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

  • Johannes M. Henn
  • Alexander V. Smirnov
  • Vladimir A. Smirnov


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 ϵ, 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 ϵ for all master integrals up to weight six.


Scattering Amplitudes Renormalization Regularization and Renormalons 


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Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Johannes M. Henn
    • 1
  • Alexander V. Smirnov
    • 2
    • 4
  • Vladimir A. Smirnov
    • 3
  1. 1.Institute for Advanced StudyPrincetonU.S.A.
  2. 2.Scientific Research Computing CenterMoscow State UniversityMoscowRussia
  3. 3.Skobeltsyn Institute of Nuclear Physics of Moscow State UniversityMoscowRussia
  4. 4.Institut für Theoretische Teilchenphysik, KITKarlsruheGermany

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