arXiv subjects

Johannes M. Henn

Publications and source records attributed to Johannes M. Henn.

At least 19 recordsLinked to original sources

arXiv2026

Bootstrapping two-loop six-gluon amplitudes in QCD

The maximally transcendental, or most complicated, terms of gauge-theory scattering amplitudes have long been singled out, following Lipatov and collaborators, as those parts of a QCD amplitude that most closely mirror maximally supersymmetric Yang--Mills theory. We report on a programme that turns this observation into a practical computational tool. We show that the rational prefactors multiplying the highest-weight special functions of planar QCD amplitudes are governed by four-dimensional leading singularities, which can be classified and evaluated using on-shell diagrams. The resulting prefactors are manifestly conformally invariant and admit compact spinor-helicity expressions that hold for arbitrary multiplicity. Combining this input with the recently established two-loop six-particle function space, we set up a symbol bootstrap and determine, for the first time, the maximal-weight symbol of the planar two-loop six-gluon amplitude in massless QCD, first for the ${-}{-}{+}{+}{+}{+}$ helicity configuration and subsequently for all MHV configurations. The answer is fixed uniquely by physical consistency conditions, requires a reduced alphabet of only $137$ symbol letters, and yields as a byproduct previously unknown two-loop triple-collinear and double-soft splitting functions. We summarise the method and the results, and outline the directions they open up.

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arXiv2026

Approximating Feynman integrals using complete monotonicity and Stieltjes properties

We present two novel approaches for the numerical evaluation of Feynman integrals based on their universal analytic properties related to positivity, namely complete monotonicity (CM) and Stieltjes properties. Building on recent results, we exploit the fact that scalar Feynman integrals in the Euclidean region are completely monotonic functions, meaning that all their derivatives have a fixed sign. Building on this observation, the CM bootstrap allows one to reconstruct integrals from differential equations without explicit boundary data, yielding rigorous bounds. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, are not only CM but in fact Stieltjes functions. This enables the use of Padé approximants with provable convergence properties in the cut complex plane, providing an efficient method for analytic continuation and fast numerical evaluation. We illustrate the method with simple examples such as the massive bubble integral and discuss applications to multi-loop integrals, including the 20-loop banana integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

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arXiv2026

Approximating Feynman Integrals Using Complete Monotonicity and Stieltjes Properties

We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives have a fixed sign, in the Euclidean kinematic region. This imposes strong constraints on the function space. Simultaneously, these integrals obey systems of linear differential equations with respect to kinematic parameters. By imposing that the solutions to these differential equations satisfy complete monotonicity across the Euclidean region, we develop an efficient and highly constraining numerical bootstrap method. We provide a proof of principle of the power of our approach by applying it to a class of multi-loop Feynman integrals with internal masses. The second method is based on a refinement of CM. We prove that Feynman integrals, within a certain range of parameters, such as dimension and propagator exponents, are not only CM but in fact Stieltjes functions. The latter can be described efficiently by Padé approximants that are known to converge in the cut complex plane. This means that these representations are valid also in analytically continued kinematics, such as physical scattering regions. These insights allow us to obtain rational approximations to Feynman integrals from minimal information, such as a Taylor expansion about a soft limit. We demonstrate the effectiveness of this method by applying it to a 20-loop banana-type Feynman integral. Finally, we comment on a number of extensions of these novel avenues for computing Feynman integrals.

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arXiv2025

All-loop Leading Singularities of Wilson Loops

We study correlators of null, $n$-sided polygonal Wilson loops with a Lagrangian insertion in the planar limit of the ${\cal N}=4$ supersymmetric Yang-Mills theory. This finite observable is closely related to loop integrands of maximally-helicity-violating amplitudes in the same theory, and, conjecturally, to all-plus helicity amplitudes in pure Yang-Mills theory. The resulting function has been observed to have an expansion in terms of functions of uniform transcendental weight, multiplied by certain rational prefactors, called leading singularities. In this work we prove several conjectures about the leading singularities: we classify and compute them at any loop order and for any number of edges of the Wilson loop, and show that they have a hidden conformal symmetry. This is achieved by leveraging the geometric definition of the loop integrand via the Amplituhedron. The leading singularities can be seen as maximal codimension residues of the integrand, and the boundary structure of the Amplituhedron geometry restricts which iterative residues are accessible. Combining this idea with a further geometric decomposition of the Amplituhedron in terms of so-called negative geometries allows us to identify the complete set of leading singularities.

