Graphical functions and single-valued multiple polylogarithms

By Oliver Schnetz. 2013

176 citations recorded by the source corpus.

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 zig-zag 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 to prove the zig-zag conjecture.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/e83e1f282bf4a383d459f3d182427dc526e71f8b

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In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

Citations in the corpus

Minimally subtracted six loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents (2017, methodology): “A particularly strong check of our results is due to Oliver Schnetz [140], who computed β and γφ to 6 loops for the case n = 1, using completely different techniques including single-valued integration and graphical functions [139].”
Closed string amplitudes as single-valued open string amplitudes (2014, background): “On the other hand, from a physical point of view SVMZVs appear in the computation of graphical functions for certain Feynman amplitudes [15].”
Feynman integrals and hyperlogarithms (2015, methodology): “So in particular the constructibility of 3-loop graphs in the sense of graphical functions [154] carries over to the position space.”