Graphical Functions by Examples

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.

• The paper introduces an example-driven framework for evaluating massless, multi-loop Feynman integrals using graphical functions.
• It leverages single-valued multiple polylogarithms and dimensional regularization to achieve high-loop computations and explicit period evaluations.
• The method bridges algebraic geometry with practical quantum field theory, offering algorithm-driven insights for precision collider and conformal studies.

Detailed Summary of "Graphical Functions by Examples" (2604.25739)

Introduction and Motivation

"Graphical Functions by Examples" delivers a comprehensive, example-driven exposition of graphical functions—a pivotal framework for evaluating massless, multi-loop Feynman integrals, especially in high-loop calculations of perturbative QFT. The authors synthesize the analytic, combinatorial, and algorithmic advances that underpin this approach, targeting applications spanning from precision collider physics (high-order Standard Model corrections) to critical phenomena in statistical and condensed matter systems. The review grew from graduate-level lectures and is shaped to address the lack of standard textbook treatments on this subject.

Graphical functions are defined as massless, three-point integrals in position space, depending on complex parameters derived from three distinct spacetime points. Their analytic properties, especially single-valuedness, render them ideally suited for explicit evaluation via iterated integration techniques and underpin their amenability to algorithmic computation. Significant advances have stemmed from this formalism, including the highest-loop results in φ³ and φ⁴ theories, and their implications for the renormalization group and conformal bootstrap programs.

Theoretical Framework and Regularization

The review formalizes graphical functions as position-space Feynman integrals that, after exploiting translational and scaling symmetries, reduce to functions of a single complex variable $z$ (and its conjugate). The external kinematics is fixed by mapping the three points to $0$, $1$, and $z$ within the complex plane $\mathbb{C}$. This identification allows full leverage of complex analysis, especially the structure of single-valued multiple polylogarithms and the role of conformal invariance.

Dimensional regularization is employed throughout, with $D = 2\lambda + 2 = d - 2$ for $d\in\{4,6,8,...\}$, focusing on even-dimensional cases where the massless scalar propagator admits the necessary analyticity. The review adopts notational conventions for (single-valued) multiple polylogarithms and multiple zeta values that are crucial for direct comparison with the computational packages and the mathematical literature.

Essential Calculational Paradigm

A cornerstone of the graphical function approach is the translation of Feynman integrals into iterated integrals of single-valued hyperlogarithmic functions. The review carefully demonstrates this for introductory cases—starting with the massless three-point one-loop integral. By exploiting duality (between momentum and position space) and symmetry reductions, the authors reformulate the integral as a unique solution to a Laplace equation on the complex plane, with boundary conditions fixed by the analyticity and single-valuedness of graphical functions.

The manipulation of such integrals is systematically reduced to recursion relations for single-valued multiple polylogarithms, solvable via explicit partial integration and by enforcing invariance under $z\rightarrow\bar{z}$ exchange. The uniqueness of these solutions in the class of graphical functions is established by rigorous symmetry and analyticity arguments.

Periods, Feynman Residues, and Stokes-Theoretic Methods

A central sequence of examples details the extraction of physically-meaningful numbers—periods—from the evaluation of vacuum and external-point Feynman diagrams. The review highlights the computation of the tetrahedral period $P_3$, illustrating:

• The systematic conversion of the original multi-dimensional integral into a single-valued complex contour integral,
• The use of Stokes' theorem for reducing the integral to "anti-residues" at critical points ($z=0,1,\infty$),
• The computation and regularization of single-valued polylogarithms at singularities (with explicit references to Deligne’s tangential base point formalism).

A representative result is $0$0, where all contributing terms are rigorously interpreted in terms of the multiple polylogarithms and their single-valued completions.

Algorithmic and Structural Advances

The review systematically progresses to more challenging instances (such as the $0$1 period), emphasizing the combinatorial complexity of words in the polylogarithmic alphabet and the correspondingly rich shuffle algebra structures. Notably, these problems are efficiently approached using computer algebra systems (e.g., the cited Maple hyperlogarithm packages), allowing the expansion, symmetrization, and regularization of polylogarithmic expressions at high weight.

A detailed example shows how the period of $0$2 reduces to a combination of 26 single-valued polylogarithms of word length 8, underscoring the exponential growth of analytic complexity with loop order and further motivating algorithmic solutions.

Extensions: Singularities, Momentum-Space Dualities, and Conformal Symmetry

The work further explores:

• The treatment of graphs in both regular and singular (exceptional) kinematic configurations and in non-integer dimensions,
• The translation between position-space and momentum-space representations, focusing on the constructive role of duality and the computation of conformal integrals,
• Self-duality properties of three-point functions and their relevance for momentum space techniques and conformal bootstrap computations.

Implications and Theoretical Significance

The graphical function approach delineates a unifying algebraic framework for massless, multi-loop Feynman integral evaluation in various QFTs. Its algorithmic viability paves the way for the construction of high-order perturbative predictions in phenomenology and critical phenomena. The methods presented have already enabled the computation of six and seven-loop results otherwise inaccessible, and provide structural insights into the periods and motives underlying quantum field theoretic amplitudes.

From a mathematical perspective, graphical functions make manifest the interplay between Feynman graph combinatorics, the theory of single-valued iterated integrals, and the Galois theory of periods. The review also connects foundational developments in the theory of multiple polylogarithms (Brown’s single-valued multiple polylogarithms, the structure of MZVs) to explicit calculations relevant to QFT.

Future Prospects

Graphical function methods are anticipated to further impact both computational quantum field theory and the theory of special values of Feynman integrals. Immediate directions include:

• The classification and algorithmic computation of periods for higher genera and face-number graphs,
• Extensions to all orders in the dimensional regulator, potentially unveiling deeper links to mixed motives and the cosmic Galois group,
• Exploration of connections to modular forms and periods beyond multiple zeta values,
• The development of efficient open-source computation tools for practitioners in both mathematics and physics.

Conclusion

"Graphical Functions by Examples" (2604.25739) provides a technically rigorous, pedagogically motivated, and computationally oriented overview of graphical functions as a transformative method for multi-loop Feynman integral evaluation. It bridges the gap between the algebraic geometry of periods and practical perturbative field theory, equipping researchers with both conceptual insight and concrete algorithms for future investigations.