arXiv subjects

Sven F. Stawinski

Publications and source records attributed to Sven F. Stawinski.

11 recordsLinked to original sources

arXiv2026

Discrete symmetries of Feynman integrals

We perform a comprehensive study of a certain class of discrete symmetries of families of Feynman integrals, defined as affine changes of variables that map different sectors of the family into each other. We show that these transformations are always encoded into permutations of the Feynman parameters that relate the Lee-Pomeransky polynomials of the two sectors, irrespective of the integral representation used to define the Feynman integrals. We then construct an affine map in loop-momentum space that encodes such a permutation. We also show that these symmetries can be naturally embedded into the framework of twisted cohomology theories, and the period and intersection parings are invariant under the symmetry transformations. If we focus on symmetries within a fixed sector, we obtain a group acting on the twisted cohomology group, and we study the decomposition of this action into irreducible representations. One of our main mathematical results is that the character of this representation is proportional to the Euler characteristic of the corresponding fixed-point set. We then study the implications for Feynman integrals, in particular for the intersection matrix in a canonical basis. We also present a formula for the number of master integrals in a given sector in the presence of a non-trivial symmetry group in terms of the Euler characteristics of fixed-point sets. As an application, we obtain the numbers of master integrals for banana integrals with up to four loops for arbitrary configurations of non-zero masses. In order to achieve our results, we had to combine tools from various different areas of mathematics, including graph theory, group theory and algebraic topology.

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arXiv2026

Intersection theory and canonical differential equations

In these proceedings we will review recent progress in applying ideas from the mathematical framework of twisted cohomology to the study of canonical differential equations for Feynman integrals. Firstly, we will show how the intersection matrix can shed some light on the nature of the canonical basis of a Feynman integral family, a concept still not fully understood in the general case. In particular we will show how the intersection matrix can detect hidden linear dependencies of the iterated integrals resulting from an $\eps$-factorized differential equation, which are difficult to find otherwise. Furthermore, we will explain how the intersection matrix can help in deriving (polynomial) relations between the transcendental functions occurring in the rotation to the canonical basis. This allows us to simplify the rotation, and furthermore leads to simplifications in the final result. The discussion we be kept as light as possible, focusing on a simple running example and deferring the technical details to the original publications.

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arXiv2025

Hypergeometry from $\mathrm{\widehat P}$-Symmetry: Feynman Integrals in One and Two Dimensions

Feynman integrals with generic propagator powers in one and two spacetime dimensions are investigated from various perspectives. In particular, we argue that the class of track integrals at any loop order is fixed by the recently found $\mathrm{\widehat P}$-symmetries of Yangian type. All track integrals up to six external points (and four loops) are bootstrapped explicitly as well as the full family of one-loop integrals at any multiplicity. Moreover, the triangle tracks at generic loop order, which constitute the most generic family of track-type integrals, are bootstrapped in this way. The results are compared to the direct evaluation via a `spectral transform' from the integrability toolbox that turns out to be particularly efficient for position-space tree integrals in lower dimensions. We prove that all $\mathrm{\widehat P}$-symmetries of these integrals can be derived from the framework of Aomoto--Gelfand hypergeometric functions, which applies to integrals in one and two dimensions. Finally, we also demonstrate the method's applicability to conformal integrals by deriving the complete results for all comb-channel conformal partial waves as well as the conformal double-box integral. We explicitly go through all examples of the above integrals in 1D and then provide a straightforward recipe for how to read off their 2D counterparts.

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arXiv2025

Non-Local Symmetries of Planar Feynman Integrals

We prove the invariance of scalar Feynman graphs of any planar topology under the Yangian level-one momentum symmetry given certain constraints on the propagator powers. The proof relies on relating this symmetry to a planarized version of the conformal simplices of Bzowski, McFadden and Skenderis. In particular, this proves a momentum-space analogue of the position-space conformal condition on propagator powers. When combined with the latter, the invariance under the level-one momentum implies full Yangian symmetry of the considered graphs. These include all scalar Feynman integrals for which a Yangian symmetry was previously demonstrated at the level of examples, e.g. the fishnet or loom graphs, as well as generalizations to graphs with massive propagators.

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arXiv2025

Canonical differential equations and intersection matrices

Differential equations are one of the main approaches to evaluate multi-loop Feynman integrals. The construction of a canonical or $\varepsilon$-factorised basis for multi-loop integrals remains a key bottleneck for this approach, especially when the differential equation involves non dlog-forms. Recently, several methods have been proposed to find $\varepsilon$-factorised differential equations. Many of them introduce new functions that are themselves defined as iterated integrals. If and when these iterated integrals can be explicitly evaluated in terms of other classes of functions remains an open problem. In this paper we elaborate on the recent proposal that one can use the fact that the intersection matrix computed in a canonical basis can be used to derive polynomial relations between these iterated integrals. On the one hand, we discuss properties of the canonical intersection matrix, in particular methods to determine the intersection matrix in a canonical basis. On the other hand we show how one can reduce the non-linear constraints on the iterated integrals to linear ones. We illustrate these ideas on examples involving Calabi-Yau varieties and higher-genus Riemann surfaces.

