Decomposition of Feynman integrals by multivariate intersection numbers
02 Mar 2021-Journal of High Energy Physics (Springer Science and Business Media LLC)-Vol. 2021, Iss: 3, pp 1-55
TL;DR: In this paper, the authors 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.
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.
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...olve huge linear systems as they are generated by IBP relations. The original formulation of this idea was based on the Baikov representation in combination with maximal cuts [527, 519, 520]. In Ref. [523], multivariate 42 intersection numbers are discussed, and several strategies for integral reduction are presented, applicable to generic parametric representations of Feynman integrals. This method ha...
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References
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TL;DR: In this paper, the authors presented a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory, and proved that their $N=4$ ansatz is correct using general properties of the relevant one-loops $n$-point integrals.
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1,043 citations
TL;DR: It is argued that a good choice of basis for (multi)loop integrals can lead to significant simplifications of the differential equations, and criteria for finding an optimal basis are proposed.
Abstract: Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi)loop integrals can lead to significant simplifications of the differential equations, and propose criteria for finding an optimal basis. This builds on experience obtained in supersymmetric field theories that can be applied successfully to generic quantum field theory integrals. It involves studying leading singularities and explicit integral representations. When the differential equations are cast into canonical form, their solution becomes elementary. The class of functions involved is easily identified, and the solution can be written down to any desired order in ϵ within dimensional regularization. Results obtained in this way are particularly simple and compact. In this Letter, we outline the general ideas of the method and apply them to a two-loop example.
979 citations