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The geometric bookkeeping guide to 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

Abstract

We report on three improvements in the context of Feynman integral reduction and $\varepsilon$-factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the $\varepsilon$-dependence of the integration-by-parts identities. Secondly, we observe that with a specific choice of order relation in the Laporta algorithm, we directly obtain a basis of master integrals, whose differential equation on the maximal cut is in Laurent polynomial form with respect to $\varepsilon$ and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an $\varepsilon$-factorised form. This provides a systematic algorithm to obtain an $\varepsilon$-factorised differential equation for any Feynman integral. Furthermore, the choices for the prefactors and the order relation significantly improve the efficiency of the reduction algorithm.

The geometric bookkeeping guide to Feynman integral reduction and $\varepsilon$-factorised differential equations

Abstract

We report on three improvements in the context of Feynman integral reduction and -factorised differential equations: Firstly, we show that with a specific choice of prefactors, we trivialise the -dependence of the integration-by-parts identities. Secondly, we observe that with a specific choice of order relation in the Laporta algorithm, we directly obtain a basis of master integrals, whose differential equation on the maximal cut is in Laurent polynomial form with respect to and compatible with a particular filtration. Thirdly, we prove that such a differential equation can always be transformed to an -factorised form. This provides a systematic algorithm to obtain an -factorised differential equation for any Feynman integral. Furthermore, the choices for the prefactors and the order relation significantly improve the efficiency of the reduction algorithm.

Paper Structure

This paper contains 5 sections, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Concepts and method
  3. Examples
  4. Conclusions
  5. End Matter

Figures (3)

  • Figure 1: A non-planar double-box integral with internal masses (indicated by red lines). The top sector has $5$ master integrals, which decompose with respect to the filtrations as shown in the right figure.
  • Figure 2: The three-loop banana graph with unequal masses. The top sector has $11$ master integrals, which decompose with respect to the filtrations as shown in the right figure.
  • Figure 3: A two-loop integral with masses (indicated by red lines). The top sector has $3$ master integrals, which decompose with respect to the $F_{\mathrm{geom}}^\bullet$-filtration and the $W_\bullet$-filtration as shown in the right figure.