On the computation of intersection numbers for twisted cocycles

Abstract

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example square roots), although the final result may be expressed without algebraic extensions. In this article I present an improvement of this algorithm, which avoids algebraic extensions.

36 pages, v2: version to be published

References in corpus (10)

• Reducing differential equations for multiloop master integrals
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Decomposition of Feynman Integrals by Multivariate Intersection Numbers
• Duals of Feynman Integrals, I: Differential Equations
• Maximal Cuts in Arbitrary Dimension
• Constructing Canonical Feynman Integrals with Intersection Theory
• On the solutions of the scattering equations
• Two-loop master integrals for the mixed QCD-electroweak corrections for $H\to b\bar{b}$ through a $Ht\bar{t}$-coupling
• Intersection numbers and twisted period relations for the generalized hypergeometric function ${}_{m+1}{F}_m$

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• Les Houches 2021: Physics at TeV Colliders: Report on the Standard Model Precision Wishlist
• Cosmology meets cohomology
• Mathematical properties of nested residues and their application to multi-loop scattering amplitudes
• Status of Intersection Theory and Feynman Integrals
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• Reduction to master integrals via intersection numbers and polynomial expansions
• Intersection Numbers from Higher-order Partial Differential Equations
• Correlation functions on the lattice and twisted cocycles
• Non-perturbative computation of lattice correlation functions by differential equations
• Factorization of denominators in integration-by-parts reductions
• The recursive structure of Baikov representations I: generics and application to symbology
• Three-loop banana integrals with four unequal masses
• Real time lattice correlation functions from differential equations
• Mixed QCD-EW corrections for Higgs leptonic decay via $H{W}^{+}{W}^{-}$ vertex
• Module Intersection and Uniform Formula for Iterative Reduction of One-loop Integrals
• Intersection Numbers from Companion Tensor Algebra
• Twisted Riemann bilinear relations and Feynman integrals
• The recursive structure of Baikov representations II: the top-down reduction with intersection theory
• Feynman integral reduction using Gröbner bases
• General Relativity from Intersection Theory
• Canonical differential equations and intersection matrices
• CALICO: Computing Annihilators from Linear Identities Constraining (differential) Operators
• IBP reduction via Gröbner bases in a rational double-shift algebra
• A stroll through the loop-tree duality
• Co-homology of Differential Forms and Feynman diagrams