour deformation into the complex plane should be such that the imaginary part of Fis negative. For real masses and Mandelstam invariants sij, the following Ansatz[775, 789, 787, 886{888] is therefore convenient: zk(~x) = xk i˝k(~x) ;˝k= xk(1 xk) @F(~x) @xk : (63)”
Direct extraction of one loop rational terms (2008, background): “Given that the integral coefficients are rational functions of the external momenta it makes sense to look for a purely algebraic procedure which avoids any explicit integration.”
Sector Decomposition (2008, background): “Solutions to this problem already have been suggested [32, 41, 42, 58, 59, 62, 64] and are subject to current research.”
Electroweak radiative corrections for collider physics (2019, background): “These basic integrals are typically reduced to the usual scalar integrals or, in particular for small Gram determinants, calculated by numerical integration [214].”
pySecDec: A toolbox for the numerical evaluation of multi-scale integrals (2017, background): “rminates due to a “sign_check_error”. A “sign_check_error” means that the chosen contour in the analytic continuation to the complex plane is invalid. Since the contour deformation is by construction [28, 29, 30, 11, 31] guaranteed to be valid with sufﬁciently small deformation parameters, you shoul”