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Analytic Structure of all Loop Banana Amplitudes

Kilian Bönisch, Fabian Fischbach, Albrecht Klemm, Christoph Nega, Reza Safari

TL;DR

The paper develops a GKZ-system/mirror-symmetry framework to fully characterize the analytic structure of banana Feynman integrals across arbitrary loop orders. It derives inhomogeneous Picard–Fuchs equations for equal masses and a comprehensive differential-system approach for non-equal masses, leveraging Calabi–Yau geometries and the $\\widehat{\\Gamma}$-class to explain leading logarithms and zeta-value patterns. The maximal-cut analysis at high energy reveals connections to modular forms and attractor points, while Batyrev coordinates and Frobenius bases enable precise, rapid numerical evaluation and master-integral organization. The work suggests deep motivic and automorphic structures underpinning multi-loop Feynman amplitudes and lays groundwork for extending these techniques to broader families of integrals.

Abstract

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $\widehat Γ$-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the $\widehat Γ$-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius $κ$-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent $p$-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

Analytic Structure of all Loop Banana Amplitudes

TL;DR

The paper develops a GKZ-system/mirror-symmetry framework to fully characterize the analytic structure of banana Feynman integrals across arbitrary loop orders. It derives inhomogeneous Picard–Fuchs equations for equal masses and a comprehensive differential-system approach for non-equal masses, leveraging Calabi–Yau geometries and the -class to explain leading logarithms and zeta-value patterns. The maximal-cut analysis at high energy reveals connections to modular forms and attractor points, while Batyrev coordinates and Frobenius bases enable precise, rapid numerical evaluation and master-integral organization. The work suggests deep motivic and automorphic structures underpinning multi-loop Feynman amplitudes and lays groundwork for extending these techniques to broader families of integrals.

Abstract

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel -class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the -class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius -constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent -adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

Paper Structure

This paper contains 26 sections, 101 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Banana Feynman integrals
  3. The $l$-loop banana amplitude and its geometrical realization
  4. Bessel function representation of $l$-loop banana integrals
  5. The maximal cut integral for large momentum
  6. The $l$-loop equal mass banana Feynman integral
  7. Inhomogeneous differential equation for the $l$-loop equal mass banana Feynman integral
  8. Analytic properties of the $l$-loop equal mass Banana graph Feynman integral
  9. Frobenius basis at the MUM point
  10. Banana Feynman integral in terms of the MUM-Frobenius basis
  11. The $\widehat{\Gamma}$-class and zeta values at the point of maximal unipotent monodromy
  12. Monodromy
  13. Special Points
  14. The $l$-loop non-equal mass banana Feynman integrals
  15. Batyrev coordinates and the maximal cut integral
  16. ...and 11 more sections

Figures (3)

  • Figure 1: The $l$-loop banana diagram with external momentum $K$ and internal masses $M_i$.
  • Figure 2: A three loop contribution to Higgs production via gluon fusion with a bottom and a top quark running in the loops (left panel). The scalar kite Feynman graph with two massless and three propapagators of equal mass (right panel).
  • Figure 3: Singularities of the Fuchsian operator $\mathcal{L}_l$. The radius of convergence of the Bessel integral representation \ref{['besselform']} of $\mathcal{F}_{\sigma_l}$ is denoted by $r_{\mathrm{BI}}$.