<p>We complete the construction of canonical differential equations (DEs) for all families of Feynman integrals appearing in the two-loop leading-color QCD amplitude for top-pair production in association with a jet at hadron colliders. This allows us to obtain analytic results for all required Feynman integrals in terms of iterated integrals with closed-form kernels. To achieve this, we study the two-loop five-point integral family for which canonical DEs were previously unavailable due to the appearance of elliptic functions and nested square roots. In addition to marking a significant step toward next-to-next-to-leading-order QCD predictions for a high-priority LHC process, this is the first time that canonical DEs are obtained for Feynman integrals where elliptic functions appear in conjunction with the high algebraic complexity of a process with more than four particles. Our results also reveal a number of new interesting analytic features, such as a “duplet” structure that generalizes the even/odd parity of square roots to nested square roots.</p>

Italy

Matteo Becchetti

Germany

Christoph Dlapa

Switzerland

Simone Zoia

Canonical differential equations for the elliptic two-loop five-point integral family relevant to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mover accent="true"><mi>t</mi><mo stretchy="false">¯</mo></mover><mo>+</mo><mtext>jet</mtext></math> production at leading color

We compute a complete set of the two-loop Feynman integrals that are required for the next-to-next-to-leading order QCD corrections to on-shell top-pair production in association with a $\textit{W}$ boson at hadron colliders in the leading colour approximation. These Feynman integrals also contribute to Higgs or $\textit{Z}$-boson production in association with a top pair. We employ the method of differential equations (DEs), facilitated by the use of finite field methods to handle the algebraic complexity stemming from the seven-scale kinematics. The presence of the top quark in the virtual propagators, in addition to the mass of the external $\textit{W}$ boson, gives rise to nested square roots and three elliptic curves. We obtain DEs that depend at most quadratically on the dimensional regulator $\textit{ϵ}$ for sectors where these analytic structures appear, and are $\textit{ϵ}$-factorised otherwise. We express the DEs in terms of a minimal set of differential one-forms, separating the logarithmic ones. We solve the DEs numerically in the physical kinematic region, with the method of generalised power series expansions.

Dhimiter Canko

Vsevolod Chestnov

Tiziano Peraro

Mattia Pozzoli

Two-loop Feynman integrals for leading colour $$ t\overline{t}W $$ production at hadron colliders

We present the first benchmark evaluation of the two-loop finite remainders for the production of a top-quark pair in association with a jet at hadron colliders in the gluon channel. We work in the leading colour approximation, and perform the numerical evaluation in the physical phase space. To achieve this result, we develop a new method for expressing the master integrals in terms of a (over-complete) basis of special functions that enables the infrared and ultraviolet poles to be cancelled analytically despite the presence of elliptic Feynman integrals. The special function basis makes it manifest that the elliptic functions appear solely in the finite remainder, and can be evaluated numerically through generalised series expansions. The helicity amplitudes are constructed using four dimensional projectors combined with finite-field techniques to perform integration-by-parts reduction, mapping to special functions and Laurent expansion in the dimensional regularisation parameter.

Simon Badger

Colomba Brancaccio

South Korea

Heribertus Hartanto

Numerical evaluation of two-loop QCD helicity amplitudes for

We present the double-virtual amplitudes contributing to the production of a Higgs boson in association with a $$ b\overline{b} $$ pair at the Large Hadron Collider. We perform the computation within the five-flavour scheme, which employs massless bottom quarks and finite bottom-Yukawa coupling, taking into account all the colour structures. We derive the analytic form of the helicity amplitudes through finite-field reconstruction techniques. The analytic expressions have been implemented in a public C++ library, and we demonstrate that evaluations are sufficiently stable and efficient for use in phenomenological studies.

Poland

Rene Poncelet

China

Zihao Wu

Yang Zhang

Full-colour double-virtual amplitudes for associated production of a Higgs boson with a bottom-quark pair at the LHC

We present the two-loop helicity amplitudes contributing to the next-to-next-to-leading order QCD predictions for $\textit{W}$-boson production in association with two photons at the Large Hadron Collider. We derived compact analytic expressions for the two-loop amplitudes in the leading colour limit, and provide numerical results for the subleading colour contributions. We employ a compact system of integration-by-part identities provided by the N IBP package, allowing for an efficient computation of the rational coefficients of the scattering amplitudes over finite fields.

