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 φ 4 theory and to order five in six-dimensional φ 3 theory. In this article we present the theory of graphical functions in even dimensions ≥ 4 with detailed reviews of known properties and full proofs whenever possible.