Graphical functions and single-valued multiple polylogarithms
Oliver Schnetz
TL;DR
The paper develops a comprehensive framework connecting graphical functions in position space to single-valued multiple polylogarithms, enabling explicit computation of Feynman periods in four-dimensional phi^4 theory. It introduces completed graphs, sequential graphs, and SVMPs to express many phi^4 periods modulo products of MZVs; it also provides a constructive algorithm for a broad class of “constructible” graphs and proves the zig-zag conjecture modulo products. By leveraging Brown’s SVMP construction, Möbius symmetries, and two-dimensional analogues, the work situates a wide array of periods within the ring of single-valued MZVs, with explicit results up to eleven loops for constructible cases and twelve loops for A/B zig-zag families. The combination of analytic techniques (graphical functions, Gegenbauer expansions, residue methods) and algebraic tools (shuffle Hopf algebras, associators, coactions) yields both exact modulo-products formulas and practical computer-implementation pathways (Polylogproc, Hyperlogproc) for high-loop computations, advancing understanding of the arithmetic structure of quantum-field-theory periods.
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.