We present compact two-loop QCD corrections in the leading-color approximation for the production of an electroweak vector boson, $\textit{V}$ = {$\textit{W}$ , $\textit{Z}$, $\textit{γ}$ }, in association with two light jets ($\textit{Vjj}$) at hadron colliders. Leptonic decays of the electroweak boson are included at the amplitude level. Working in the analytic-reconstruction approach, we develop two techniques to build compact partial-fraction forms for individual rational functions. One approach exploits their analytic structure. In the other, we iteratively construct subtraction terms that match the rational functions in singular limits. Moreover, we show how the singular behavior of the rational functions can be systematically used to find a more compact basis of the space that they span. We apply our techniques to the $\textit{Vjj}$ amplitudes, yielding a representation that is three orders of magnitude smaller than previous results. We then use these compact expressions to provide an efficient and stable C++ numerical implementation suitable for phenomenological applications.

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Giuseppe Laurentis

Switzerland

Harald Ita

Belgium

Ben Page

Vasily Sotnikov

Compact two-loop QCD corrections for $\textit{Vjj}$ production in proton collisions

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

Simone Zoia

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

We compute the two-loop helicity amplitudes for the scattering of five gluons, including all contributions beyond the leading-color approximation. The analytic expressions are represented as linear combinations of transcendental functions with rational coefficients, which we reconstruct from finite-field samples obtained with the numerical unitarity method. Guided by the requirement of removing unphysical singularities, we find a remarkably compact generating set of rational coefficients, which we display entirely in the manuscript. We implement our results in a public code, which provides efficient and reliable numerical evaluations for phenomenological applications.

Giuseppe De Laurentis

Germany

Maximillian Klinkert

Double-virtual NNLO QCD corrections for five-parton scattering. I. The gluon channel

We complete the computation of two-loop helicity amplitudes required to obtain next-to-next-to-leading order QCD corrections for three-jet production at hadron colliders, including all contributions beyond the leading-color approximation. The analytic expressions are reconstructed from finite-field samples obtained with the numerical unitarity method. We find that the reconstruction is significantly facilitated by exploiting the overlaps between rational coefficient functions of quark and gluon processes, and we display their compact generating sets. We implement our results in a public code, and demonstrate its suitability for phenomenological applications.

Double-virtual NNLO QCD corrections for five-parton scattering. II. The quark channels

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.

United States

Wladimir Tschernow

Italy

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

The scale evolution of parton distributions is determined by universal splitting functions. As a milestone towards the computation of these functions to four-loop order in QCD, we compute all contributions to the pure-singlet quark-quark splitting functions that involve two closed fermion loops. The splitting functions are extracted from the pole terms of off-shell operator matrix elements, and the workflow for their calculation is outlined. We reproduce known results for the non-singlet four-loop splitting functions and validate our new pure-singlet results against fixed Mellin moments.

Thomas Gehrmann

Andreas Manteuffel

Tong-Zhi Yang

Complete   <math> <msubsup> <mi>N</mi> <mi>f</mi> <mn>2</mn> </msubsup> </math>  $$ {N}_f^2 $$  contributions to four-loop pure-singlet splitting functions

We report a new result for the <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mspace width="0.2em"></mspace><msubsup><mrow><mi>C</mi></mrow><mrow><mi>F</mi></mrow><mrow><mn>3</mn></mrow></msubsup></math> contribution to the four-loop anomalous dimensions of non-singlet, twist-two operators in Quantum Chromodynamics. This result is obtained through computations of off-shell operator matrix elements. Employing integration-by-parts reductions and differential equations with respect to a tracing parameter allowed us to derive analytic results valid for arbitrary Mellin moment n.

Andreas von Manteuffel

The Contribution to the Non-singlet Splitting Function at Four-loop Order

We present analytic results for all planar two-loop Feynman integrals contributing to five-particle scattering amplitudes with one external massive leg. We express the integrals in terms of a basis of algebraically-independent transcendental functions, which we call one-mass pentagon functions. We construct them by using the properties of iterated integrals with logarithmic kernels. The pentagon functions are manifestly free of unphysical branch cuts, do not require analytic continuation, and can be readily evaluated over the whole physical phase space of the massive-particle production channel. We develop an efficient algorithm for their numerical evaluation and present a public implementation suitable for direct phenomenological applications.

Pentagon functions for one-mass planar scattering amplitudes

We present next-to-next-to-leading-order (NNLO) QCD corrections to the production of three isolated photons in hadronic collisions at the fully differential level. We employ <math><msub><mrow><mi>q</mi></mrow><mrow><mi>T</mi></mrow></msub></math> subtraction within Matrix and an efficient implementation of analytic two-loop amplitudes in the leading-colour approximation to achieve the first on-the-fly calculation for this process at NNLO accuracy. Numerical results are presented for proton–proton collisions at energies ranging from 7 TeV to 100 TeV. We find full agreement with the 8 TeV results of Ref. [1] and confirm that NNLO corrections are indispensable to describe ATLAS 8 TeV data. In addition, we demonstrate the significance of NNLO corrections for future precision studies of triphoton production at higher collision energies.