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On genera of curves from high-loop generalized unitarity cuts

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Abstract

Generalized unitarity cut of a Feynman diagram generates an algebraic system of polynomial equations. At high-loop levels, these equations may define a complex curve or a (hyper-)surface with complicated topology. We study the curve cases, i.e., a 4-dimensional L-loop diagram with (4L−1) cuts. The topology of a complex curve is classified by its genus. Hence in this paper, we use computational algebraic geometry to calculate the genera of curves from two and three-loop unitarity cuts. The global structure of degenerate on-shell equations under some specific kinematic configurations is also sketched. The genus information can also be used to judge if a unitary cut solution could be rationally parameterized.

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Article Open access 10 May 2021

Hjalte Frellesvig, Cristian Vergu, … Matt von Hippel

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Authors and Affiliations

  1. Niels Bohr International Academy and Discovery Center, The Niels Bohr Institute, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark

    Rijun Huang & Yang Zhang

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  1. Rijun Huang
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  2. Yang Zhang
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Correspondence to Yang Zhang.

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ArXiv ePrint: 1302.1023

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Huang, R., Zhang, Y. On genera of curves from high-loop generalized unitarity cuts. J. High Energ. Phys. 2013, 80 (2013). https://doi.org/10.1007/JHEP04(2013)080

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  • DOI: https://doi.org/10.1007/JHEP04(2013)080

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