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The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods

Albrecht Klemm, Christoph Nega, Reza Safari

TL;DR

This work reframes l-loop banana Feynman integrals with general mass dependence as GKZ period problems on Barth-Nieto Calabi–Yau $(l-1)$-folds, exploiting maximal unipotent monodromy to organize holomorphic and logarithmic periods. By extending the GKZ system to relative periods, the authors derive a complete homogeneous Picard–Fuchs ideal for the maximal-cut integrals and then construct inhomogeneous differential equations to capture boundary contributions, yielding the full mass-dependent amplitudes up to three loops. They explicitly implement the method for the bubble, sunset, and three-loop banana graphs, obtaining new analytic mass-dependent expressions and confirming consistency with known equal-mass results. The approach connects Feynman integrals to mirror symmetry, periods, and, in higher cases, automorphic structures, suggesting a path toward higher-loop computations and deeper mathematical structure in perturbative quantum field theory.

Abstract

We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau $(l-1)$-folds in order to solve the $l$-loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.

The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods

TL;DR

This work reframes l-loop banana Feynman integrals with general mass dependence as GKZ period problems on Barth-Nieto Calabi–Yau -folds, exploiting maximal unipotent monodromy to organize holomorphic and logarithmic periods. By extending the GKZ system to relative periods, the authors derive a complete homogeneous Picard–Fuchs ideal for the maximal-cut integrals and then construct inhomogeneous differential equations to capture boundary contributions, yielding the full mass-dependent amplitudes up to three loops. They explicitly implement the method for the bubble, sunset, and three-loop banana graphs, obtaining new analytic mass-dependent expressions and confirming consistency with known equal-mass results. The approach connects Feynman integrals to mirror symmetry, periods, and, in higher cases, automorphic structures, suggesting a path toward higher-loop computations and deeper mathematical structure in perturbative quantum field theory.

Abstract

We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau -folds in order to solve the -loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.

Paper Structure

This paper contains 21 sections, 95 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. $l$-loop Banana Diagram in the Toric Approach
  3. $l$-loop banana diagram
  4. Geometry associated to $l$-loop banana diagram
  5. Calabi-Yau hypersurfaces in toric ambient spaces
  6. Period integrals on $M$ and maximal cut amplitude
  7. GKZ systems and Picard Fuchs differential ideal
  8. Geometrical and physical periods
  9. The complete banana diagram and inhomogeneous differential equations
  10. Examples
  11. Example 1: The Bubble Graph
  12. Example 2: The Sunset Graph
  13. The sunset maximal cut integral
  14. Extension to inhomogeneous differential operators
  15. Comparison with the equal mass case and other known results
  16. ...and 6 more sections

Figures (4)

  • Figure 2.1: The l-loop banana diagram
  • Figure 3.1: The toric diagram for the bubble graph
  • Figure 3.2: Toric diagram for the sunset graph
  • Figure 3.3: Toric diagram for the three-loop banana graph