Graphical Functions by Examples

TL;DR

Graphical functions are a versatile framework for evaluating multi-loop Feynman integrals, revealing rich structures and enabling automated computations in quantum field theory and conformal field theory.

Contribution

This review introduces the core concepts of graphical functions, including their analytic structures, computational methods, and connections to other areas, providing a comprehensive entry point.

Findings

Enables automatic computation of many graphical functions.

Reveals rich analytic structures and applications in quantum field theory.

Connects graphical functions to momentum space and self-duality.

Abstract

Graphical functions have emerged as a powerful framework for evaluating multi-loop Feynman integrals in perturbative quantum field theory. Defined as massless three-point position-space integrals, they reveal rich analytic structures and have enabled major advances, including the highest-loop results currently known in several quantum field theories. Their role extends to conformal field theory, and recent algorithmic developments now allow many graphical functions to be computed automatically. This review, based on graduate-level lectures held by O.S. in 2025/26 at the University of Hamburg, introduces the central ideas behind graphical functions, covering periods, Feynman residues, and the treatment of regular and singular cases in both integer and non-integer dimensions. It also discusses connections to momentum space and self-duality, and provides guidance for further study, offering a coherent entry point into a topic not addressed in standard textbooks.