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Improving integration-by-parts and 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 talk addresses the computational bottlenecks in Feynman integral reduction and in constructing $\varepsilon$-factorised differential equations. It introduces a geometry-driven Laporta ordering based on the maximal-cut Baikov representation and twisted cohomology, complemented by three filtrations, to guide master-integral selection and to decompose the integral space. A new extended order relation significantly reduces reduction sizes and can be combined with finite-field and other optimisations; separately, the basis obtained from this ordering yields differential equations on the maximal cut that are $\varepsilon$-factorised via a transforming matrix $R_2$, with a procedure to handle sub-sectors and rational $\varepsilon$-dependence. Importantly, the framework works without requiring explicit prior knowledge of the underlying geometry (e.g., elliptic or higher-genus structures) and relies on concepts from twisted cohomology and Hodge theory to achieve robust, algorithmic construction of $\varepsilon$-factorised differential equations. These advances promise improvements in both analytic and numerical perturbative quantum-field-theory calculations.

Abstract

In this talk, we discuss how ideas from geometry help to improve Feynman integral reduction and the construction of $\varepsilon$-factorised differential equations. In particular, we outline a systematic procedure to obtain an $\varepsilon$-factorised differential equation for any Feynman integral.

Improving integration-by-parts and differential equations

TL;DR

The talk addresses the computational bottlenecks in Feynman integral reduction and in constructing -factorised differential equations. It introduces a geometry-driven Laporta ordering based on the maximal-cut Baikov representation and twisted cohomology, complemented by three filtrations, to guide master-integral selection and to decompose the integral space. A new extended order relation significantly reduces reduction sizes and can be combined with finite-field and other optimisations; separately, the basis obtained from this ordering yields differential equations on the maximal cut that are -factorised via a transforming matrix , with a procedure to handle sub-sectors and rational -dependence. Importantly, the framework works without requiring explicit prior knowledge of the underlying geometry (e.g., elliptic or higher-genus structures) and relies on concepts from twisted cohomology and Hodge theory to achieve robust, algorithmic construction of -factorised differential equations. These advances promise improvements in both analytic and numerical perturbative quantum-field-theory calculations.

Abstract

In this talk, we discuss how ideas from geometry help to improve Feynman integral reduction and the construction of -factorised differential equations. In particular, we outline a systematic procedure to obtain an -factorised differential equation for any Feynman integral.
Paper Structure (4 sections, 27 equations, 4 figures)

This paper contains 4 sections, 27 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Integration-by-parts
  3. Differential equations
  4. Conclusions

Figures (4)

  • Figure 1: A non-planar double box integral.
  • Figure 2: A mixed geometry: Inside a surface of dimension two there can be curves of dimension one and points of dimension zero.
  • Figure 3: The decomposition of $H_{\mathrm{geom}}^2$ into spaces $H_{\mathrm{geom}}^{p,q}$ with the help of the filtrations $W_\bullet$ and $F_{\mathrm{geom}}^\bullet$.
  • Figure 4: Examples of decompositions of $V^{n}$ into spaces $V^{p,q}$ with the help of the filtrations $W_\bullet$ and $F_{\mathrm{geom}}^\bullet$. Coloured lines correspond to massive propagators.