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Deriving canonical differential equations for Feynman integrals from a single uniform weight integral

Christoph Dlapa, Johannes Henn, Kai Yan

TL;DR

This work delivers an algorithm to construct canonical differential equations for Feynman integrals starting from a single UT integral, by linking the Picard–Fuchs equation to a canonical first-order system and solving for constant matrices that define the basis transformation. The approach both produces canonical forms and provides a UT-test, enabling efficient handling of large, multi-loop, multi-variable systems. The authors validate the method on multiple cutting-edge examples, including planar and non-planar sectors up to four loops and a four-variable two-loop five-point case, and release a public Mathematica tool (INITIAL) for implementation. The work significantly simplifies obtaining canonical forms and broadens the applicability of the UT/DE framework to complex integral families.

Abstract

Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to Hoschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-planar two-loop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper.

Deriving canonical differential equations for Feynman integrals from a single uniform weight integral

TL;DR

This work delivers an algorithm to construct canonical differential equations for Feynman integrals starting from a single UT integral, by linking the Picard–Fuchs equation to a canonical first-order system and solving for constant matrices that define the basis transformation. The approach both produces canonical forms and provides a UT-test, enabling efficient handling of large, multi-loop, multi-variable systems. The authors validate the method on multiple cutting-edge examples, including planar and non-planar sectors up to four loops and a four-variable two-loop five-point case, and release a public Mathematica tool (INITIAL) for implementation. The work significantly simplifies obtaining canonical forms and broadens the applicability of the UT/DE framework to complex integral families.

Abstract

Differential equations are a powerful tool for evaluating Feynman integrals. Their solution is straightforward if a transformation to a canonical form is found. In this paper, we present an algorithm for finding such a transformation. This novel technique is based on a method due to Hoschele et al. and relies only on the knowledge of a single integral of uniform transcendental weight. As a corollary, the algorithm can also be used to test the uniform transcendentality of a given integral. We discuss the application to several cutting-edge examples, including non-planar four-loop HQET and non-planar two-loop five-point integrals. A Mathematica implementation of our algorithm is made available together with this paper.

Paper Structure

This paper contains 13 sections, 57 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of the method
  3. From the Picard-Fuchs equation of a uniform weight Feynman integral to a canonical system of first-order equations
  4. Ansatz for canonical differential equations, and determination of unknowns
  5. Generalization to multi-variable case
  6. Special cases with degenerate Psi-matrix and algebraic letters
  7. Examples and applications
  8. Full differential equation for planar three-loop integrals
  9. New result for a four-loop four-point integral
  10. Canonical form for non-planar four-loop sector with 17 master integrals
  11. Four-variable example: Non-planar double pentagon integrals
  12. Public implementation
  13. Conclusion and outlook

Figures (4)

  • Figure 1: Planar three-loop four-point integrals. The number of MI is 26 for (a) and 41 for (b).
  • Figure 2: Planar four-loop four-point integral. The number of MI is 19.
  • Figure 3: Non-planar four-loop integral appearing in the calculation of the cusp anomalous dimension. The sector shown has $17$ coupled master integrals, according to FIRE6.
  • Figure 4: The non-planar double-pentagon integral family.