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Mathematical Structures in Feynman Integrals

Feb 13 – 16, 2023
Emmy Noether Campus
Europe/Berlin timezone

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1. Welcome
Jan Piclum (University of Siegen), Mohamed Barakat (University of Siegen), Robin Brüser (University Siegen), Tobias Huber (Siegen U)
2/13/23, 2:00 PM
2. Wilhelm Killing -- Life and mathematical achievements
Wolfgang Hein
2/13/23, 2:10 PM

February 11, 2023 marks the 100th anniversary of the death of Wilhelm Killing. Far from all scientific centers, Killing formulated a research program that is still relevant today and has significantly influenced mathematical research for a century, namely the structure theory of Lie algebras. The 100th anniversary of Killing's death is an appropriate opportunity to honour Killing's merits...
3. 2D fishnet integrals and Calabi-Yau geometries
Claude Duhr
2/13/23, 3:00 PM

We argue that Yangian-invariant l-loop fishnet integrals in 2 dimensions are closely related to a family of Calabi-Yau l-folds.
This allows us to reduce the computation of these integrals to the computation of the Calabi-Yau periods. The periods are solutions of the Picard-Fuchs differential equations, which in turn are determined by the Yangian symmetry. Finally, we show that the values of...
4. Epsilon-factorised differential equations for non-trivial geometries
Stefan Weinzierl
2/13/23, 4:30 PM

The method of differential equations is a popular method to compute Feynman integrals. It is particular powerful if the differential equation can be cast into an epsilon-factorised form. In this talk I will discuss how this can be achieved for Feynman integrals which are related to non-trivial geometries like Calabi-Yau manifolds.
5. Constraining the Analytic Properties of Feynman Integrals
Andrew McLeod
2/13/23, 5:30 PM

The analytic properties of Feynman integrals are heavily constrained by basic physical principles such as causality and locality; however, the specific implications of these principles remain only partially known. In this talk, I will describe two methods for deriving concrete constraints on the analytic structure of Feynman integrals. The first of these methods leverages information about the...
6. Interpolation, Rational Reconstruction and Modular Algorithms
Claus Fieker
2/14/23, 9:00 AM

I will give a survey about interpolation, evaluation, and
reconstruction methods for multivariate polynomial and rational problems with applications in linear algebra and Groebner basis. This will include both asymptotically fast as well as practical algorithms.
7. The ice cone family and iterated integrals for Calabi-Yau varieties
Christoph Nega
2/14/23, 10:00 AM

In this talk I will explain how one can compute ice cone integrals in two dimensions for arbitrary loop order and which mathematical structures show up in this procedure. Using a leading singularity analysis we can find two copies of the banana graph which are related to period integrals on Calabi-Yau varieties. This observation allows us to express also the ice cone through iterated...
8. Recent progress in the reconstruction of multi-loop amplitudes
Tiziano Peraro
2/14/23, 11:15 AM

I will discuss recent progress in techniques for generating and reconstructing multi-loop scattering amplitudes. I will focus on novel developments in physical projectors, reduction to master integrals and partial fraction decomposition, as well as their combination with finite fields and rational reconstruction methods.
9. Computing intersection numbers with a rational algorithm
Gaia Fontana
2/14/23, 11:45 AM

Intersection theory allows to exploit the vector space structure obeyed by Feynman integrals, turning the decomposition to master integrals into the calculation of the projection of a vector into a basis, via scalar products called intersection numbers. In this talk I will discuss how the calculation of intersection numbers can be achieved via a purely rational algorithm, through the...
10. A tale of two packages, pfd-parallel and NeatIBP, for the multi-loop computations
Yang Zhang
2/14/23, 2:30 PM

In this talk, we are glad to introduce two packages, pfd-parallel and NeatIBP, based on computational algebraic geometry, aiming at cutting-edge computations for multi-loop Feynman integrals. The former one, powered by Singular and the massive parallelization framework GPI/space, implements multivariate partial fraction algorithms for simplifying analytic coefficients in multi-loop...
11. Massively parallel computer algebra and its application to the computation of Feynman integrals
Janko Böhm
2/14/23, 3:00 PM

In this talk, I will first discuss the Singular/GPI-Space project which aims to bring together computer algebra systems and the workflow management system GPI-Space to perform massive parallel computations. I will illustrate how GPI-Space leverages a coordination language based on the idea of Petri nets to efficiently model algorithms. Focusing on specific use cases, I will demonstrate how...
12. Integration-By-parts-Reduction using syzygies
Bakul Agarwal (KIT)
2/14/23, 4:30 PM

Integration By parts Reduction to master integrals is an essential step in most modern multi-loop scattering amplitude calculations. In recent years, IBP reduction based on syzygy methods has become increasingly popular, allowing the calculation of many challenging processes. We will discuss our approach to calculate these syzygies using linear algebra and finite field based methods, as well...
25. Faster modular IBP solving via Ratracer and tricks
Vitaly Magerya
2/14/23, 5:00 PM

