Geometric Singularities of Feynman Integrals

• Felix Tellander

Room: Seminar Room bctp I will present a new method to calculate the full microlocal description of the singularities of Feynman integrals. This is done by associating a unique constructible function to the system of partial differential equations (PDEs) annihilating the integral and from this function the singularities can directly be read-off. This function can be constructed explicitly even if the system of PDEs is unknown and describes both the location of the singularities and the number of (generalized) master integrals on them. The framework is flexible enough to perform the calculation in any of the Lee-Pomeransky, Feynman, or momentum representations.

Higher-Genus Multiple Zeta Values

• Konstantin Baune

Room: Seminar Room bctp At genus zero and one, multiple zeta values (MZVs) and elliptic MZVs are defined as special values of (elliptic) multiple polylogarithms and they have been important ingredients in the calculation of amplitudes in both QFT and string theory.

Electroweak Double-Box Integrals for Møller Scattering with Three Z Bosons

• Iris Bree

Room: Seminar Room bctp We compute planar and nonplanar twoloop doublebox integrals required for NNLO electroweak corrections to Møller scattering, with three massive Z bosons exchanged between the fermion lines. These integrals involve integration kernels tied to elliptic, K3 and genus two geometries. By applying a new εfactorization method for the associated differential equations, we obtain closedform results in terms of iterated integrals over those kernels.

Full Classification of Feynman Integral Geometries at Two Loops

• Hjalte Frellesvig

Room: Seminar Room bctp In this talk I will discuss ongoing work on the complete classification of the Feynman Integral geometries that can appear in two-loop corrections to scattering amplitudes in any quantum field theory, including the standard model. We systematically categorize all graphs that may contribute in four dimensions, finding 79 in total. We then investigate them for generic mass configurations, in order to find their corresponding integral geometry. We approach this both with maximal cuts performed in the Baikov representation, as well as using Picard-Fuchs operators. We uncover a plethora of geometries, such as elliptic curves, hyperelliptic curves, and K3 manifolds, of which some are known, but many have not been investigated until now.

Higher-Loop Scattering Amplitude Computations Beyond Polylogarithms — From Feynman Integrals to Numbers

• Christoph Nega

Room: Seminar Room bctp In my talk, I will discuss strategies to compute scattering amplitudes in particle physics that evaluate to functions beyond polylogarithms. A crucial step in calculating these amplitudes is selecting the appropriate scalar Feynman integrals. It turns out that certain integrals known as canonical master integrals reveal the transcendental structure of the amplitude and are also simpler to calculate. In the first part of my talk, I will explain a method for identifying these canonical integrals, particularly when the underlying geometry is of elliptic type or even beyond. In the second part, I will then provide several examples where this technique has been applied to compute elliptic scattering amplitudes. In particular, I will discuss strategies for evaluating the appearing elliptic iterated integrals and, therefore, also the final scattering amplitude in an efficient and fast way. Moreover, this process can be enhanced by using Bernoulli-like variables, which accelerate and improve the convergence of the resulting series expansions.

Lessons from Elliptic Leading Singularities

• Vasily Sotnikov

Room: Seminar Room bctp A good choice of Feynman integral bases plays an essential role in the computation of loop scattering amplitudes. While such choice is well understood when the integrals can be expressed as iterated integrals with logarithmic kernels, the situation remains less clear when elliptic curves get involved. A common approach is to construct integrals that satisfy differential equations in which the dependence on dimensional regulator factorizes completely. Is it possible, instead, to define a good integral basis purely from geometric considerations at the integrand level? We propose a generalization of dlog integrands with unit leading singularities to the case of integration on elliptic curves, which we argue provides such a definition. We observe that the corresponding Feynman integrals satisfy a previously unnoticed form of differential equations, whose solutions evaluate to pure functions. We further argue that certain properties of this basis may offer advantages in multi-scale amplitude calculations.

 Based on [arXiv:2504.20897].

Equivariant Iterated Eisenstein Integrals

• Franca Lippert

Room: Seminar Room bctp We study equivariant primitives of Eisenstein series for principal congruence subgroups and show that they are precisely the corresponding non-holomorphic Eisenstein series. We present closed formulas that naturally generalise existing results for the full modular group. Building on this, we construct an equivariant analogue of double Eisenstein integrals for congruence subgroups. This construction generalizes the one for the full modular group. For the specific case of the group Γ(2), we further study the single-valued Gauss hypergeometric function and its relation to iterated integrals of weights 2 and 4.

