2012/05/31 by Yang Zhang · 116 citations
Computer Science · Mathematics · Medicine · Physics and Astronomy · #Algebraic geometry #Algebraic number #Algorithm #Applied mathematics #Basis (linear algebra) #Cancer Treatment and Pharmacology #Combinatorics #Field (mathematics) #Geometry #Gröbner basis #Ideal (ethics) #Loop (graph theory) #Mathematical analysis #Mathematics #Numerical methods for differential equations #Particle physics #Physics #Polynomial #Polynomial and algebraic computation #Pure mathematics #Reduction (mathematics) #Unitarity #hep-ph #hep-th
paper · pdf · doi:10.1007/jhep09(2012)042
published in Journal of High Energy Physics 2012(9) (Springer Nature) · published version: typos corrected; more examples added
openalex publication_date 2012/09/01 · arxiv created 2012/09/20 · arxiv updated 2015/03/19 · openalex created_date 2020/11/23 · openalex updated_date 2026/08/05
We present an algorithm for the integrand-level reduction of multi-loop amplitudes of renormalizable field theories, based on computational algebraic geometry. This algorithm uses (1) the Gröbner basis method to determine the basis for integrand-level reduction, (2) the primary decomposition of an ideal to classify all inequivalent solutions of unitarity cuts. The resulting basis and cut solutions can be used to reconstruct the integrand from unitarity cuts, via polynomial fitting techniques. The basis determination part of the algorithm has been implemented in the Mathematica package, BasisDet. The primary decomposition part can be readily carried out by algebraic geometry softwares, with the output of the package BasisDet. The algorithm works in both D=4 and D=4-2ε dimensions, and we present some two and three-loop examples of applications of this algorithm.