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Oliver Schnetz

Publications and source records attributed to Oliver Schnetz.

At least 19 recordsLinked to original sources
arXiv2026

HyperFORM -- a FORM package for parametric integration with hyperlogarithms

HyperFORM brings the parametric integration of hyperlogarithms, weighted by rational prefactors, into the symbolic-manipulation system FORM. It ports the capabilities of Erik Panzer's Maple package HyperInt, capitalizing on FORM's speed with bulky algebraic input and on its ability to spread a single calculation across many processor cores. We keep the description of the method brief and concentrate instead on how the package is organized and driven: a fully self-contained program for the three-loop zigzag period serves as a worked illustration, and timing measurements for zigzags through six loops gauge its present reach. HyperFORM is released openly and applies to a broad class of problems, the evaluation of Feynman integrals prominently among them.

hep-ph↗
arXiv2026

Graphical Functions by Examples

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.

hep-th↗
arXiv2026

Self-duality of massless scalar three-point amplitudes

We prove that off-shell massless scalar three-point Feynman integrals are self-dual under Fourier transformation. This implies that a momentum space integral can be expressed as the position space integral of the same Feynman graph with transformed edge-weights (not the dual graph) if external vertices are labeled accordingly. In particular, any off-shell massless scalar three-point Feynman integral can be expressed as a graphical function. The result follows immediately from a theorem by M. Golz, E. Panzer and the author on parametric representations of position space integrals (2015), but it was only observed by X. Jiang in 2025 in the context of four-dimensional $\mathcal{N}=4$ Super-Yang-Mills theory. We generalize Jiang's result and discuss the consequences of the self-duality in the context of graphical functions. In particular, we derive a new identity for graphical functions and a new twist relation for scalar integrals (Feynman periods) in $\phi^4$ theory.

hep-th↗
arXiv2025

HyperFORM -- a FORM package for parametric integration with hyperlogarithms

We present an implementation of algorithms for the symbolic integration of hyperlogarithms multiplied by rational functions in the computer algebra system FORM. This implementation encompasses cases where hyperlogarithms have rational letters or a rational argument. It complements the previous implementation, HyperInt, in MAPLE by Erik Panzer, utilizing the advantages of FORM in the efficient handling of large symbolic expressions. Among a wide range of applications, this approach enables the computation of many Feynman integrals.

hep-ph↗
arXiv2025

LinApart2: efficient parallel partial fraction decomposition algorithm for denominators with polynomials of general degree

We present LinApart2, a major update to the LinApart algorithm for univariate partial fraction decomposition. Unlike its predecessor, LinApart2 can handle denominators of arbitrary polynomial degree without explicit factorization, while retaining the efficiency and parallelizability of the Laurent series method. Benchmarks show substantial speedups in both runtime and memory usage compared to Mathematica's built-in routine Apart and to the Euclidean algorithm, enabling computations that were previously intractable.

cs.SC↗
arXiv2025

$\phi^3$ theory at six loops

We present the renormalization functions of dimensionally regularized $\phi^3$ theory in six dimensions up to loop order six in the minimal subtraction scheme.

hep-th↗
arXiv2025

The five-twist identity for Feynman periods

We prove a new identity for Feynman periods that acts on five-vertex cuts of completed primitive Feynman graphs. It is shown that in $\phi^4$ theory this identity is independent from existing identities which are the twist, the Fourier identity and the Fourier split.

hep-th↗
arXiv2025

Graphical functions with spin

The theory of graphical functions is generalized from scalar theories to theories with spin, leading to a numerator structure in Feynman integrals. The main part of this article treats the case of positive integer spin, which is obtained from spin $1/2$ theories after the evaluation of $\gamma$ traces. As an application (in this article used mainly to prove consistency and efficiency of the method), we calculate primitive Feynman integrals in Yukawa-$\phi^4$ (Gross-Neveu-Yukawa) theory up to loop order eight.

hep-th↗
arXiv2025

Notes on color reductions and $\gamma$ traces

We present efficient algorithms to calculate the color factors for the $SU(N)$ gauge group and to evaluate $\gamma$ traces. The aim of these notes is to give a self-contained, proved account of the basic results with particular emphasis on color reductions. We fine tune existing algorithms to make calculations at high loop orders possible.

