Communications in Number Theory and Physics

Volume 16 (2022)

Number 3

Graphical functions in even dimensions

Pages: 515 – 614



Michael Borinsky (Nikhef Theory Group, Amsterdam, The Netherlands)

Oliver Schnetz (Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen, Germany)


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

2010 Mathematics Subject Classification

Primary 81Q15, 81Q30. Secondary 81T99.

M.B. was supported by NWO Vidi grant 680-47-551.

O.S. was supported by DFG grant SCHN 1240/3.

Received 11 May 2021

Accepted 4 May 2022

Published 4 October 2022