Analytic results for one-loop integrals in dimensional regularisation

TL;DR

This paper introduces a method to analytically evaluate one-loop integrals in quantum field theory using polylogarithms, hyperbolic geometry, and an algorithmic approach, enabling calculations for multiple scales and orders in the dimensional regulator.

Contribution

It provides a novel algorithmic framework to compute one-loop integrals analytically in terms of polylogarithms, connecting hyperbolic geometry with dimensional regularisation.

Findings

All integrals up to five external legs can be evaluated algorithmically.

The method allows for arbitrary order in the Laurent expansion in epsilon.

It enables analytic continuation across different kinematic regions.

Abstract

We present a method to obtain analytic results in terms of multiple polylogarithms for one-loop triangle, box and pentagon integrals depending on an arbitrary number of scales and to any desired order in the Laurent expansion in the dimensional regulator $\varepsilon$. Our method leverages the fact that for $\varepsilon=0$ one-loop integrals compute volumes of simplices in hyperbolic spaces, which can always be evaluated in terms of polylogarithms using an algorithm recently introduced in pure mathematics. The higher orders in $\varepsilon$ can then be expressed as a one-fold integral involving the result for $\varepsilon=0$. Remarkably, we find that for up to five external legs, all integrals can be evaluated algorithmically in terms of polylogarithms using direct integration techniques, which, in particular, requires us to rationalise all appearing square roots. We also discuss how we can use the connection to hyperbolic geometry to perform the analytic continuation from the Euclidean region to other kinematic regions.