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Decomposition of Feynman integrals by multivariate intersection numbers. (English) Zbl 1461.81044
Summary: We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

##### MSC:
 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 81V05 Strong interaction, including quantum chromodynamics 81U05 $$2$$-body potential quantum scattering theory
##### Software:
CutTools; Axodraw; Kira; FIRE5; JaxoDraw; FIRE; Reduze; LiteRed
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