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arXiv2024

Analytic evaluation of the three-loop three-point form factor of $\operatorname{tr}ϕ^3$ in $\mathcal{N}=4$ sYM

We compute analytically the three-loop correlation function of the local operator $\text{tr} \, ϕ^3$ inserted into three on-shell states, in maximally supersymmetric Yang-Mills theory. The result is expressed in terms of Chen iterated integrals. We also present our result using generalised polylogarithms, and evaluate them numerically, finding agreement with a previous numerical result in the literature. We observe that the result depends on fewer kinematic singularities compared to individual Feynman integrals. Furthermore, upon choosing a suitable definition of the finite part, we find that the latter satisfies powerful symbol adjacency relations similar to those previously observed for the $\text{tr} \, ϕ^2$ case.

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arXiv2024

A computation of two-loop six-point Feynman integrals in dimensional regularization

We compute three families of two-loop six-point massless Feynman integrals in dimensional regularization, namely the double-box, the pentagon-triangle, and the hegaxon-bubble family. This constitutes the first analytic computation of two-loop master integrals with eight scales. We use the method of canonical differential equations. We describe the corresponding integral basis with uniform transcendentality, the relevant function alphabet, and analytic boundary values at a particular point in the Euclidean region up to the fourth order in the regularization parameter $ε$. The results are expressed as one-fold integrals over classical polylogarithms suitable for fast and high-precision evaluation.

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arXiv2024

Integrated negative geometries in ABJM

We study, in the context of the three-dimensional ${\cal N}=6$ Chern-Simons-matter (ABJM) theory, the infrared-finite functions that result from performing $L-1$ loop integrations over the $L$-loop integrand of the logarithm of the four-particle scattering amplitude. Our starting point are the integrands obtained from the recently proposed all-loop projected amplituhedron for the ABJM theory. Organizing them in terms of negative geometries ensures that no divergences occur upon integration if at least one loop variable is left unintegrated. We explicitly perform the integrations up to $L=3$, finding both parity-even and -odd terms. Moreover, we discuss a prescription to compute the cusp anomalous dimension $Γ_{\rm cusp}$ of ABJM in terms of the integrated negative geometries, and we use it to reproduce the first non-trivial order of $Γ_{\rm cusp}$. Finally, we show that the leading singularities that characterize the integrated results are conformally invariant.

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arXiv2023

Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry

The Amplituhedron provides, via geometric means, the all-loop integrand of scattering amplitudes in maximally supersymmetric Yang-Mills theory. Unfortunately, dimensional regularization, used conventionally for integration, breaks the beautiful geometric picture. This motivates us to propose a 'deformed' Amplituhedron. Focusing on the four-particle amplitude, we introduce two deformation parameters, which can be interpreted as particle masses. We provide evidence that the mass pattern corresponds to a specific choice of vacuum expectation values on the Coulomb branch. The deformed amplitude is infrared finite, making the answer well-defined in four dimensions. Leveraging four-dimensional integration techniques based on differential equations, we compute the amplitude up to two loops. In the limit where the deformation parameters are taken to zero, we recover the known Bern-Dixon-Smirnov amplitude. In the limit where only one deformation parameter is taken to zero, we find a connection to the angle-dependent cusp anomalous dimension.

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arXiv2023

A new method for calculating the soft anomalous dimension matrix for massive particle scattering

The general structure of infrared divergences in the scattering of massive particles is captured by the soft anomalous dimension matrix. The latter can be computed from a correlation function of multiple Wilson lines. The state-of-the-art two-loop result has a tantalizingly simple structure that is not manifest in the calculations. We argue that the complexity in intermediate steps of the known calculations comes from spurious, regulator-dependent terms. Based on this insight we propose a different infrared regulator that is associated to only one of the Wilson lines. We demonstrate that this streamlines obtaining the two-loop result: computing the required Feynman integrals via the differential equations method, only multiple polylogarithmic functions appear (to all orders in the dimensional regulator), as opposed to elliptic polylogarithms. We show that the new method is promising for higher-loop applications by computing a three-loop diagram of genuine complexity, and provide the answer in terms of multiple polylogarithms. The relatively simple symbol alphabet we obtain may be of interest for bootstrap approaches.

