Pentagon integrals to arbitrary order in the dimensional regulator

We analytically calculate one-loop five-point Master Integrals, pentagon integrals, with up to one off-shell leg to arbitrary order in the dimensional regulator in d = 4-2? 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 Mathematica package 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.

Automated summary

We analytically calculated one-loop five-point Master Integrals, pentagon integrals, with one off-shell leg to arbitrary order in dimensional regulator. The study constructed a pure basis of Master Integrals for the pentagon family with one off-shell leg and provided relevant boundary terms in closed form. The solutions of the canonical differential equation were obtained in terms of Goncharov Polylogartihms of arbitrary transcendental weight.