Reduction schemes for one-loop tensor integrals

By Ansgar Denner, Stefan Dittmaier. 2005

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Abstract We present new methods for the evaluation of one-loop tensor integrals which have been used in the calculation of the complete electroweak one-loop corrections to e + e − → 4 fermions. The described methods for 3-point and 4-point integrals are, in particular, applicable in the case where the conventional Passarino–Veltman reduction breaks down owing to the appearance of Gram determinants in the denominator. One method consists of different variants for expanding tensor coefficients about limits of vanishing Gram determinants or other kinematical determinants, thereby reducing all tensor coefficients to the usual scalar integrals. In a second method a specific tensor coefficient with a logarithmic integrand is evaluated numerically, and the remaining coefficients as well as the standard scalar integral are algebraically derived from this coefficient. For 5-point tensor integrals, we give explicit formulas that reduce the corresponding tensor coefficients to coefficients of 4-point integrals with tensor rank reduced by one. Similar formulas are provided for 6-point functions, and the generalization to functions with more internal propagators is straightforward. All the presented methods are also applicable if infrared (soft or collinear) divergences are treated in dimensional regularization or if mass parameters (for unstable particles) become complex.

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Citations in the corpus

Gluon-induced Higgs-strahlung at next-to-leading order QCD (2012, unknown): “[48–53].”
On-the-fly reduction of open loops (2017, methodology): “Thanks to a tensorial representation that retains the loop-momentum dependence of all building blocks, this approach can be used in combination with reduction techniques based on tensor integrals [4] or with the OPP reduction method [5], resulting in both cases in very fast computer code [9].”
A Systematic and Efficient Method to Compute Multi-loop Master Integrals (2017, unknown): “At the one-loop order, thanks to the improvement of traditional tensor reduction [1] and the development of unitarity-based reduction [2–4], one can efficiently express scattering amplitudes in terms of linear combinations of master integrals (MIs).”
Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report (2017, methodology): “Combined with the Collier tensor-reduction library [546],Denner:2016kdg, which implements the Denner–Dittmaier reduction techniques [552,553] and the scalar integrals of Ref.”
Handbook of LHC Higgs Cross Sections: 3. Higgs Properties (2013, methodology): “Virtual corrections are computed with O PENLOOPS[164], an automated generator of N LO QCD amplitudes for SM processes, which uses the C OLLIER library for the numerically stable evaluation of tensor integrals [171, 172] and scalar integrals [173].”
Cut-constructible part of QCD amplitudes (2006, methodology): “ [3] as regarding QCD corrections and six [4] as for electroweak. The most recent theoretical eﬀorts for tackling the one-loop multi-leg amplitudes have been using algebraic/semi-numerical approaches [5]. For the analysis of jet-events produced at the high energies of the LHC, it is mandatory to ove”
Electroweak corrections to W + jet hadroproduction including leptonic W-boson decays (2009, background): “[43] in exceptional phase-space regions where small Gram determinants appear.”
Two-loop integrand decomposition into master integrals and surface terms (2015, methodology): “At one-loop level one can highlight explicit [16] and implicit methods [9, 17–22] for reducing (tensor) loop integrals to a standard set of master integrals.”
Implications of conformal invariance in momentum space (2013, unknown): “In this case we can write [47]”
QCD matrix elements + parton showers. The NLO case (2012, background): “These developments range from the (near) automation of cross section calculations for multi-particle final states at the next-to leading order [1–16] to the construction of algorithms that allow to use multi-particle calculations at leading and next-to-leading order to drive event simulation at the ”
Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report (2016, methodology): “Combined with the Collier tensor-reduction library [250], which implements the Denner–Dittmaier reduction techniques [295, 296] and the scalar integrals of Ref.”