hep-ph↗
arXiv2024

The wheel classes in the locally finite homology of $\mathrm{GL}_n(\mathbb{Z})$, canonical integrals and zeta values

We compute the canonical integrals associated to wheel graphs, and prove that they are proportional to odd zeta values. From this we deduce that wheel classes define explicit non-zero classes in: the locally finite homology of the general linear group $\GL_n(\ZZ)$ in both odd and even ranks, the homology of the moduli spaces of tropical curves, and the moduli space of tropical abelian varieties. We deduce the existence of a doubly infinite family of auxiliary classes in the even commutative graph complex.

math.NT↗
arXiv2023

Geometries in perturbative quantum field theory

In perturbative quantum field theory one encounters certain, very specific geometries over the integers. These perturbative quantum geometries determine the number contents of the amplitude considered. In the article `Modular forms in quantum field theory' F. Brown and the author report on a first list of perturbative quantum geometries using the $c_2$-invariant in $ϕ^4$ theory. A main tool was denominator reduction which allowed the authors to examine graphs up to loop order (first Betti number) 10. We introduce an improved quadratic denominator reduction which makes it possible to extend the previous results to loop order 11 (and partially orders 12 and 13). For comparison, also non-$ϕ^4$ graphs are investigated. Here, we extend the results from loop order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously 157) -- while being consistent with all major $c_2$-conjectures -- leads to a more refined picture of perturbative quantum geometries. In the appendix, Friedrich Knop proves a Chevalley-Warning-Ax theorem for double covers of affine space.

math-ph↗
arXiv2022

Seven loops $ϕ^4$

We give a detailed account of the theory of position space renormalization using graphical functions in the case of dimensionally regularized $ϕ^4$ theory in four dimensions. In this theory we calculate the beta function, the mass gamma function, and the self energy to seven loops in the minimal subtraction scheme. The anomalous dimension $γ$ is calculated to loop order eight. When possible, we generalize to even dimensions $\geq4$ with particular focus on $ϕ^3$ theory in six dimensions. In this theory we calculate the anomalous dimension $γ$ to loop order six.

hep-th↗
arXiv2022

Recursive computation of Feynman periods

Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in $ϕ^3$ theory up to six loops and 561 of 607 Feynman periods at seven loops. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that $ϕ^3$ and $ϕ^4$ theory share the same number content.

hep-th↗
arXiv2021

Generalized single-valued hyperlogarithms

Single-valued hyperlogarithms are generalized to include primitives of differential forms $\mathrm{d} z/(az\overline{z}+bz+c\overline{z}+d)$, $a,b,c,d\in\mathbb{C}$, where $\overline{z}$ is the complex conjugate of the variable $z\in\mathbb{C}$. The construction of these generalized single-valued hyperlogarithms (GSVHs) relies on a commutative hexagon which allows one to express primitives as anti-primitives. The article provides a proved constructive theory of GSVHs.

math-ph↗
arXiv2021

$c_2$ invariants of hourglass chains via quadratic denominator reduction

We introduce families of four-regular graphs consisting of chains of hourglasses which are attached to a finite kernel. We prove a formula for the $c_2$ invariant of these hourglass chains which only depends on the kernel. For different kernels these hourglass chains typically give rise to different $c_2$ invariants. An exhaustive search for the $c_2$ invariants of hourglass chains with kernels that have a maximum of ten vertices provides Calabi-Yau manifolds with point-counts which match the Fourier coefficients of modular forms whose weights and levels are [4,8], [4,16], [6,4], and [9,4]. Assuming the completion conjecture, we show that no modular form of weight 2 and level $\leq1000$ corresponds to the $c_2$ of such hourglass chains. This provides further evidence in favour of the conjecture that curves are absent in $c_2$ invariants of $ϕ^4$ quantum field theory.

math-ph↗
arXiv2021

Graphical functions in even dimensions

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.

hep-th↗
arXiv2018

Further investigations into the graph theory of $ϕ^4$-periods and the $c_2$ invariant

A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are via graph operations which are period invariant and other graph quantities which predict aspects of the Feynman period, one notable example is known as the $c_2$ invariant. We give results and computations in both these directions, proving a new period identity and computing its consequences up to 11 loops in $ϕ^4$-theory, proving a $c_2$ invariant identity, and giving the results of a computational investigation of $c_2$ invariants at 11 loops.

hep-th↗