Two-loop amplitudes for $$ \mathcal{O}\left({\alpha}_s^2\right) $$ corrections to $\textit{Wγγ}$ production at the LHC

We consider the complete set of planar two-loop five-point Feynman integrals with two off-shell external legs. These integrals are relevant, for instance, for the calculation of the second-order QCD corrections to the production of two heavy vector bosons in association with a jet or a photon at a hadron collider. We construct pure bases for these integrals and reconstruct their analytic differential equations in canonical form through numerical sampling over finite fields. The newly identified symbol alphabet, one of the most complex to date, provides valuable data for bootstrap methods. We then apply our results to initiate the study of double Lagrangian insertions in a four-cusp Wilson loop in planar maximally supersymmetric Yang-Mills theory, computing it through two loops. We observe that it is finite, conformally invariant in four dimensions, and of uniform transcendentality. Furthermore, we provide numerical evidence for its positivity within the amplituhedron region through two loops.

CERN

United Kingdom

Samuel Abreu

France

Dmitry Chicherin

Vasily Sotnikov

Two-loop five-point two-mass planar integrals and double Lagrangian insertions in a Wilson loop

We perform an exploratory study of a new approach for evaluating Feynman integrals numerically. We apply the recently-proposed framework of physics-informed deep learning to train neural networks to approximate the solution to the differential equations satisfied by the Feynman integrals. This approach relies neither on a canonical form of the differential equations, which is often a bottleneck for the analytical techniques, nor on the availability of a large dataset, and after training yields essentially instantaneous evaluation times. We provide a proof-of-concept implementation within the PyTorch framework, and apply it to a number of one- and two-loop examples, achieving a mean magnitude of relative difference of around 1% at two loops in the physical phase space with network training times on the order of an hour on a laptop GPU.

Francesco Calisto

Ryan Moodie

Learning Feynman integrals from differential equations with neural networks

We compute the differential equations for the two remaining integral topologies contributing to the leading colour two-loop amplitudes for pp →   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mi>j</mi> </math>  $$ t\overline{t}j $$ . We derive differential equations for the master integrals by solving the integration-by-parts identities over finite fields. Of the two systems of differential equations, one is presented in canonical ‘d log’ form, while the other is found to have an elliptic sector. For the elliptic topology we identify the relevant elliptic curve, and present the differential equations in a more general form which depends quadratically on ε and contains non-logarithmic one-forms in addition to the canonical d log’s. We solve the systems of differential equations numerically using generalised series expansions with the boundary terms obtained using the auxiliary mass flow method. A summary of all one-loop and two-loop planar topologies is presented including the list of alphabet letters for the ‘d log’ form systems and high-precision boundary values.

Nicolò Giraudo

Two-loop integrals for   <math> <mi>t</mi> <mover> <mi>t</mi> <mo>¯</mo> </mover> </math>  $$ t\overline{t} $$ +jet production at hadron colliders in the leading colour approximation

We compute the complete set of two-loop master integrals for the scattering of four massless particles and a massive one. Our results are ready for phenomenological applications, removing a major obstacle to the computation of complete next-to-next-to-leading order QCD corrections to processes such as the production of a <math><mrow><mi>H</mi><mo>/</mo><mi>Z</mi><mo>/</mo><mi>W</mi></mrow></math> boson in association with two jets at the LHC. Furthermore, they open the door to new investigations into the structure of quantum-field theories and provide precious analytic data for studying the mathematical properties of Feynman integrals.

Harald Ita

Ben Page

United States

Wladimir Tschernow

All Two-Loop Feynman Integrals for Five-Point One-Mass Scattering

We compute the two-loop QED helicity amplitudes for the scattering of a lepton pair with an off-shell and an on-shell photon, 0 →   <math> <mi>ℓ</mi> <mover> <mi>ℓ</mi> <mo>¯</mo> </mover> <mi>γγ</mi> </math>  $$ \ell \overline{\ell}\gamma \gamma $$  *, using the approximation of massless leptons. We express all master integrals relevant for the scattering of four massless particles with a single external off-shell leg up to two loops in a basis of algebraically independent multiple polylogarithm, which guarantees an efficient numerical evaluation and compact analytic representations of the amplitudes. Analytic forms of the amplitudes are reconstructed from numerical evaluations over finite fields. Our results complete the amplitude-level ingredients contributing to the N3LO predictions of electron-muon scattering eμ → eμ, which are required to meet the precision goal of the future MUonE experiment.

Jakub Kryś

Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED