Graphical Functions in Quantum Field Theory

• Graphical functions are massless three-point Feynman integrals defined on the punctured complex plane with fixed external vertices 0, 1, and z.
• They use Schwinger-parametric representations and Symanzik polynomials to ensure convergence and real-analytic properties via rigorous power-counting conditions.
• Generalizations incorporating spin introduce tensorial structures and single-valued polylogarithms, facilitating high-loop computations in quantum field theoretic models.

Searching arXiv for the core literature on graphical functions in quantum field theory to ground the article in the cited papers. Searching for foundational papers on scalar graphical functions, parametric representations, even-dimensional theory, and spin generalizations. Graphical functions are special massless three-point position-space Feynman integrals associated with a connected graph carrying three distinguished external vertices, usually labeled $0$, $1$, and $z$. After fixing the external configuration by a Euclidean motion so that $x_0=0$, $x_1=1$, and $x_z=z\in\mathbb C\setminus\{0,1\}$, the resulting integral becomes a function on the punctured complex plane. In even dimensions $d=2\lambda+2>2$, and under the usual infrared and ultraviolet power-counting conditions, scalar graphical functions define single-valued real-analytic functions on $\mathbb C\setminus\{0,1\}$; in the scalar case they are moreover positive functions $\mathbb C\setminus\{0,1\}\to\mathbb R_+$ (Golz et al., 2015, Borinsky et al., 2021). Their theory sits at the intersection of position-space quantum field theory, Symanzik-polynomial techniques, Möbius symmetry, single-valued multiple polylogarithms, and high-loop period calculations, and it has recently been generalized from scalar theories to theories with spin by incorporating numerator structures and spin-dependent differential operators (Schnetz, 2013, Schnetz, 8 Apr 2025).

1. Definition and normalization conventions

Let $G$ be a connected graph with three distinguished external vertices and internal vertex set $1$0. In the scalar setting, each edge $1$1 carries a weight $1$2, and one sets

$1$3

In Euclidean dimension $1$4, the position-space Feynman integral is

$1$5

with the external points fixed at $1$6, $1$7, and $1$8 (Golz et al., 2015). In the four-dimensional normalization emphasized in later work, one equivalently writes

$1$9

and then pulls back to a function of $z$0 by fixing $z$1, $z$2, $z$3 (Borinsky et al., 2021).

The literature uses more than one divergence convention. In the parametric treatment of Panzer and Schnetz, the superficial degree of divergence is

$z$4

whereas in the spin-generalized formulation one writes

$z$5

and factors out an overall scale so that the graphical function depends only on the conformal ratio $z$6 (Golz et al., 2015, Schnetz, 8 Apr 2025). These are notationally different but serve the same structural purpose: they isolate the dimensionally correct three-point object.

A basic point is that graphical functions are “graphical” in the sense that, after the external frame is fixed, the integral depends only on the combinatorics of the graph and the assigned edge weights. In scalar theories with convergent integrals, they are special three-point massless Feynman integrals and form a distinguished subclass of position-space amplitudes (Golz et al., 2015).

2. Parametric representation and convergence

A central structural result is the projective Schwinger-parametric representation. Introducing one Schwinger parameter $z$7 for each edge, one defines the first and second Symanzik polynomials

$z$8

and

$z$9

where $x_0=0$0 is the spanning-forest polynomial obtained from two-forest decompositions separating $x_0=0$1 from $x_0=0$2 (Golz et al., 2015).

Under the convergence conditions, one obtains

$x_0=0$3

where $x_0=0$4 is the positive projective domain and $x_0=0$5 is the standard projective measure (Golz et al., 2015). In the compact Lam–Nakanishi notation this becomes

$x_0=0$6

with $x_0=0$7, $x_0=0$8, $x_0=0$9, and $x_1=1$0.

The convergence assumptions are the usual power-counting constraints. Infrared convergence requires that for every subgraph $x_1=1$1 with any internal vertex,

$x_1=1$2

and ultraviolet convergence requires that for every induced $x_1=1$3 with at most one external vertex,

$x_1=1$4

Under these hypotheses the scalar integral converges absolutely (Golz et al., 2015).

The parametric form also yields analyticity. The key observation is that $x_1=1$5 is affine-linear in the squared distances $x_1=1$6, with positive real parts in the appropriate domain. Combined with combinatorial degree bounds such as

$x_1=1$7

and analogous estimates for forest polynomials, this permits differentiation under the integral and proves real analyticity on $x_1=1$8 via Morera’s theorem or dominated convergence for holomorphic parameters (Golz et al., 2015).

