Graphical functions in even dimensions

Abstract

Graphical functions are special position space Feynman integrals, which can be used to calculate Feynman periods and one- or two-scale processes at high loop orders. With graphical functions, renormalization constants have been calculated to loop orders seven and eight in four-dimensional $\ddot{I}{•}^4$ theory and to order five in six-dimensional $\ddot{I}{•}^3$ theory. In this article we present the theory of graphical functions in even dimensions $\ge 4$ with detailed reviews of known properties and full proofs whenever possible.

67 pages, 36 figures; v3: incorporated technical corrections from the erratum 'Commun. Number Theory Phys. 20 (2026) 461--462' (doi: 10.4310/CNTP.260606021019)