On $\varepsilon$-factorised bases and pure Feynman integrals

TL;DR

This paper explores the properties of $\varepsilon$-factorised differential equations in Feynman integrals, revealing limitations in their relation to uniform transcendental weight and the global definition of purity, especially beyond multiple polylogarithms.

Contribution

It clarifies that $\varepsilon$-factorisation alone does not guarantee uniform transcendental weight and highlights issues with the global definition of purity in Feynman integrals.

Findings

$\varepsilon$-factorised differential equations do not always imply uniform transcendental weight.

A proposed definition of purity is only valid locally, not globally.

The study extends understanding beyond multiple polylogarithm integrals.

Abstract

We investigate $\varepsilon$-factorised differential equations, uniform transcendental weight and purity for Feynman integrals. We are in particular interested in Feynman integrals beyond the ones which evaluate to multiple polylogarithms. We show that a $\varepsilon$-factorised differential equation does not necessarily lead to Feynman integrals of uniform transcendental weight. We also point out that a proposed definition of purity works locally, but not globally.