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 ${\varphi }^4$ theory and to order five in six-dimensional ${\varphi }^3$ 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)

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Cited by in corpus (6)

• Five loop renormalization of ${\varphi }^3$ theory with applications to the Lee-Yang edge singularity and percolation theory
• Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: ${\varphi }^3$ QFT in 6 Dimensions
• Flow-oriented perturbation theory
• Invariant Differential Forms on Complexes of Graphs and Feynman Integrals
• Generalized single-valued hyperlogarithms
• $c_2$ invariants of hourglass chains via quadratic denominator reduction