Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a procedure to derive $ε$-factorised differential equations beyond polylogarithms

Lennard Görges, Christoph Nega, Lorenzo Tancredi, Fabian J. Wagner

TL;DR

This work develops a practical procedure to derive $\epsilon$-factorised differential equations for multi-scale, multi-loop Feynman integrals that evaluate to non-polylogarithmic functions, including elliptic and Calabi–Yau geometries. The method builds canonical $\epsilon$-forms via a sector-by-sector rotation of the master-integral basis, using the semi-simple part of the period matrix and systematic cleanup to expose the iterated-integral structure. Through detailed applications to the equal-mass sunrise, single- and multi-parameter elliptic triangles, and multi-curve cases like the ice cone and banana graphs, the authors show how to introduce new kernels $G_i$ when necessary to capture residues and higher-genus effects. This framework advances the ability to obtain transparent, analytically tractable representations of complex Feynman integrals and points toward extensions to even richer geometries.

Abstract

In this manuscript, we elaborate on a procedure to derive $ε$-factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out $ε$-factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain $ε$-factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.

On a procedure to derive $ε$-factorised differential equations beyond polylogarithms

TL;DR

This work develops a practical procedure to derive -factorised differential equations for multi-scale, multi-loop Feynman integrals that evaluate to non-polylogarithmic functions, including elliptic and Calabi–Yau geometries. The method builds canonical -forms via a sector-by-sector rotation of the master-integral basis, using the semi-simple part of the period matrix and systematic cleanup to expose the iterated-integral structure. Through detailed applications to the equal-mass sunrise, single- and multi-parameter elliptic triangles, and multi-curve cases like the ice cone and banana graphs, the authors show how to introduce new kernels when necessary to capture residues and higher-genus effects. This framework advances the ability to obtain transparent, analytically tractable representations of complex Feynman integrals and points toward extensions to even richer geometries.

Abstract

In this manuscript, we elaborate on a procedure to derive -factorised differential equations for multi-scale, multi-loop classes of Feynman integrals that evaluate to special functions beyond multiple polylogarithms. We demonstrate the applicability of our approach to diverse classes of problems, by working out -factorised differential equations for single- and multi-scale problems of increasing complexity. To start we are reconsidering the well-studied equal-mass two-loop sunrise case, and move then to study other elliptic two-, three- and four-point problems depending on multiple different scales. Finally, we showcase how the same approach allows us to obtain -factorised differential equations also for Feynman integrals that involve geometries beyond a single elliptic curve.
Paper Structure (17 sections, 126 equations, 7 figures)

This paper contains 17 sections, 126 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Description of the procedure
  3. Application to the two-loop sunrise family
  4. The sunrise with two equal non-vanishing masses and a vanishing mass
  5. The sunrise with three equal non-vanishing masses
  6. The sunrise with three non-vanishing masses of which two are equal
  7. Applications to Feynman integrals with a single elliptic curve
  8. A single-scale elliptic three-point function
  9. A second single-scale elliptic three-point function
  10. Two-parameter triangle
  11. Three-parameter double box
  12. Cases beyond a single elliptic curve
  13. The three-loop ice cone
  14. The three-loop banana
  15. Conclusions and outlook
  16. ...and 2 more sections

Figures (7)

  • Figure 1: The two-loop sunrise graph.
  • Figure 2: The graph of non-planar triangle number 1.
  • Figure 3: The graph of non-planar triangle number 2.
  • Figure 4: The double box graph.
  • Figure 5: Feynman graph of the elliptic sector inside the double box.
  • ...and 2 more figures