Communications in Number Theory and Physics

Volume 8 (2014)

Number 4

Graphical functions and single-valued multiple polylogarithms

Pages: 589 – 675



Oliver Schnetz (Department Mathematik, Friedrich-Alexander Universität (FAU) Erlangen-Nürnberg, Erlangen, Germany)


Graphical functions are single-valued complex functions which arise from Feynman amplitudes. We study their properties and use their connection to multiple polylogarithms to calculate Feynman periods. For the zigzag and two more families of $\phi^4$ periods we give exact results modulo products. These periods are proved to be expressible as integer linear combinations of single-valued multiple polylogarithms evaluated at one. For the larger family of “constructible” graphs, we give an algorithm that allows one to calculate their periods by computer algebra. The theory of graphical functions is used in [19] to prove the zig-zag conjecture.

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