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An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

Christoph Dlapa, Johannes M. Henn, Fabian J. Wagner

TL;DR

This work extends the canonical differential-equation framework to elliptic Feynman integrals by adapting the algorithmic approach that yields ε-forms. It introduces an Ψ1-based ansatz for elliptic kernels, validates the method on multiple non-trivial examples (kite, Higgs-phase-space, gravity, banana), and shows that some cases require augmenting the ansatz with additional elliptic data (e.g., F2) to achieve a true ε-form. The authors demonstrate the practical viability of the approach and release an updated INITIAL package capable of handling enhanced ansätze, enabling systematic computation of elliptic Feynman integrals. These developments advance automated, analytic computation of higher-loop integrals beyond multiple polylogarithms.

Abstract

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.

An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

TL;DR

This work extends the canonical differential-equation framework to elliptic Feynman integrals by adapting the algorithmic approach that yields ε-forms. It introduces an Ψ1-based ansatz for elliptic kernels, validates the method on multiple non-trivial examples (kite, Higgs-phase-space, gravity, banana), and shows that some cases require augmenting the ansatz with additional elliptic data (e.g., F2) to achieve a true ε-form. The authors demonstrate the practical viability of the approach and release an updated INITIAL package capable of handling enhanced ansätze, enabling systematic computation of elliptic Feynman integrals. These developments advance automated, analytic computation of higher-loop integrals beyond multiple polylogarithms.

Abstract

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.
Paper Structure (16 sections, 54 equations, 4 figures)

This paper contains 16 sections, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Description of the method
  3. Review of the algorithm
  4. Ansatz for the canonical form
  5. Examples and applications
  6. Kite integral family
  7. Definitions and differential equations
  8. Ansatz for the eps-form
  9. The resulting eps-form
  10. N3LO Higgs-production phase-space integrals
  11. Three-loop gravitational potential integrals
  12. Three-loop equal-mass banana graph
  13. Public implementation
  14. Conclusions and outlook
  15. Elliptic functions appearing in the sunrise graph
  16. ...and 1 more sections

Figures (4)

  • Figure 1: The kite integral family. Thick lines denote massive propagators. If one pinches the two massless lines, one recovers the sunrise graph which is the source of elliptic integrals.
  • Figure 2: Elliptic sector in the phase-space integrals for Higgs boson production at N$^3$LO in QCD. The thick line denotes the massive Higgs line. Lines crossing the dashed line denote cut propagators.
  • Figure 3: A three-loop integral sector that gives rise to elliptic functions in the gravitational potential of non-spinning binaries. Double lines denote linear propagators.
  • Figure 4: The three-loop equal-mass banana graph. The mass of the internal lines is denoted by $m$.