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arXiv2023

An algorithmic approach to finding canonical differential equations for elliptic Feynman integrals

In recent years, differential equations have become the method of choice to compute multi-loop Feynman integrals. Whenever they can be cast into canonical form, their solution in terms of special functions is straightforward. Recently, progress has been made in understanding the precise canonical form for Feynman integrals involving elliptic polylogarithms. In this article, we make use of an algorithmic approach that proves powerful to find canonical forms for these cases. To illustrate the method, we reproduce several known canonical forms from the literature and present examples where a canonical form is deduced for the first time. Together with this article, we also release an update for INITIAL, a publicly available Mathematica implementation of the algorithm.

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arXiv2023

First look at the evaluation of three-loop non-planar Feynman diagrams for Higgs plus jet production

We present new computations for Feynman integrals relevant to Higgs plus jet production at three loops, including first results for a non-planar class of integrals. The results are expressed in terms of generalised polylogarithms up to transcendental weight six. We also provide the full canonical differential equations, which allows us to make structural observations on the answer. In particular, we find a counterexample to previously conjectured adjacency relations, for a planar integral of the tennis-court type. Additionally, for a non-planar triple ladder diagram, we find two novel alphabet letters. This information may be useful for future bootstrap approaches.

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arXiv2022

One-loop hexagon integral to higher orders in the dimensional regulator

The state-of-the-art in current two-loop QCD amplitude calculations is at five-particle scattering. Computing two-loop six-particle processes requires knowledge of the corresponding one-loop amplitudes to higher orders in the dimensional regulator. In this paper we compute analytically the one-loop hexagon integral via differential equations. In particular we identify its function alphabet for general $D$-dimensional external states. We also provide integral representations for all one-loop integrals up to weight four. With this, the one-loop integral basis is ready for two-loop amplitude applications. We also study in detail the difference between the conventional dimensional regularization and the four-dimensional helicity scheme at the level of the master integrals and their function space.

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arXiv2022

Symmetry properties of Wilson loops with a Lagrangian insertion

Null Wilson loops in $\mathcal{N}=4$ super Yang-Mills are dual to planar scattering amplitudes. This duality implies hidden symmetries for both objects. We consider closely related infrared finite observables, defined as the Wilson loop with a Lagrangian insertion, normalized by the Wilson loop itself. Unlike ratio and remainder functions studied in the literature, this observable is non-trivial already for four scattered particles and bears close resemblance to (finite parts of) scattering processes in non-supersymmetric Yang-Mills theory. We study the general structure of the Wilson loop with a Lagrangian insertion, focusing in particular on its leading singularities and their (hidden) symmetry properties. We find evidence that the leading singularities can be written as certain Grassmannian integrals. The latter are manifestly dual conformal. They also have a conformal symmetry, up to total derivatives. We find that, surprisingly, the conformal symmetry becomes an invariance in the frame where the Lagrangian insertion point is sent to infinity. Furthermore, we use integrability methods to study how higher Yangian charges act on the Grassmannian integral. We evaluate the $n$-particle observable both at tree- and at one-loop level, finding compact analytic formulas. These results are explicitly written in the form of conformal leading singularities, multiplied by transcendental functions. We then compare these formulas to known expressions for all-plus amplitudes in pure Yang-Mills theory. We find a remarkable new connection: the Wilson loop with Lagrangian insertion in $\mathcal{N}=4$ super Yang-Mills appears to predict the maximal weight terms of the planar pure Yang-Mills all-plus amplitude. We test this relationship for the two-loop $n$-point Yang-Mills amplitude, as well as for the three-loop four-point amplitude.

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arXiv2021

Maximal transcendental weight contribution of scattering amplitudes

Feynman integrals in quantum field theory evaluate to special functions and numbers that are usefully described by the notion of transcendental weight. In this paper, we propose a way of projecting a given dimensionally-regularised Feynman integral, for example contributing to a scattering amplitudes, onto its maximal weight part. The method uses insights into the singularity structure of space-time loop integrands, and is complementary to usual generalised unitarity approaches. We describe the method and give a proof-of-principle application to the two-loop scattering amplitudes $gg \to H$ in the heavy top-quark mass limit, which involves both planar and non-planar Feynman integrals. We also comment on further possible applications and discuss subtleties related to evanescent integrand terms.

