arXiv subjects

Claude Duhr

Publications and source records attributed to Claude Duhr.

At least 37 records · Page 2Linked to original sources

arXiv2025

Infrared singularities and the collinear limits of multi-leg scattering amplitudes

Scattering amplitudes are expected to admit a factorised structure in special kinematic limits, such as the Regge, soft and collinear limits. However, less is known about the precise mechanisms through which factorisation of $n$-particle scattering amplitudes is realised at high perturbative orders, where more complex structures arise. Starting with the soft anomalous dimension, in this work we investigate the multi-particle collinear limits of massless amplitudes at three- and four-loop orders. Using colour conservation and rescaling symmetry, we show how strict collinear factorisation of multiple massless final-state coloured particles is realised, and provide results for the corresponding splitting amplitude soft anomalous dimensions. In particular, we demonstrate through four loops that the conditions on the structure of the soft anomalous dimension that are required by strict collinear factorisation in all two-particle collinear limits, are sufficient to guarantee such factorisation also in any multiple collinear limit. Then, assuming that strict collinear factorisation of massless partons holds also for amplitudes containing massive coloured particles, we derive new constraints on the soft anomalous dimension from multi-collinear limits.

hep-ph↗

arXiv2025

Double spacelike collinear limits from multi-Regge kinematics

We study scattering amplitudes and form factors in planar $\mathcal{N}=4$ Super Yang-Mills theory in the limit where two pairs of gluons become collinear. We find that, when the virtualities of both collinear pairs are spacelike, the collinear factorisation of the amplitude involves a generalised splitting amplitude that correlates the two collinear directions, confirming a recent proposal in the literature. Remarkably, we find that our generalised splitting amplitude agrees with the Bern-Dixon-Smirnov (BDS) subtracted six-point amplitude in multi-Regge kinematics. The latter can be explicitly evaluated using integrability to all orders in the coupling. We also present compelling evidence for the universality of the generalised splitting amplitude, by showing that the same function governs the double spacelike collinear limit of scattering amplitudes with up to 8 particles and 3 loops and form factors with up to 6 particles and 2 loops.

hep-th↗

arXiv2025

Antipodal self-duality of square fishnet graphs

In strongly-deformed planar ${\cal N}=4$ super-Yang-Mills theory, or fishnet theory, a point-split single-trace correlation function of four dimension-$m$ scalar operators is given by a single Feynman integral, which involves integrating over locations of a $m\times m$ grid of points. We show that for any integer $m$ this square fishnet graph is invariant under the combined action of a kinematic map and the antipode map of the Hopf algebra on multiple polylogarithms, i.e. it possesses an antipodal self-duality.

hep-th↗

arXiv2025

Aspects of canonical differential equations for Calabi-Yau geometries and beyond

We show how a method to construct canonical differential equations for multi-loop Feynman integrals recently introduced by some of the authors can be extended to cases where the associated geometry is of Calabi-Yau type and even beyond. This can be achieved by supplementing the method with information from the mixed Hodge structure of the underlying geometry. We apply these ideas to specific classes of integrals whose associated geometry is a one-parameter family of Calabi-Yau varieties, and we argue that the method can always be successfully applied to those cases. Moreover, we perform an in-depth study of the properties of the resulting canonical differential equations. In particular, we show that the resulting canonical basis is equivalent to the one obtained by an alternative method recently introduced in the literature. We apply our method to non-trivial and cutting-edge examples of Feynman integrals necessary for gravitational wave scattering, further showcasing its power and flexibility.

hep-th↗

arXiv2025

Two-loop renormalisation of quark and gluon fields in the SMEFT

We compute the contributions of CP-conserving operators in the dimension-six SMEFT to the two-loop renormalisation constants of quark and gluon fields in the $\overline{\textrm{MS}}$-scheme. Specifically, we consider the top-quark chromomagnetic operator and the triple gluon operator. We work with the background-field method, which allows us to extract the contribution of these operators to the two-loop running of the top mass and the strong coupling constant. We discuss in detail the mixing with the unphysical operators required for the renormalisation, and we present analytic results for the renormalisation constants of all relevant operators.

hep-ph↗

arXiv2025

Modular forms for three-loop banana integrals

We study periods of multi-parameter families of K3 surfaces, which are relevant to compute the maximal cuts of certain classes of Feynman integrals. We focus on their automorphic properties, and we show that generically the periods define orthogonal modular forms. Using exceptional isomorphisms between Lie groups of small rank, we show how one can use the intersection product on the periods to identify K3 surfaces whose periods can be expressed in terms of other classes of modular forms that have been studied in the mathematics literature. We apply our results to maximal cuts of three-loop banana integrals, and we show that depending on the mass configuration, the maximal cuts define ordinary modular forms or Hilbert, Siegel or hermitian modular forms.

