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Aspects of canonical differential equations for Calabi-Yau geometries and beyond

Claude Duhr, Sara Maggio, Christoph Nega, Benjamin Sauer, Lorenzo Tancredi, Fabian J. Wagner

TL;DR

This work extends the canonical differential equations program beyond polylogarithms to Calabi–Yau geometries by incorporating the mixed Hodge structure of maximal-cut varieties. It develops a robust roadmap: identify the geometry at maximal cuts, align a distinguished master basis with the MHS, split the period matrix into semi-simple and unipotent parts, realign transcendental weights, and perform ε-factorisation to obtain canonical forms. The authors demonstrate the method on one-parameter CY families, proving equivalence with other approaches and exploring properties such as integrality and weight structure, then apply it to challenging gravitational-wave related integrals and beyond-CY examples like ice-cone graphs and CY3 sectors. The results show that canonical, ε-form differential equations can be constructed for a broad class of multi-loop integrals, offering powerful, geometry-driven tools for both analytic and numerical evaluation in quantum field theory and gravitational scattering contexts.

Abstract

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

Aspects of canonical differential equations for Calabi-Yau geometries and beyond

TL;DR

This work extends the canonical differential equations program beyond polylogarithms to Calabi–Yau geometries by incorporating the mixed Hodge structure of maximal-cut varieties. It develops a robust roadmap: identify the geometry at maximal cuts, align a distinguished master basis with the MHS, split the period matrix into semi-simple and unipotent parts, realign transcendental weights, and perform ε-factorisation to obtain canonical forms. The authors demonstrate the method on one-parameter CY families, proving equivalence with other approaches and exploring properties such as integrality and weight structure, then apply it to challenging gravitational-wave related integrals and beyond-CY examples like ice-cone graphs and CY3 sectors. The results show that canonical, ε-form differential equations can be constructed for a broad class of multi-loop integrals, offering powerful, geometry-driven tools for both analytic and numerical evaluation in quantum field theory and gravitational scattering contexts.

Abstract

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

Paper Structure

This paper contains 50 sections, 260 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Feynman integrals and their differential equations
  3. A motivational example
  4. The polylogarithmic case: $M^2 = 0$
  5. The elliptic case: $M^2 \neq 0$
  6. Integrand analysis on the maximal cut
  7. Canonical differential equations in the elliptic case
  8. The splitting of the Wronskian and the transcendental weight.
  9. Realignment of the transcendental weight and the $\epsilon$-expansion.
  10. $\epsilon$-factorisation of the full $3\times 3$ system.
  11. Transformation behaviour under analytic continuation.
  12. A roadmap to more complicated geometries
  13. Towards geometries beyond elliptic curves
  14. Identifying the geometries associated with the maximal cuts.
  15. Identifying a distinguished basis compatible with the mixed Hodge structure.
  16. ...and 35 more sections

Figures (5)

  • Figure 1: The two-loop sunrise graph with two different masses.
  • Figure 2: The contours for the two-mass sunrise integral.
  • Figure 3: The four-loop ice cone graph
  • Figure 4: A K3 sector at 5PM-2SF. The dotted lines represent the worldlines and translate on the upper line to $\delta(k_i \cdot v_1)$ and on the bottom line to $\delta(k_i \cdot v_2)$. The wiggly lines represent gravitons and translate to the propagators $D_9-D_{22}$.
  • Figure 5: The graphs related to a family of CY threefolds.