Graphical functions and single-valued multiple polylogarithms

TL;DR

This paper explores the properties of graphical functions from Feynman amplitudes, connecting them to multiple polylogarithms, and provides exact calculations and algorithms for specific families of Feynman periods.

Contribution

It establishes a link between graphical functions and single-valued multiple polylogarithms, proving the zig-zag conjecture and developing computational methods for Feynman periods.

Findings

Exact results for zig-zag and phi^4 periods modulo products

Proof that these periods are expressible as integer combinations of single-valued multiple polylogarithms

An algorithm for computing periods of 'constructible' graphs

Abstract

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.