<p>We present a new class of evolution equations that govern the high-energy behavior of power-suppressed scattering amplitudes. The equations can be viewed as a renormalization group flow with respect to the relevant effective field theory cutoff. A distinct feature of the method is in the use of a multidimensional cutoff to separate the relevant scales in problems characterized by a complex factorization structure. By adjusting the renormalization group variables to the geometry of the effective theory modes, our method naturally extends to a broad spectrum of physical problems that include massive, massless, small, and wide-angle scattering. We present applications to the benchmark processes of electron-positron forward annihilation and light-quark-mediated Higgs boson production and decays.</p>

CERN

Maximilian Delto

Canada

Alexander Penin

Germany

Lorenzo Tancredi

High-Energy Evolution of Power-Suppressed Amplitudes

In this paper, we present analytical results for the two-loop QCD corrections to the production of two partons or a photon and a parton in hadronic collisions, mediated by loops of massive quarks. These amplitudes involve Feynman integrals defined on an elliptic curve. We compute them by generalizing our recent results for the production of two photons to include additional crossings of the corresponding master integrals, which we compute in terms of the same basis of independent iterated integrals. We discuss the analytical properties of the amplitudes, highlighting the cancellations of a large number of elliptic differential forms in their finite remainders. Finally, we elaborate on a strategy for their numerical evaluation based on generalized series expansions at singular points of the physical amplitude, through the introduction of suitable sets of variables that allow us to resolve all singularities.

Belgium

Federico Coro

Christoph Nega

Fabian Wagner

Analytic two-loop amplitudes for di-jet and $\textit{γ}$+jet production mediated by a heavy-quark loop

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

Sara Ditsch

Philipp Kreer

Mattia Pozzoli

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

Benjamin Sauer

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.

Spain

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

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