Analytic Results for Elliptic Two-Loop QCD Amplitudes

• Fabian Wagner

Room: Seminar Room bctp Two-loop four-point QCD amplitudes receive elliptic contributions stemming from internal, massive quark loops. In this talk, I will explain how these contributions can be expressed in terms of iterated integrals over a set of independent kernels utilizing canonical differential equations. Further, I will demonstrate how the corresponding amplitudes can be evaluated numerically fast and reliable via appropriate series expansions.

Coactions for Genus One Integrals

• Axel Kleinschmidt

Room: Seminar Room bctp The motivic coaction of multiple zeta values and multiple polylogarithms encodes both structural insights on and computational methods for scattering amplitudes in a variety of quantum field theories and in string theory. I will report on joint work with Franziska Porkert and Oliver Schlotterer where we propose coaction formulae for iterated integrals over holomorphic Eisenstein series that arise from configuration-space integrals at genus one. Our proposal is motivated by formal similarities between the motivic coaction and the single-valued map of multiple polylogarithms at genus zero that are exposed in their recent reformulations via zeta generators. The proposed genus-one coaction is constructed by analogy with the construction of single-valued iterated Eisenstein integrals via zeta generators at genus one and subjected to various checks.

Motivic Coaction for Hypergeometric Functions

• Deepak Kamlesh

Room: Seminar Room bctp In this talk we will recall a conjecture that posits that special functions such as hypergeometric functions may have a Galois theory associated with them which is in fact compatible with the Galois theory of periods. We will also discuss recent advances on the problem.

Wall Crossing Structure from Quantum Phenomena to Feynman Integrals

• Anthony Massidda

Room: Seminar Room bctp A growing body of evidence suggests that the complexity of Feynman integrals is best understood through geometry. Recent mathematical developments have shed light on the role of exponential integrals as periods of twisted de Rham cocycles over Betti cycles, offering a structured approach to tackle this problem in a wide range of cases. In this talk, based on arxiv 2506.03252, I will introduce these concepts and show how families of physically relevant integrals, ranging from exponentials to logarithmic multivalued functions, can be recast as twisted periods of differential forms over homology cycles. Focusing on the case of holomorphic exponents, I will use the Pearcey integral as a guiding example to present its explicit decomposition via thimble expansion, unveiling a geometric wall-crossing structure underlying the analytic continuation in parameter space. We will briefly explore how the generalization to multivalued functions provides the right framework to describe Feynman integrals in the Baikov representation, where the multivaluedness is governed by the logarithm of the Baikov polynomial. In this context, the thimble decomposition aligns with the decomposition into Master Integrals. I will highlight how the analysis of wall-crossing structure allows for a sharp count of independent Master Integrals (or periods), circumventing complications arising from Stokes phenomena.

Geometric Bookkeeping for ε -factorised Differential Equations

• Pouria Mazloumi

Room: Seminar Room bctp In this talk we will present our new method to compute the ε-factorised differential equations for arbitrary Feynman integrals. The fresh strategy is based on two steps: in the first step, we study the twisted cohomology (and its filtrations) associated with a given integral family (on the maximal cut) and introduce a particular order relation in the Laporta algorithm, such that the resulting differential equation is a Laurent polynomial in ε. In the second step, we explain how systematically achieve an ε-factorised form by solving a set differential constraints. Furthermore, we will also comment on the possible benefits of the newly introduced ideas in the context of Feynman integral reduction.

Recent Developments for Multiple Eisenstein Series

• Henrik Bachmann

Room: Seminar Room bctp Multiple Eisenstein series, introduced in 2006 by Gangl-Kaneko-Zagier, form a bridge between multiple zeta values and classical Eisenstein series. In this talk, I will discuss recent developments and conjectures concerning these series, with a focus on a possible sl2-algebra structure. By this, we mean an algebra A equipped with an injective Lie algebra homomorphism from the three-dimensional Lie algebra sl2 into the derivation algebra Der(A). A classical example is the algebra of quasimodular forms, which carries three natural derivations satisfying the sl2 commutation relations. The algebra of multiple Eisenstein series contains the quasimodular forms as a subalgebra. I will present a conjectural framework suggesting that the sl2-structure on quasimodular forms extends naturally to this larger algebra. Finally, I will outline a dimension conjecture for the space of multiple Eisenstein series, which indicates that the presence of an sl2-structure might be the key feature distinguishing them from multiple zeta values.

