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Feynman integrals and intersection theory. (English) Zbl 1411.81093

Summary: 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.

MSC:

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81U20 \(S\)-matrix theory, etc. in quantum theory

Software:

Reduze; LiteRed; Kira; Azurite
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References:

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