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arXiv · 1306.2799

Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation

Johannes M. Henn·Alexander V. Smirnov·Vladimir A. Smirnov
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Abstract

We apply a recently suggested new strategy to solve differential equations for master integrals for families of Feynman integrals. After a set of master integrals has been found using the integration-by-parts method, the crucial point of this strategy is to introduce a new basis where all master integrals are pure functions of uniform transcendentality. In this paper, we apply this method to all planar three-loop four-point massless on-shell master integrals. We explicitly find such a basis, and show that the differential equations are of the Knizhnik-Zamolodchikov type. We explain how to solve the latter to all orders in the dimensional regularization parameter epsilon, including all boundary constants, in a purely algebraic way. The solution is expressed in terms of harmonic polylogarithms. We explicitly write out the Laurent expansion in epsilon for all master integrals up to weight six.

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Johannes M. Henn, Alexander V. Smirnov, Vladimir A. Smirnov. 2014-02-05. Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation. https://doi.org/10.1007/jhep07(2013)128

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