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Integrability for Feynman Integrals

Florian Loebbert

Abstract

We give a brief overview of the Yangian symmetry of Feynman integrals. After a short introduction to the Yangian and integrability, we motivate the emergence of integrable structures for Feynman integrals via the fishnet limit of AdS/CFT. We discuss the resulting Yangian differential equations for massless fishnets in four dimensions as well as generalizations to massive propagators and generic dimensions. We also comment on the relation to momentum space conformal symmetry and on examples in dimensional regularization. Finally we sketch the recent application to fishnet integrals in two spacetime dimensions and the curious identification of Yangian invariants with period integrals of Calabi-Yau geometries.

Integrability for Feynman Integrals

Abstract

We give a brief overview of the Yangian symmetry of Feynman integrals. After a short introduction to the Yangian and integrability, we motivate the emergence of integrable structures for Feynman integrals via the fishnet limit of AdS/CFT. We discuss the resulting Yangian differential equations for massless fishnets in four dimensions as well as generalizations to massive propagators and generic dimensions. We also comment on the relation to momentum space conformal symmetry and on examples in dimensional regularization. Finally we sketch the recent application to fishnet integrals in two spacetime dimensions and the curious identification of Yangian invariants with period integrals of Calabi-Yau geometries.
Paper Structure (14 sections, 34 equations, 1 figure)

This paper contains 14 sections, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integrability and the Yangian
  3. AdS/CFT and Fishnet Theories
  4. Integrability for Feynman Integrals
  5. Example: 4D Cross (or Box) Integral.
  6. Generalizations: Dimensions, Masses, Momentum Space,…
  7. Fishnet Structures and Dimensions.
  8. Yangian Symmetry for the Masses.
  9. Momentum Space Conformal Symmetry.
  10. Dimensional Regularization.
  11. Soft and Collinear Anomalies.
  12. Yangian Symmetry and Calabi--Yau Geometry
  13. Example: 2D Cross (or Box) Integral.
  14. Conclusion

Figures (1)

  • Figure 1: Factorized scattering in two dimensions. For consistency the quantum Yang--Baxter equation (right equality) has to hold: $\mathrm{S}_{12}S_{13}S_{23}=S_{23}S_{13}S_{12}$. A similar equation holds for the generating function $T(u)$ of the Yangian generators together with the quantum R-matrix: $RTT=TTR.$