Abstract
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.
Highlights
Feynman integrals in dimensional regularization admit parametric integral representations which expose their nature as Aomoto-Gel’fand integrals, thereby enabling a novel form of investigation of their algebraic structure by means of intersection theory of twisted de Rhamhomology for general hypergeometric functions [1,2,3]
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
Univariate intersection numbers, as shown in the original studies [1, 2], were sufficient to validate a novel method based on intersection theory for deriving integral relations, which was used for the direct derivation of contiguity relations for Lauricella FD functions, as well as for Feynman integrals on maximal cuts, i.e. with on-shell internal lines, that admit a one-fold integral representations
Summary
Feynman integrals in dimensional regularization admit parametric integral representations which expose their nature as Aomoto-Gel’fand integrals, thereby enabling a novel form of investigation of their algebraic structure by means of intersection theory of twisted de Rham (co)homology for general hypergeometric functions [1,2,3]. A recursive algorithm for computing multivariate intersection numbers was proposed in [14] and later refined and applied to a few paradigmatic cases of Feynman integral decomposition [3]. This recursive algorithm was developed in order to compute intersection numbers for twisted cohomologies associated to n-forms, which in the general case may contain poles that are not necessarily simple. In the case of logarithmic (dlog) differential forms, owing to the presence of simple poles only, the computation of the intersection numbers is known to be simpler [6, 12]
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