From Black Hole Scattering to Calabi-Yau Manifolds

• Gustav Uhre Jakobsen

Room: Seminar Room bctp I will present recent four-loop results on black hole scattering observables. These unprecedented results shed new light on the classic gravitational two-body problem, achieve new levels of precision required in gravitational wave physics, and, strikingly, involve Calabi–Yau periods in their analytic description. Although classical and free of quantum effects, these results are deeply informed by techniques originally developed in quantum chromodynamics and particle physics, encapsulated in the worldline quantum field theory (WQFT) formalism. In my talk, I will introduce the basics of WQFT and its Feynman diagrammatic expansion of the classical black hole scattering system, highlighting both the physical aspects and the mathematical structures. Examples of the former include dissipative, self-force, and spin effects; of the latter, loop integrals, differential equations, and special functions. Finally, as a perspective on applications in gravitational wave physics, I will compare the perturbative four-loop results and their resummation with numerical relativity.

Canonical Differential Equations and Twisted Cohomology

• Sven Stawinski

Room: Seminar Room bctp Finding the rotation to the canonical form for Feynman integrals associated with non-trivial geometries typically requires the introduction of not only periods of the geometry, but also (iterated) integrals thereof. In this talk I will describe how the twisted cohomology intersection matrix can „detect“ the canonical form and how this can be used to constrain these integrals of periods. In particular this allows one to evaluate many (linear combinations of) these integrals in terms of periods and algebraic functions, which greatly reduces the complexity of the resulting canonical differential equations.

Three Loop Unequal Masses Banana Integral

• Sara Maggio

Room: Seminar Room bctp In this talk, I will demonstrate how combining techniques such as twisted cohomology and canonical differential equations, enables the analytical computation of multi-scale loop integrals. As a central example, I will focus on the three-loop banana integrals with all distinct internal masses. This method, however, is broadly applicable and offers a promising framework for tackling a wide class of multi-scale Feynman integrals.

Bethe Colloquium: The Landau Bootstrap

• Andrew McLeod

Room: Seminar Room bctp Physical principles such as locality and unitarity are believed to impose strong constraints on the mathematical structure of scattering amplitudes. However, we remain incapable of working out the full implications of these principles in concrete examples. In this talk, I will give an overview of recent progress towards working out the constraints these principles place on individual Feynman integrals. I will also describe how these constraints can be fed into a bootstrap approach, via which the functional form of Feynman integrals can be determined from just knowledge of their singular behavior.

Surprising Evasions of Elliptic Obstructions

• David Broadhurst

Room: Seminar Room bctp In 1968, I met an elliptic integral when studying the decay of the eta meson. Then it was a surprise to find that elliptic integrals are not needed for the magnetic moment of the electron at two loops. In 1998, I expected elliptic obstructions when evaluating three-loop massive vacuum diagrams. Yet empirical fits to numerical integrations gave results in terms of polylogarithms. Recently I have found more examples, where integrals of elliptic integrals multiplied by polylogarithms are empirically reducible to classical tetralogarithms. I shall describe how Steven Charlton and I have proved these results. The basic idea is to arrive at a double integral of dilogarithms and then remove an obstructing square root of a quartic polynomial by a pair of Euler transformations.

Periods of fibre product of elliptic surfaces and the Gamma conjecture

• Eric Pichon-Pharabod

Room: Seminar Room bctp Given two rational elliptic surfaces over the projective line, one may construct a Calabi-Yau threefold by considering their fibre product, following a construction of Schoen. For specific pairs of elliptic surfaces, deforming the parameter of one of the elliptic surfaces in a specific manner one obtains a family of such threefolds called the Hadamard product. These Hadamard products carry a motive of type (1,1,1,1) and the associated Picard-Fuchs equation is a Calabi-Yau operator. I will describe a method for computing numerical approximations of the periods of such threefolds with very high precision, relying on an explicit description of their homology. This computation allows to probe into the Gamma class formula for these Calabi-Yau operators: we find a formula that seems to fit all example of the Calabi-Yau database.