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arXiv2021

What can we learn about QCD and collider physics from N=4 super Yang-Mills?

Tremendous ongoing theory efforts are dedicated to developing new methods for QCD calculations. Qualitative rather than incremental advances are needed to fully exploit data still to be collected at the LHC. The maximally supersymmetric Yang-Mills theory (${\mathcal N}=4$ sYM) shares with QCD the gluon sector, which contains the most complicated Feynman graphs, but at the same time has many special properties, and is believed to be solvable exactly. It is natural to ask what we can learn from advances in ${\mathcal N}=4$ sYM for addressing difficult problems in QCD. With this in mind, we review here several remarkable developments and highlights of recent results in ${\mathcal N}=4$ sYM. This includes all-order results for certain scattering amplitudes, novel symmetries, surprising geometrical structures of loop integrands, novel tools for the calculation of Feynman integrals, and bootstrap methods. While several insights and tools have already been carried over to QCD and have contributed to state-of-the-art calculations for LHC physics, we argue that there is a host of further fascinating ideas waiting to be explored.

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arXiv2021

Cluster algebras for Feynman integrals

We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster adjacency relations that restrict the allowed function space. By embedding $C_{2}$ inside the $A_3$ cluster algebra, we identify these adjacencies with the extended Steinmann relations for six-particle massless scattering. The cluster algebra connection we find restricts the functions space for vector boson or Higgs plus jet amplitudes, and for form factors recently considered in $\mathcal{N}=4$ super Yang-Mills. We explain general procedures for studying relationships between alphabets of generalized polylogarithmic functions and cluster algebras, and use them to provide various identifications of one-loop alphabets with cluster algebras. In particular, we show how one can obtain one-loop alphabets for five-particle scattering from a recently discussed dual conformal eight-particle alphabet related to the $G(4,8)$ cluster algebra.

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arXiv2020

The full angle-dependence of the four-loop cusp anomalous dimension in QED

The angle-dependent cusp anomalous dimension governs divergences coming from soft gluon exchanges between heavy particles, such as top quarks. We focus on the matter-dependent contributions and compute the first truly non-planar terms. They appear at four loops and are proportional to a quartic Casimir operator in color space. Specializing our general gauge theory result to U(1), we obtain the full QED four-loop angle-dependent cusp anomalous dimension. While more complicated functions appear at intermediate steps, the analytic answer depends only on multiple polylogarithms with singularities at fourth roots of unity. It can be written in terms of four rational structures, and contains functions of up to maximal transcendental weight seven. Despite this complexity, we find that numerically the answer is tantalizingly close to the appropriately rescaled one-loop formula, over most of the kinematic range. We take several limits of our analytic result, which serves as a check and allows us to obtain new, power-suppressed terms. In the anti-parallel lines limit, which corresponds to production of two massive particles at threshold, we find that the subleading power correction vanishes. Finally, we compute the quartic Casimir contribution for scalars in the loop. Taking into account a supersymmetric decomposition, we derive the first non-planar corrections to the quark anti-quark potential in maximally supersymmetric gauge theory.

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arXiv2019

The full four-loop cusp anomalous dimension in $\mathcal{N}=4$ super Yang-Mills and QCD

We present the complete formula for the cusp anomalous dimension at four loops in QCD and in maximally supersymmetric Yang-Mills. In the latter theory it is given by \begin{equation} Γ^{\rm}_{\rm cusp}\Big|_{α_s^4} = -\left( \frac{α_s N}π\right)^4 \left[ \frac{73 π^6}{20160} + \frac{ ζ_{3}^2}{8} + \frac{1}{N^2} \left( \frac{31π^6}{5040} + \frac{9 ζ_3^2}{4} \right) \right] \,.\nonumber \end{equation} Our approach is based on computing the correlation function of a rectangular light-like Wilson loop with a Lagrangian insertion, normalized by the expectation value of the Wilson loop. In maximally supersymmetric Yang-Mills theory, this ratio is a finite function of a cross-ratio and the coupling constant. We compute it to three loops, including the full colour dependence. Integrating over the position of the Lagrangian insertion gives the four-loop Wilson loop. We extract its leading divergence, which determines the four-loop cusp anomalous dimension. Finally, we employ a supersymmetric decomposition to derive the last missing ingredient in the corresponding QCD result.

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