Feynman Integrals and Intersection Theory

Abstract

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop.

13 pages, published version

References in corpus (12)

• Algorithm FIRE -- Feynman Integral REduction
• Scattering amplitudes over finite fields and multivariate functional reconstruction
• Reducing differential equations for multiloop master integrals
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Scattering Amplitudes from Intersection Theory
• Combinatorics and Topology of Kawai-Lewellen-Tye Relations
• A first look at two-loop five-gluon scattering in QCD
• Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections
• Feynman integral relations from parametric annihilators
• Maximal Cuts in Arbitrary Dimension

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