Journal of High Energy Physics

, 2015:136 | Cite as

Global structure of curves from generalized unitarity cut of three-loop diagrams

  • Jonathan D. Hauenstein
  • Rijun HuangEmail author
  • Dhagash Mehta
  • Yang Zhang
Open Access
Regular Article - Theoretical Physics


This paper studies the global structure of algebraic curves defined by generalized unitarity cut of four-dimensional three-loop diagrams with eleven propagators. The global structure is a topological invariant that is characterized by the geometric genus of the algebraic curve. We use the Riemann-Hurwitz formula to compute the geometric genus of algebraic curves with the help of techniques involving convex hull polytopes and numerical algebraic geometry. Some interesting properties of genus for arbitrary loop orders are also explored where computing the genus serves as an initial step for integral or integrand reduction of three-loop amplitudes via an algebraic geometric approach.


Scattering Amplitudes Differential and Algebraic Geometry 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2015

Authors and Affiliations

  • Jonathan D. Hauenstein
    • 1
  • Rijun Huang
    • 2
    Email author
  • Dhagash Mehta
    • 1
  • Yang Zhang
    • 3
  1. 1.Department of Applied and Computational Mathematics and StatisticsUniversity of Notre DameNotre DameU.S.A.
  2. 2.Institut de Physique Théorique, CEA-SaclayGif-sur-Yvette CedexFrance
  3. 3.Niels Bohr International Academy and Discovery Center, The Niels Bohr InstituteUniversity of CopenhagenCopenhagenDenmark

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