Recursive numerical calculus of one-loop tensor integrals

2004/04/30 by F. del Águila, F. del Aguila, R. Pittau · 8 citations
Mathematics · Physics and Astronomy · #Algorithm #Applied mathematics #Black Holes and Theoretical Physics #Combinatorics #Cosmology and Gravitation Theories #Geometry #Integral equation #Inverse #Mathematical analysis #Mathematics #Numerical integration #Order of integration (calculus) #Particle physics theoretical and experimental studies #Pure mathematics #Rank (graph theory) #Recursion (computer science) #Scalar (mathematics) #Slater integrals #Tensor (intrinsic definition) #Volume integral #hep-ph

paper · pdf · doi:10.1088/1126-6708/2004/07/017

published as JHEP 0407:017,2004 · Typo corrected in formula 79. 22 pages, Latex, 1 figure, uses axodraw.sty

openalex publication_date 2004/07/13 · arxiv created 2005/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

A numerical approach to compute tensor integrals in one-loop calculations is presented. The algorithm is based on a recursion relation which allows to express high rank tensor integrals as a function of lower rank ones. At each level of iteration only inverse square roots of Gram determinants appear. For the phase-space regions where Gram determinants are so small that numerical problems are expected, we give general prescriptions on how to construct reliable approximations to the exact result without performing Taylor expansions. Working in 4+epsilon dimensions does not require an analytic separation of ultraviolet and infrared/collinear divergences, and, apart from trivial integrals that we compute explicitly, no additional ones besides the standard set of scalar one-loop integrals are needed.

Citations

• Reduction method for dimensionally regulatedone-loop N-point Feynman integrals
• Reduction of one-loop tensor 5-point integrals
• Connection between Feynman integrals having different values of the space-time dimension
• One-loop tensor integrals in dimensional regularisation
• A Calculational Formalism for One-Loop Integrals
• ALPGEN, a generator for hard multiparton processes in hadronic collisions

Cited by

• Automated one-loop calculations: a proof of concept
• Solving for tadpole coefficients in one-loop amplitudes
• Scattering amplitudes from unitarity-based reduction algorithm at the integrand-level
• Rational terms of UV origin at two loops
• One-Loop Calculations with BlackHat
• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• The rational parts of one-loop QCD amplitudes I: The general formalism
• Tensorial reconstruction at the integrand level