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Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms.

Jacob L Bourjaily1, Yang-Hui He2,3,4, Andrew J McLeod1

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

Physical Review Letters
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Summary

We introduce Feynman integrals crucial for quantum field theory calculations. These integrals, linked to Calabi-Yau manifolds, are essential for representing amplitudes in theories like maximally supersymmetric Yang-Mills.

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Area of Science:

  • Theoretical Physics
  • High Energy Physics
  • String Theory

Background:

  • Feynman integrals are fundamental in quantum field theory for calculating scattering amplitudes.
  • Elliptically fibered varieties and Calabi-Yau manifolds are advanced geometric objects with connections to theoretical physics.

Purpose of the Study:

  • To describe a new family of finite, four-dimensional, L-loop Feynman integrals.
  • To connect these integrals to specific geometric structures, namely Calabi-Yau manifolds.

Main Methods:

  • Integration of weight-(L+1) hyperlogarithms over (L-1)-dimensional elliptically fibered varieties.
  • Explicit identification of the K3 manifold at three loops.
  • Providing evidence for a Calabi-Yau threefold at four loops.

Main Results:

  • A family of L-loop Feynman integrals is characterized.
  • The geometric interpretation involves Calabi-Yau manifolds, specifically K3 at three loops and a Calabi-Yau threefold at four loops.
  • These integrals are demonstrated to be necessary for amplitude representation in various theories.

Conclusions:

  • The study establishes a novel class of Feynman integrals with deep connections to algebraic geometry.
  • These findings are relevant for understanding amplitudes in diverse quantum field theories, including maximally supersymmetric Yang-Mills theory.