We present a computation of the one-loop QCD corrections to top-quark pair production in association with a $\textit{W}$ boson, including terms up to order $\textit{ε}$ in dimensional regularization. Providing a first glimpse into the complexity of the corresponding two-loop amplitude, this result is a first step towards a description of this process at next-to-next-to-leading order (NNLO) in QCD. We perform a tensor decomposition and express the corresponding form factors in terms of a basis of independent special functions with compact rational coefficients, providing a structured framework for future developments. In addition, we derive an explicit analytic representation of the form factors, valid up to order $\textit{ε}$ , expressed in terms of logarithms and dilogarithms. For the complete set of special functions required, we obtain a semi-numerical solution based on generalized power series expansion.

Italy

Matteo Becchetti

Switzerland

Germany

Maximilian Delto

Sara Ditsch

Philipp Kreer

Mattia Pozzoli

Lorenzo Tancredi

One-loop QCD corrections to $\textit{ūd}$ → $$ t\overline{t}W $$ at $$ \mathcal{O}\left({\varepsilon}^2\right) $$

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.

Claude Duhr

Sara Maggio

Christoph Nega

Benjamin Sauer

Fabian Wagner

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

We address the analytic computation of the two-loop scattering amplitudes for the production of two photons in parton-parton scattering, mediated by loops of heavy quarks. Due to the presence of integrals of elliptic type, both partonic channels have been previously computed using semi-numerical methods. In this paper, leveraging new advances in the theory of differential equations for elliptic Feynman integrals, we derive a canonical basis for all integrals involved and compute them in terms of independent iterated integrals over elliptic and polylogarithmic differential forms. We use this representation to showcase interesting cancellations in the physical expressions for the scattering amplitudes. Furthermore, we address their numerical evaluation by producing series expansion representations for the whole amplitudes, which we demonstrate to be fast and numerically reliable across a large region of the phase space.

Belgium

Spain

Federico Coro

Analytic two-loop amplitudes for $$ q\overline{q}\to \gamma \gamma $$ and $\textit{gg}$ → $\textit{γγ}$ mediated by a heavy-quark loop

We compute the photon self-energy to three loops in Quantum Electrodynamics. The method of differential equations for Feynman integrals and a complete -factorization of the former allow us to obtain fully analytical results in terms of iterated integrals involving integration kernels related to a K3 geometry. We argue that our basis has the right properties to be a natural generalization of a canonical basis beyond the polylogarithmic case and we show that many of the kernels appearing in the differential equations, cancel out in the final result to finite order in . We further provide generalized series expansions that cover the whole kinematic space so that our results for the self-energy may be easily evaluated numerically for all values of the momentum squared. From the local solution at = 0, we extract the photon wave function renormalization constant in the on-shell scheme to three loops and confirm its agreement with previously obtained results.

Felix Forner

On the photon self-energy to three loops in QED

We describe an algorithm to organize Feynman integrals in terms of their infrared properties. Our approach builds upon the theory of Landau singularities, which we use to classify all configurations of loop momenta that can give rise to infrared divergences. We then construct bases of numerators for arbitrary Feynman integrals, which cancel all singularities and render the integrals finite. Through the same analysis, one can also classify so-called evanescent and evanescently finite Feynman integrals. These are integrals whose vanishing or finiteness relies on properties of dimensional regularization. To illustrate the use of these integrals, we display how to obtain a simpler form for the leading-color two-loop four-gluon scattering amplitude through the choice of a suitable basis of finite integrals. In particular, when all gluon helicities are equal, we show that with our basis the most complicated double-box integrals do not contribute to the finite remainder of the scattering amplitude.

United Kingdom

Giulio Gambuti

France

CERN

David A. Kosower

Pavel P. Novichkov

Finite Feynman integrals

Several recent works have demonstrated the powerful algebraic simplifications that can be achieved for scattering amplitudes through a systematic grading of transcendental quantities. We develop these concepts to construct a minimal basis of functions tailored to a scattering amplitude in a general way. Starting with formal solutions for all master integral topologies, we organise the appearing functions by properties such as their symbol alphabet or letter adjacency. We rotate the basis such that functions with spurious features appear in the least possible number of basis elements. Since their coefficients must vanish for physical quantities, this approach avoids complex cancellations. As a first application, we evaluate all integral topologies relevant to the three-loop $\textit{Hggg}$ and $$ Hgq\overline{q} $$ amplitudes in the leading-colour approximation and heavy-top limit. We describe the derivation of canonical differential equation systems and present a method for fixing boundary conditions without the need for a full functional representation. Using multiple numerical reductions, we test the maximal transcendentality conjecture for Hggg and identify a new letter which appears in functions of weight 4 and 5. In addition, we provide the first direct analytic computation of a three-point form factor of the operator Tr($\textit{ϕ}$ ) in planar $$ \mathcal{N} $$ = 4 sYM and find agreement with numerical and bootstrapped results.

Thomas Gehrmann

Johannes Henn

Petr Jakubčík

Jungwon Lim

Cesare Mella

Nikolaos Syrrakos

William Torres Bobadilla

Graded transcendental functions: an application to four-point amplitudes with one off-shell leg

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 $\textit{ϵ}$-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 $\textit{ϵ}$-resummed results for the self-energy in the on-shell limit $\textit{p}$ = $\textit{m}$ , which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Federico Gasparotto

Stefan Weinzierl

On the electron self-energy to three loops in QED

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.

Yu Jiao Zhu

Two-Loop QED Corrections to the Scattering of Four Massive Leptons

We compute all helicity amplitudes for the scattering of five partons in two-loop QCD in all the relevant flavor configurations, retaining all contributing color structures. We employ tensor projection to obtain helicity amplitudes in the ’t Hooft-Veltman scheme starting from a set of primitive amplitudes. Our analytic results are expressed in terms of massless pentagon functions, and are easy to evaluate numerically. These amplitudes provide important input to investigations of soft-collinear factorization and to studies of the high-energy limit.

Bakul Agarwal

Federico Buccioni

Federica Devoto

United States

Andreas von Manteuffel

Five-parton scattering in QCD at two loops

The ATLAS and CMS collaborations at the LHC have recently announced evidence for the rare Higgs boson decay into a Z boson and a photon. We analyze the interference between the process <math><mi>g</mi><mi>g</mi><mo>→</mo><mi>H</mi><mo>→</mo><mi>Z</mi><mi>γ</mi></math> induced by loops of heavy particles, which is by far the dominant contribution to the signal, and the continuum <math><mi>g</mi><mi>g</mi><mo>→</mo><mi>Z</mi><mi>γ</mi></math> QCD background process mediated by light quark loops. This interference modifies the event yield, the resonance line-shape and the apparent mass of the Higgs boson. We calculate the radiative corrections to this interference beyond the leading-order approximation in perturbative QCD and find that, while differing numerically from the corresponding effects on the more studied <math><mi>g</mi><mi>g</mi><mo>→</mo><mi>γ</mi><mi>γ</mi></math> signal, they are generally rather small. As such, they do not impact significantly the interpretation of the present measurements of the <math><mi>H</mi><mo>→</mo><mi>Z</mi><mi>γ</mi></math> decay mode.

Abdelhak Djouadi

John Ellis

Jérémie Quevillon

Interference effects in → → Beyond Leading Order