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

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.