Rationalisation of multiple square roots in Feynman integrals

TL;DR

This paper explores conditions under which multiple square roots in Feynman integrals can be rationalized to express results in terms of multiple polylogarithms, including a novel example involving a one-loop pentagon with massive legs.

Contribution

It introduces a method to use different rationalisations in different parts of Feynman integral calculations based on parameterisations, expanding the class of integrals expressible in polylogarithms.

Findings

Different rationalisations can be used if they correspond to different parameterisations of the same path.

The technique is demonstrated on a one-loop pentagon with three massive external legs.

The method allows expressing certain integrals in terms of multiple polylogarithms.

Abstract

Feynman integrals are very often computed from their differential equations. It is not uncommon that the $\varepsilon$-factorised differential equation contains only dlog-forms with algebraic arguments, where the algebraic part is given by (multiple) square roots. It is well-known that if all square roots are simultaneously rationalisable, the Feynman integrals can be expressed in terms of multiple polylogarithms. This is a sufficient, but not a necessary criterium. In this paper we investigate weaker requirements. We discuss under which conditions we may use different rationalisations in different parts of the calculation. In particular we show that we may use different rationalisations if they correspond to different parameterisations of the same integration path. We present a non-trivial example -- the one-loop pentagon function with three adjacent massive external legs involving seven square roots -- where this technique can be used to express the result in terms of multiple polylogarithms.