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New algorithms for Feynman integral reduction and $\varepsilon$-factorised differential equations

Iris Bree, Federico Gasparotto, Antonela Matijašić, Pouria Mazloumi, Dmytro Melnichenko, Sebastian Pögel, Toni Teschke, Xing Wang, Stefan Weinzierl, Konglong Wu, Xiaofeng Xu

TL;DR

The paper presents a constructive two-step framework for Feynman integral reduction that first selects master integrals via a geometry-informed Laporta ordering producing $F^{\bullet}$-compatible Laurent-$\varepsilon$ differential equations, and then converts these into an $\varepsilon$-factorised form through a systematic rotation. The first step relies on twisted cohomology and Baikov-represented integrands to define a robust, geometry-agnostic master basis; the second step removes $\varepsilon$-dependence using differential constraints, even when algebraic or transcendental functions appear. The approach is validated across polylogarithmic, elliptic, and Calabi–Yau-type geometries through a spectrum of examples, including one-, two-, three-, and four-loop integrals, demonstrating broad applicability and potential for further mathematical insights. The methodology promises improvements in analytical and numerical computations of perturbative QFT, and opens avenues for rigorous proofs, symmetry-aware extensions in twisted cohomology, and self-dual formulations of $\varepsilon$-factorised systems.

Abstract

In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and $\varepsilon$-factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations are of a Laurent polynomial form in the regularisation parameter $\varepsilon$ and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to $\varepsilon$-factorise the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.

New algorithms for Feynman integral reduction and $\varepsilon$-factorised differential equations

TL;DR

The paper presents a constructive two-step framework for Feynman integral reduction that first selects master integrals via a geometry-informed Laporta ordering producing -compatible Laurent- differential equations, and then converts these into an -factorised form through a systematic rotation. The first step relies on twisted cohomology and Baikov-represented integrands to define a robust, geometry-agnostic master basis; the second step removes -dependence using differential constraints, even when algebraic or transcendental functions appear. The approach is validated across polylogarithmic, elliptic, and Calabi–Yau-type geometries through a spectrum of examples, including one-, two-, three-, and four-loop integrals, demonstrating broad applicability and potential for further mathematical insights. The methodology promises improvements in analytical and numerical computations of perturbative QFT, and opens avenues for rigorous proofs, symmetry-aware extensions in twisted cohomology, and self-dual formulations of -factorised systems.

Abstract

In this paper, we give a detailed account of the algorithm outlined in [1] for Feynman integral reduction and -factorised differential equations. The algorithm consists of two steps. In the first step, we use a new geometric order relation in the integration-by-parts reduction to obtain a basis of master integrals, whose differential equations are of a Laurent polynomial form in the regularisation parameter and compatible with a filtration. This step works entirely with rational functions. In a second step, we provide a method to -factorise the aforementioned Laurent differential equations. The second step may introduce algebraic and transcendental functions. We illustrate the versatility of the algorithm by applying it to different examples with a wide range of complexity.

Paper Structure

This paper contains 49 sections, 7 theorems, 363 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Notation, definitions and review
  3. Summary of the notation
  4. Feynman integrals
  5. Integration-by-parts and the Laporta algorithm
  6. The method of differential equations
  7. The Baikov representation
  8. The democratic Baikov representation
  9. The loop-by-loop Baikov representation
  10. Good Baikov representations
  11. Projective space
  12. Twisted cohomology
  13. The set-up
  14. Objects
  15. Vector spaces
  16. ...and 34 more sections

Key Result

Lemma 1

Let $A$ be a matrix with terms of $B$-order $\{k,k+1,\dots,1\}$ (with $k \ge -n$) and $R_2^{(l)}$ with $l \in \{k,k+1,\dots,0\}$ as above. Then the products $A R_2^{(l)}$ and $R_2^{(l)} A$ have again only terms of $B$-order $\{k,k+1,\dots,1\}$.

Figures (15)

  • Figure 1: The Feynman graph for sector $93$. Massive propagators are indicated by red lines.
  • Figure 2: The Feynman graph for sector $79$. Massive propagators are indicated by red lines.
  • Figure 3: The Feynman graph for sector $15$. Massive propagators are indicated by red lines.
  • Figure 4: The sector of the banana integral together with the relevant super-sectors.
  • Figure 5: Sector $123$ (left), sector $127$ (middle) and sector $511$ (right).
  • ...and 10 more figures

Theorems & Definitions (29)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 19 more