Abstract
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\U0001d716 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.
Highlights
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
We find that all relevant boundary terms required for the solution of the canonical differential equation are given in closed form
The ultimate check that a candidate pure basis of Master Integrals has the desired properties is the derivation of its differential equation [14]
Summary
For the construction of the top sector basis element we follow the consensus developed in [9, 24], expressed here in the language of the Baikov representation of Feynman Integrals [25, 26]. By studying the maximal cut of (2.6) in the Baikov represenation [27], we can see that it has the desired properties of being pure and universally transcendental This result is a strong indicator that the uncut (2.6) will satisfy a canonical differential equation. When one has identified an integral, or a combination of integrals, which in its maximal cut is expressed in terms of pure functions of uniform weight, this integral, or combination of integrals, is a strong candidate for an element of the desired pure basis. Both the pure basis g as well as our choice of Feynman Integrals G are given in appendix B
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