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Top-Down Decomposition: A Cut-Based Approach to Integral Reductions

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Anti-Differentiation and the Calculation of Feynman Amplitudes

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Abstract

In this contribution we will discuss a new approach to the derivation of linear relations between Feynman integrals. This new approach uses the mathematical object known as the intersection number to define what amounts to an inner product between Feynman integrals, which can be used to project directly unto the basis of master integrals. In particular we will discuss one perspective to this intersection-based method, which we name the top-down approach. This approach can be seen as an integral level version of the algorithms based on integrand reduction and generalized unitarity cuts, that were revolutionizing NLO scattering computations in the early 2000s.

Based on [1].

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Notes

  1. 1.

    Equation (1) is brushing a lot under the rug. Besides the spurious terms discussed below, it does not include the rational term \(\mathcal {R}\) containing the contributions not captured by four-dimensional cuts, nor does it include the extraction of pentagon-terms.

  2. 2.

    We note that this step has nothing to do with intersection theory, and was done for instance in ref. [45].

  3. 3.

    We note that had we replaced the subtraction of the box-coefficient in Eq. (38) with a free subtraction term (c 1 → κ 1), this κ-fitting step would give a solution only if κ 1 = c 1, so treating κ 1 on equal footing with the other κs, is another approach to fixing the box-coefficient.

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Acknowledgements

I would like to thank my collaborators on the work presented here, Federico Gasparotto, Stefano Laporta, Manoj K. Mandal, Pierpaolo Mastrolia, Luca Mattiazzi, and Sebastian Mizera. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 ‘INTERACTIONS’. The work of HF has been partially supported by a Carlsberg Foundation Reintegration Fellowship.

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Frellesvig, H. (2021). Top-Down Decomposition: A Cut-Based Approach to Integral Reductions. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_8

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