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Feynman integrals and intersection theory

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  • Published: 21 February 2019
  • volume 2019, Article number: 139 (2019)
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Feynman integrals and intersection theory
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  • Pierpaolo Mastrolia  ORCID: orcid.org/0000-0001-9711-77981,2 &
  • Sebastian Mizera3,4 

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A preprint version of the article is available at arXiv.

Abstract

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.

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Authors and Affiliations

  1. Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131, Padova, Italy

    Pierpaolo Mastrolia

  2. INFN, Sezione di Padova, Via Marzolo 8, 35131, Padova, Italy

    Pierpaolo Mastrolia

  3. Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada

    Sebastian Mizera

  4. Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Sebastian Mizera

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  1. Pierpaolo Mastrolia
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Correspondence to Pierpaolo Mastrolia.

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ArXiv ePrint: 1810.03818

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Mastrolia, P., Mizera, S. Feynman integrals and intersection theory. J. High Energ. Phys. 2019, 139 (2019). https://doi.org/10.1007/JHEP02(2019)139

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  • Received: 31 October 2018

  • Revised: 22 January 2019

  • Accepted: 11 February 2019

  • Published: 21 February 2019

  • DOI: https://doi.org/10.1007/JHEP02(2019)139

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Keywords

  • Scattering Amplitudes
  • Differential and Algebraic Geometry
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