3. Analytic structure, singularities, and symmetries

In even dimensions $x_1=1$9, the scalar graphical function $x_z=z\in\mathbb C\setminus\{0,1\}$0 is real-valued on $x_z=z\in\mathbb C\setminus\{0,1\}$1, invariant under complex conjugation, and single-valued real-analytic there (Borinsky et al., 2021). In the scalar theory with general even dimension $x_z=z\in\mathbb C\setminus\{0,1\}$2, Panzer–Schnetz theory further states that $x_z=z\in\mathbb C\setminus\{0,1\}$3 is a single-valued real-analytic function with possible logarithmic singularities at $x_z=z\in\mathbb C\setminus\{0,1\}$4, $x_z=z\in\mathbb C\setminus\{0,1\}$5, and $x_z=z\in\mathbb C\setminus\{0,1\}$6 (Schnetz et al., 2024).

A refined local statement is available in $x_z=z\in\mathbb C\setminus\{0,1\}$7 with all $x_z=z\in\mathbb C\setminus\{0,1\}$8. At each singular point $x_z=z\in\mathbb C\setminus\{0,1\}$9, $d=2\lambda+2>2$0 admits a single-valued log-Laurent expansion, with an explicit bound

$d=2\lambda+2>2$1

where $d=2\lambda+2>2$2 is the sum of edge weights attached to $d=2\lambda+2>2$3 (Borinsky et al., 2021). This places graphical functions naturally on the punctured Riemann sphere.

Their symmetry group is larger than complex conjugation. After “completion,” which adjoins the point $d=2\lambda+2>2$4 and balances external weighted valences, a permutation $d=2\lambda+2>2$5 of $d=2\lambda+2>2$6 induces a Möbius transformation $d=2\lambda+2>2$7 of the cross-ratio variable, and one has

$d=2\lambda+2>2$8

This $d=2\lambda+2>2$9-cross-ratio symmetry is one of the main organizing principles of the theory (Schnetz, 2013, Borinsky et al., 2021).

Graphical functions also satisfy exact factorization identities. If removing the external vertices disconnects the graph into components, then the graphical function factorizes as the product of the graphical functions of the components (Borinsky et al., 2021). In the spin-generalized setting the same factorization persists in tensorial form,

$\mathbb C\setminus\{0,1\}$0

when the graph disconnects upon removal of $\mathbb C\setminus\{0,1\}$1 (Schnetz et al., 2024).

4. Single-valued polylogarithms and recursive computation

A major advance in the subject is the identification of graphical functions with explicit spaces of single-valued polylogarithmic functions. Brown’s single-valued multiple polylogarithms $\mathbb C\setminus\{0,1\}$2 are built from ordinary iterated integrals $\mathbb C\setminus\{0,1\}$3 by imposing trivial monodromy around $\mathbb C\setminus\{0,1\}$4, $\mathbb C\setminus\{0,1\}$5, and $\mathbb C\setminus\{0,1\}$6, with $\mathbb C\setminus\{0,1\}$7 and $\mathbb C\setminus\{0,1\}$8 (Schnetz, 2013). Many graphical functions can then be represented in the form

$\mathbb C\setminus\{0,1\}$9

where $\mathbb C\setminus\{0,1\}\to\mathbb R_+$0 is the shuffle algebra generated by the $\mathbb C\setminus\{0,1\}\to\mathbb R_+$1 up to suitable weight (Schnetz, 2013).

This representation underlies an effective recursive calculus. In even dimensions, if $\mathbb C\setminus\{0,1\}\to\mathbb R_+$2 is obtained from $\mathbb C\setminus\{0,1\}\to\mathbb R_+$3 by attaching an edge of weight $\mathbb C\setminus\{0,1\}\to\mathbb R_+$4 at the external point $\mathbb C\setminus\{0,1\}\to\mathbb R_+$5, then the two graphical functions satisfy a Laplace-type equation

$\mathbb C\setminus\{0,1\}\to\mathbb R_+$6

with

$\mathbb C\setminus\{0,1\}\to\mathbb R_+$7

Solving this PDE inside the single-valued function space yields the append-edge algorithm (Borinsky et al., 2021).

The algorithmic version for constructible graphs proceeds by completion, decomposition after removing $\mathbb C\setminus\{0,1\}\to\mathbb R_+$8, sequential-function building by repeated edge appending, evaluation of the period integral by the residue theorem inside the polylogarithmic algebra, and reduction to a standard MZV basis; the Maple package polylog_procedures implements this workflow (Schnetz, 2013). Later expositions place the same strategy in the broader algebra of generalized single-valued hyperlogarithms and emphasize automated implementations such as HyperlogProcedures and HyperFORM (Chakraborty et al., 28 Apr 2026).

The one-loop three-star already displays the characteristic analytic structure. In $\mathbb C\setminus\{0,1\}\to\mathbb R_+$9,

$G$0

equivalently

$G$1

where $G$2 is the Bloch–Wigner dilogarithm (Borinsky et al., 2021). A 2026 review further formulates period extraction in terms of “anti-residues” at $G$3, $G$4, and $G$5, making Stokes-theoretic reduction explicit (Chakraborty et al., 28 Apr 2026).