hep-th↗

arXiv2025

Feynman integrals, elliptic integrals and two-parameter K3 surfaces

The three-loop banana integral with three equal masses and the conformal two-loop five-point traintrack integral in two dimensions are related to a two-parameter family of K3 surfaces. We compute the corresponding periods and the mirror map, and we show that they can be expressed in terms of ordinary modular forms and functions. In particular, we find that the maximal cuts of the three-loop banana integral with three equal masses can be written as a product of two copies of the maximal cuts of the two-loop equal-mass sunrise integral. Our computation reveals a hidden symmetry of the banana integral not manifest from the Feynman integral representation, which corresponds to exchanging the two copies of the sunrise elliptic curve.

hep-th↗

arXiv2025

Equivariant primitives of Eisenstein series for congruence subgroups

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. We also focus on Eisenstein series of weight two in the case where the modular curve has genus zero. We show that in those cases the non-holomorphic Eisenstein series of weight two can be written as single-valued logarithms whose argument is a rational function of the Hauptmodul.

math.NT↗

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.

hep-th↗

arXiv2024

On the electron self-energy to three loops in QED

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an $ε$-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by product, we also obtain $ε$-resummed results for the self-energy in the on-shell limit $p^2 = m^2$, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

hep-th↗

arXiv2024

Rational terms of UV origin to all loop orders

Numerical approaches to computations typically reconstruct the numerators of Feynman diagrams in four dimensions. In doing so, certain rational terms arising from the (D-4)-dimensional part of the numerator multiplying ultraviolet (UV) poles in dimensional regularisation are not captured and need to be obtained by other means. At one-loop these rational terms of UV origin can be computed from a set of process-independent Feynman rules. Recently, it was shown that this approach can be extended to two loops. In this paper, we show that to all loop orders it is possible to compute rational terms of UV origin through process-independent vertices that are polynomial in masses and momenta.

hep-ph↗

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.

hep-th↗

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.

hep-th↗

arXiv2024

Geometry from Integrability: Multi-Leg Fishnet Integrals in Two Dimensions

We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders.

hep-th↗

arXiv2024

Some conjectures around magnetic modular forms

We study a class of meromorphic modular forms characterised by Fourier coefficients that satisfy certain divisibility properties. We present new candidates for these so-called magnetic modular forms, and we conjecture properties that these functions should obey. In particular, we conjecture that magnetic modular forms are closed under the standard operators acting on spaces of modular forms (SL$_2(\mathbb{Z})$ action, Hecke and Atkin-Lehner operators), and that they are characterised by algebraic residues and vanishing period polynomials. We use our conjectures to construct examples of real-analytic modular forms with poles.

math.NT↗

arXiv2024

The Basso-Dixon Formula and Calabi-Yau Geometry

We analyse the family of Calabi-Yau varieties attached to four-point fishnet integrals in two dimensions. We find that the Picard-Fuchs operators for fishnet integrals are exterior powers of the Picard-Fuchs operators for ladder integrals. This implies that the periods of the Calabi-Yau varieties for fishnet integrals can be written as determinants of periods for ladder integrals. The representation theory of the geometric monodromy group plays an important role in this context. We then show how the determinant form of the periods immediately leads to the well-known Basso-Dixon formula for four-point fishnet integrals in two dimensions. Notably, the relation to Calabi-Yau geometry implies that the volume is also expressible via a determinant formula of Basso-Dixon type. Finally, we show how the fishnet integrals can be written in terms of iterated integrals naturally attached to the Calabi-Yau varieties.

hep-th↗

arXiv2024

Cutting-Edge Tools for Cutting Edges

We review different notions of cuts appearing throughout the literature on scattering amplitudes. Despite similar names, such as unitarity cuts or generalized cuts, they often represent distinct computations and distinct physics. We consolidate this knowledge, summarize how cuts are used in various computational strategies, and explain their relations to other quantities including imaginary parts, discontinuities, and monodromies. Differences and nuances are illustrated on explicit examples.

hep-th↗

arXiv2023

Two-loop QED corrections to the scattering of four massive leptons

We study two-loop corrections to the scattering amplitude of four massive leptons in quantum electrodynamics. These amplitudes involve previously unknown elliptic Feynman integrals, which we compute analytically using the differential equation method. In doing so, we uncover the details of the elliptic geometry underlying this scattering amplitude and show how to exploit its properties to obtain compact, easy-to-evaluate series expansions that describe the scattering of four massive leptons in QED in the kinematical regions relevant for Bhabha and Møller scattering processes.

hep-ph↗