I'd like to present Rational Tracer, a solver of systems of Integration-By-Parts relations (and more) based on modular arithmetic methods, and explain how using new programming ideas we can achieve lower evaluation times and higher flexibility for practical IBP solving.
13. Elliptic substructure of 2-loop kites and 3-loop tadpoles
David Broadhurst
2/15/23, 9:00 AM

The generic 2-loop kite integral has 5 internal masses. Its
completion by a sixth propagator gives a 3-loop tadpole
whose substructure involves 12 elliptic curves. I shall show
how to compute all such kites and their tadpoles, with 200
digit precision achieved in seconds, thanks to the procedure
of the arithmetic-geometric mean for complete elliptic
integrals of the third kind. The number...
14. Computing motives: an approach to irrationality proofs for periods
Daniel Juteau
2/15/23, 10:00 AM

Following Francis Brown, I will explain why certain questions in number theory, like irrationality of zeta values, make it desirable to be able to compute some mixed Tate motives of moduli spaces. Then I will explain an algorithm devised by Clément Dupont in his Ph.D. thesis for that purpose, and its implementation in the package MotivesForBiarrangements, which uses CAP (Categories,...
15. Amplitudes, Ansätze and Algebraic Geometry
Ben Page
2/15/23, 11:30 AM

Multi-particle scattering amplitudes beyond tree level are complicated functions of many variables. In practical computations, amplitudes are organized as a linear combination of transcendental functions with rational functions as coefficients. The computation of these rational functions in cutting-edge cases poses a significant challenge. In the modern approach, these rational functions are...
16. The Function Space of N=4 Scattering Amplitudes in the Regge Limit to all Orders
Robin Marzucca
2/15/23, 12:00 PM

I will review a recent mathematical technique to efficiently compute scattering amplitudes in the forward scattering limit of N=4 SYM and explain how it can be used to fully determine the function space of scattering amplitudes in this limit to all orders in perturbation theory.
17. One-loop hexagon integral to higher orders in the dimensional regulator
Antonela Matijašić
2/15/23, 2:30 PM

The state–of–the–art in current two–loop QCD amplitude calculations is at five-particle scattering. In contrast, very little is known at present about two–loop six–particle scattering processes. Computing two–loop six–particle processes requires knowledge of the corresponding one–loop amplitudes to higher orders in the dimensional regulator. In this talk, I will show the analytic results for...
18. Computer Algebra Methods for Feynman Integrals
Carsten Schneider
2/15/23, 3:00 PM

We report on a collection of computer algebra algorithms and their implementations that have been used in recent calculations of massive and massless 3-loop Feynman integrals. Among them we will focus on symbolic summation and integration methods, (partial) linear differential and difference equation solvers and the large moment method. A specific feature of all the techniques is that the...
19. Mathematical Structures in Massive Operator Matrix Elements
Johannes Blümlein
2/15/23, 4:30 PM

In this talk we will elaborate, among others, on root-valued alphabets and elliptic structures in Massive Operator Matrix Elements, together with concrete phenomenological results at O(a_s^3).
20. Antipodal (Self)-Duality in Planar N=4 Super-Yang-Mills Theory
Lance Dixon
2/16/23, 9:00 AM

In planar N=4 super-Yang-Mills theory, some scattering amplitudes and form factors that evaluate to multiple polylogarithms can be bootstrapped, or constructed without knowing the precise Feynman integrals, in some cases through eight loops. This allows a window into high orders of perturbation theory for certain scattering processes. Remarkably, the 6-gluon amplitude and the 3-gluon form...
21. Motivic Galois theory for Feynman integrals via twisted cohomology
Clément Dupont
2/16/23, 10:00 AM

I will report on ongoing joint work with Francis Brown, Javier Fresán, and Matija Tapušković, in which we prove that dimensionally regularized Feynman integrals are closed under the action of the motivic Galois group, termwise in the epsilon expansion. This fits into a larger framework of motivic Galois theory for algebraic Mellin transforms, where the main protagonists are formal versions of...
22. Progress on 3-loop Feynman integrals for 4-point 1-mass processes
Nikolaos Syrrakos
2/16/23, 11:30 AM
23. Intersection Numbers from Electromagnetism to Quantum Field Theory
Pierpaolo Mastrolia
2/16/23, 12:00 PM

Elaborating on the recent developments concerning Feynman integrals' calculus, I shall introduce Intersection Theory for twisted de Rham co-homology, and mainly discuss the impact of Intersection Numbers on the Vector-Space structure of special functions appearing in Mathematics and Physics.
24. tba
Antonela Matijašić
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