5. Planar duality and momentum-space correspondences

For externally planar graphs there is an exact planar-duality theorem. If $G$6 embeds in the plane with $G$7, $G$8, and $G$9 on the outer face, one constructs its dual $1$00 by adjoining an extra vertex connected to $1$01, $1$02, and $1$03, dualizing, and then removing the dual triangle. When

$1$04

the dual graph also satisfies $1$05, and the parametric polynomials obey

$1$06

Consequently,

$1$07

This theorem generalizes the familiar momentum-position duality for off-shell three-point integrals and is especially effective in four dimensions, where many dual graphs reduce to ladder or completed graphs with known closed forms (Golz et al., 2015).

A related formulation in the general even-dimensional theory is Fourier or position/momentum duality for planar graphs. If each propagator $1$08 is replaced by its Fourier-transformed mate with weight $1$09 satisfying

$1$10

then the graphical function becomes proportional to that of the planar dual graph $1$11; in parametric language this is associated with the Cremona transformation on Schwinger parameters (Borinsky et al., 2021).

Recent expository work pushes this correspondence further. The one-loop triangle is described as self-dual, and the general identity

$1$12

is presented as the three-point manifestation of momentum-space/position-space self-duality (Chakraborty et al., 28 Apr 2026).

6. Periods, exact evaluations, and high-loop applications

Graphical functions are a principal tool for computing Feynman periods. For a completed primitive $1$13 graph with four external edges $1$14, $1$15, $1$16, and $1$17, the period is

$1$18

and if a graph has only two external points, or one isolated external point, the corresponding graphical function is constant and coincides with the usual Feynman period (Schnetz, 2013, Borinsky et al., 2021).

The period theory is especially developed for sequential or zig-zag families. For the $1$19-loop zig-zag graph $1$20, the period was reduced to special values of sequential functions, then to combinations of $1$21, and finally evaluated in closed form by applying Zagier’s theorem: $1$22 This establishes the Broadhurst–Kreimer zig-zag conjecture (Schnetz, 2013).

The same framework yields exact formulas modulo products for two further $1$23 families $1$24 and $1$25, and it extends algorithmically to a larger class of constructible graphs (Schnetz, 2013). In even dimensions the method also covers fishnet ladder integrals $1$26, whose graphical functions in $1$27 are expressible in terms of single-valued classical polylogarithms of weight $1$28 (Borinsky et al., 2021).

These structural results translate directly into perturbative calculations. Graphical-function methods have been used to compute renormalization constants and anomalous dimensions in four-dimensional $1$29 theory to seven and eight loops, and in six-dimensional $1$30 theory to five loops (Borinsky et al., 2021). A later review presents graphical functions as a framework that also handles singular cases, dimensional regularization in non-integer dimension, and automated high-loop examples (Chakraborty et al., 28 Apr 2026).

7. Generalization to theories with spin

The scalar theory has been generalized to theories with spin by introducing numerator propagators carrying multi-indices. In dimension $1$31, the spin-$1$32 numerator propagator is

$1$33

with equivalent differential form

$1$34

A general spinful Feynman integral is obtained by assigning such propagators to edges and contracting repeated indices with the Euclidean metric $1$35 (Schnetz et al., 2024, Schnetz, 8 Apr 2025).

In the three-point case one defines

$1$36

and expands the tensor structure as

$1$37

where $1$38 partitions the spin indices into metric-pairs and indices carried by the external unit vectors $1$39 and $1$40. The components $1$41 are single-valued functions of $1$42 with at worst logarithmic singularities at $1$43, $1$44, and $1$45 (Schnetz et al., 2024).

The formalism has two notable algebraic consequences. First, nondegenerate bilinear pairings in the Brauer algebra allow spin-$1$46 periods and two-point amplitudes to be reduced to linear combinations of scalar periods of augmented graphs (Schnetz et al., 2024). Second, the action of differentiation with respect to the external point $1$47 is encoded by a block-triangular effective Laplace operator $1$48. Appending a scalar edge of weight $1$49 at $1$50 produces the differential equation

$1$51

which can be inverted recursively sector by sector in the number of $1$52-factors, imposing single-valuedness and excluding poles of order $1$53 at $1$54 (Schnetz et al., 2024).

The function space remains remarkably stable under this generalization. Three-point graphical functions with numerators still lie in the same space of single-valued combinations of generalized hyperlogarithms, which preserves the feasibility of high-loop computations without Laporta IBP (Schnetz et al., 2024). The 2025 full spin treatment applies these ideas to primitive periods in Yukawa–$1$55 theory up to eight loops; one completed ten-loop graph, corresponding to eight loops before completion, was evaluated numerically as

$1$56

in $1$57 minutes on a single desktop core using the Maple package HyperlogProcedures (Schnetz, 8 Apr 2025). This places spinful graphical functions within the same computational paradigm that had previously proved effective for scalar $1$58 and $1$59 theories.