Graphical functions in even dimensions

TL;DR

This paper develops the theory of graphical functions in even dimensions greater than or equal to four, providing detailed properties, proofs, and applications to high-loop Feynman integrals in quantum field theories.

Contribution

It extends the theory of graphical functions to all even dimensions ≥4, including proofs and known properties, facilitating calculations of high-loop Feynman integrals.

Findings

Calculated renormalization constants up to 8 loops in 4D $\,\phi^4$ theory

Computed renormalization constants up to 5 loops in 6D $\,\phi^3$ theory

Provided comprehensive review and proofs of 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 $\phi^4$ theory and to order five in six-dimensional $\phi^3$ theory. In this article we present the theory of graphical functions in even dimensions $\geq4$ with detailed reviews of known properties and full proofs whenever possible.