Pentagon integrals to arbitrary order in the dimensional regulator

TL;DR

The paper addresses analytic evaluation of one-loop pentagon integrals with up to one off-shell leg in $d=4-2\epsilon$. It constructs a pure basis and solves a single-variable canonical differential equation within the Simplified Differential Equations framework, yielding Goncharov Polylogarithms up to weight four and a closed-form hypergeometric boundary term expandable to arbitrary $\epsilon$ order. The massless pentagon is recovered as the $x\to1$ limit, and results are extended to arbitrary weight by the same approach. Explicit weight-four expressions are provided, and the authors supply open-source data and code to facilitate applications to two-loop pentabox calculations and LHC phenomenology.

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 $d=4-2ε$ 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.