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Canonical differential equations beyond polylogs

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

TL;DR

The paper addresses constructing canonical differential equations for Feynman integrals whose geometries extend beyond the Riemann sphere, using the two-loop sunrise as a paradigmatic example. It analyzes both polylogarithmic ($M^2=0$) and elliptic ($M^2\neq0$) cases, showing how leading singularities, the Baikov representation, and period-matrix methods can identify appropriate master integrals and basis choices. The main contribution is an algorithm that builds a canonical basis by separating the period matrix into semisimple and unipotent parts, realigning transcendental weight, and performing basis rotations to achieve an $\varepsilon$-factorised form $\mathrm{d}\boldsymbol{J}=\varepsilon\boldsymbol{A}(\boldsymbol{s})\,\boldsymbol{J}$, valid beyond MPLs. The framework generalises to higher-genus curves and Calabi–Yau geometries, with potential applications to precision collider physics and gravitational-wave computations.

Abstract

Feynman integrals whose associated geometries extend beyond the Riemann sphere, such as elliptic curves and Calabi-Yau varieties, are increasingly relevant in modern precision calculations. They arise not only in collider cross-section calculations, but also in the post-Minkowskian expansion of gravitational-wave scattering. A powerful approach to compute integrals of this type is via differential equations, particularly when cast in a canonical form, which simplifies their $\varepsilon$-expansion and makes analytic properties manifest. In these proceedings, we will present a method to systematically construct canonical differential equations even for integrals that evaluate beyond multiple polylogarithms. The discussion is kept as light as possible, focusing on the two-loop sunrise integral, deferring the technical details to the original publications.

Canonical differential equations beyond polylogs

TL;DR

The paper addresses constructing canonical differential equations for Feynman integrals whose geometries extend beyond the Riemann sphere, using the two-loop sunrise as a paradigmatic example. It analyzes both polylogarithmic () and elliptic () cases, showing how leading singularities, the Baikov representation, and period-matrix methods can identify appropriate master integrals and basis choices. The main contribution is an algorithm that builds a canonical basis by separating the period matrix into semisimple and unipotent parts, realigning transcendental weight, and performing basis rotations to achieve an -factorised form , valid beyond MPLs. The framework generalises to higher-genus curves and Calabi–Yau geometries, with potential applications to precision collider physics and gravitational-wave computations.

Abstract

Feynman integrals whose associated geometries extend beyond the Riemann sphere, such as elliptic curves and Calabi-Yau varieties, are increasingly relevant in modern precision calculations. They arise not only in collider cross-section calculations, but also in the post-Minkowskian expansion of gravitational-wave scattering. A powerful approach to compute integrals of this type is via differential equations, particularly when cast in a canonical form, which simplifies their -expansion and makes analytic properties manifest. In these proceedings, we will present a method to systematically construct canonical differential equations even for integrals that evaluate beyond multiple polylogarithms. The discussion is kept as light as possible, focusing on the two-loop sunrise integral, deferring the technical details to the original publications.
Paper Structure (10 sections, 29 equations, 2 figures)

This paper contains 10 sections, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Canonical differential equations and geometries
  3. Integrals that evaluate to multiple polylogarithms.
  4. Integrals that evaluate to special functions beyond MPLs.
  5. Two-loop sunrise integral
  6. The polylogarithmic case: $M^2 = 0$.
  7. The elliptic case: $M^2 \neq 0$.
  8. The algorithm
  9. Beyond elliptic curves.
  10. Conclusions

Figures (2)

  • Figure 1: The two-loop sunrise graph. The thin and thick lines represent propagators with mass $M$ and $m$, respectively.
  • Figure 2: The contours for the two-mass sunrise integral.