Pentagon integrals to arbitrary order in the dimensional regulator

TL;DR

This paper analytically computes one-loop pentagon integrals with up to one off-shell leg to arbitrary order in the dimensional regulator, providing explicit solutions in terms of Goncharov Polylogarithms.

Contribution

It introduces a pure basis of master integrals satisfying a canonical differential equation, enabling solutions to arbitrary order in the dimensional regulator.

Findings

Explicit solutions for pentagon integrals up to weight four.

Closed-form boundary terms including hypergeometric functions.

Analytic results for massless pentagon family.

Abstract

We analytically calculate one-loop five-point Master Integrals, \textit{pentagon integrals}, with up to one off-shell leg to arbitrary order in the dimensional regulator in $d=4-2\epsilon$ space-time dimensions. A pure basis of Master Integrals is constructed for the pentagon family with one off-shell leg, satisfying a single-variable canonical differential equation in the Simplified Differential Equations approach. The relevant boundary terms are given in closed form, including a hypergeometric function which can be expanded to arbitrary order in the dimensional regulator using the \texttt{Mathematica} package \texttt{HypExp}. Thus one can obtain solutions of the canonical differential equation in terms of Goncharov Polylogartihms of arbitrary transcendental weight. As a special limit of the one-mass pentagon family, we obtain a fully analytic result for the massless pentagon family in terms of pure and universally transcendental functions. For both families we provide explicit solutions in terms of Goncharov Polylogartihms up to weight four.