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arXiv2025

Three-loop banana integrals with four unequal masses

We present a system of canonical differential equations satisfied by the three-loop banana integrals with four distinct non-zero masses in $D = 2-2\eps$ dimensions. Together with the initial condition in the small-mass limit, this provides all the ingredients to find analytic results for three-loop banana integrals in terms of iterated integrals to any desired order in the dimensional regulator. To obtain this result, we rely on recent advances in understanding the K3 geometry underlying these integrals and in how to construct rotations to an $\eps$-factorized basis. This rotation typically involves the introduction of objects defined as integrals of (derivatives of) K3 periods and rational functions. We apply and extend a method based on results from twisted cohomology to identify relations among these functions, which allows us to reduce their number considerably. We expect that the methods that we have applied here will prove useful to compute further multiloop multiscale Feynman integrals attached to non-trivial geometries.

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arXiv2024

Canonical Differential Equations Beyond Genus One

We discuss for the first time canonical differential equations for hyperelliptic Feynman integrals. We study hyperelliptic Lauricella functions that include in particular the maximal cut of the two-loop non-planar double box, which is known to involve a hyperlliptic curve of genus two. We consider specifically three- and four-parameter Lauricella functions, each associated to a hyperelliptic curve of genus two, and construct their canonical differential equations. Whilst core steps of this construction rely on existing methods $\unicode{x2014}$ that we show to be applicable in the higher-genus case $\unicode{x2014}$ we use new ideas on the structure of the twisted cohomology intersection matrix associated to the integral family in canonical form to obtain a better understanding of the appearing new functions. We further observe the appearance of Siegel modular forms in the $\varepsilon$-factorized differential equation matrix, nicely generalizing similar observations from the elliptic case.

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arXiv2024

Conformal Four-Point Integrals: Recursive Structure, Toda Equations and Double Copy

We consider conformal four-point Feynman integrals to investigate how much of their mathematical structure in two spacetime dimensions carries over to higher dimensions. In particular, we discuss recursions in the loop order and spacetime dimension. This results e.g. in new expressions for conformal ladder integrals with generic propagator powers in all even dimensions and allows us to lift results on 2d Feynman integrals with underlying Calabi-Yau geometry to higher dimensions. Moreover, we demonstrate that the Basso-Dixon generalizations of these integrals obey different variants of the Toda equations of motion, thus establishing a connection to classical integrability and the family of so-called tau-functions. We then show that all of these integrals can be written in a double copy form that combines holomorphic and anti-holomorphic building blocks. Here integrals in higher dimensions are constructed from an intersection pairing of two-dimensional "periods" together with their derivatives. Finally, we comment on extensions to higher-point integrals which provide a richer kinematical setup.

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arXiv2024

Self-duality from twisted cohomology

Recently a notion of self-duality for differential equations of maximal cuts was introduced, which states that there should be a basis in which the matrix for an ε-factorised differential equation is persymmetric. It was observed that the rotation to this special basis may introduce a Galois symmetry relating different integrals. We argue that the proposed notion of self-duality for maximal cuts stems from a very natural notion of self-duality from twisted cohomology. Our main result is that, if the differential equations and their duals are simultaneously brought into canonical form, the cohomology intersection matrix is a constant. Furthermore, we show that one can associate quite generically a Lie algebra representation to an ε-factorised system. For maximal cuts, this representation is irreducible and self-dual. The constant intersection matrix can be interpreted as expressing the equivalence of this representation and its dual, which in turn results in constraints for the differential equation matrix. Unlike the earlier proposal, the most natural symmetry of the differential equation matrix is defined entirely over the rational numbers and is independent of the basis choice.

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arXiv2024

Twisted Riemann bilinear relations and Feynman integrals

Using the framework of twisted cohomology, we study twisted Riemann bilinear relations (TRBRs) satisfied by multi-loop Feynman integrals and their cuts in dimensional regularisation. After showing how to associate to a given family of Feynman integrals a period matrix whose entries are cuts, we investigate the TRBRs satisfied by this period matrix, its dual and the intersection matrices for twisted cycles and co-cycles. For maximal cuts, the non-relative framework is applicable, and the period matrix and its dual are related in a simple manner. We then find that the TRBRs give rise to quadratic relations that generalise quadratic relations that have previously appeared in the literature. However, we find that the TRBRs do not allow us to obtain quadratic relations for non-maximal cuts or completely uncut Feynman integrals. This can be traced back to the fact that the TRBRs are not quadratic in the period matrix, but separately linear in the period matrix and its dual, and the two are not simply related in the case of a relative cohomology theory, which is required for non-maximal cuts.

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arXiv2024

An Elliptic One-Loop Amplitude in Anti-de-Sitter Space

We present full analytic results for the four-point one-loop amplitude of a conformally coupled scalar in four-dimensional Anti-de-Sitter space dual to a primary operator with scaling dimension 1. The computation is based on an intriguing recent discovery, connecting Witten diagrams and flat-space Feynman integrals, which led to an expression of the amplitude of interest as a pure combination of single-valued multiple polylogarithms and an integral which cannot be reduced to multiple polylogarithms. We explicitly evaluate that integral in terms of elliptic multiple polylogarithms, finding that it is not manifestly single-valued unlike the polylogarithmic contributions to the amplitude. Further we compute the symbol of the integral and observe similar structures as for (elliptic) flat-space amplitudes. The result presented here adds to the relatively short list of explicitly known position space curved-space amplitudes beyond tree level, and constitutes the first curved-space amplitude evaluated in terms of elliptic multiple polylogarithms.

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