Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

Abstract

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2{F}_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to $n$-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

115 pages, 29 figures; references added; additional examples added; matches published version

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• Scattering Equations: From Projective Spaces to Tropical Grassmannians
• Monodromy relations from twisted homology
• Positive Geometries for all Scalar Theories from Twisted Intersection Theory
• A first look at the function space for planar two-loop six-particle Feynman integrals
• Gauge and Gravity Amplitudes on the Celestial Sphere
• On Polytopes and Generalizations of the KLT Relations
• The Stratification of Rigidity
• Co-homology of Differential Forms and Feynman diagrams
• The diagrammatic coaction beyond one loop