Algebraic algorithms for multiloop calculations The first 15 years. What's next?

1996/09/30 by F.V. Tkachov, Fyodor V. Tkachov · 3 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic number #Algorithm #Applied mathematics #Black Holes and Theoretical Physics #Combinatorics #Computer science #Cosmology and Gravitation Theories #Feynman diagram #Feynman integral #Mathematical analysis #Mathematical physics #Mathematics #Numerical integration #Particle physics theoretical and experimental studies #Pure mathematics #Topology (electrical circuits) #hep-ph #hep-th

paper · pdf · doi:10.1016/s0168-9002(97)00110-1

published as Nucl.Instrum.Meth.A389:309-313,1997 · 5 pages, PS; 7-11-98: maintenance (PS, misprints, etc.)

openalex publication_date 1997/04/01 · arxiv created 1998/11/07 · arxiv updated 2011/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The ideas behind the concept of algebraic ("integration-by-parts") algorithms for multiloop calculations are reviewed. For any topology and mass pattern, a finite iterative algebraic procedure is proved to exist which transforms the corresponding Feynman-parametrized integrands into a form that is optimal for numerical integration, with all the poles in D-4 explicitly extracted.

Citations

• EUCLIDEAN ASYMPTOTIC EXPANSIONS OF GREEN FUNCTIONS OF QUANTUM FIELDS (I) EXPANSIONS OF PRODUCT OF SINGULAR FUNCTIONS
• Integration by parts: The algorithm to calculate β-functions in 4 loops
• Connection between Feynman integrals having different values of the space-time dimension
• Explicit solutions of the n--loop vacuum integral recurrence relations

Cited by

• Two-loop tensor integrals in quantum field theory
• Computation of form factors in massless QCD with finite master integrals
• Bernstein–Sato polynomial [wikipedia]