We compute all planar two-loop six-point Feynman integrals entering scattering observables in massless gauge theories such as QCD. A central result of this paper is the formulation of the differential-equations method under the algebraic constraints stemming from four-dimensional kinematics, which in this case leaves only 8 independent scales. We show that these constraints imply that one must compute topologies with only up to 8 propagators, instead of the expected 9. This leads to the decoupling of entire classes of integrals that do not contribute to scattering amplitudes in four dimensional gauge theories. We construct a pure basis and derive their canonical differential equations, of which we discuss the numerical solution. This work marks an important step towards the calculation of massless 2 → 4 scattering processes at two loops.

Switzerland

not found

Samuel Abreu

Pier Monni

Belgium

Ben Page

Germany

Johann Usovitsch

Planar six-point Feynman integrals for four-dimensional gauge theories

High-energy factorization of 2 → 2 amplitudes in QCD has been recently pushed to the next-to-next-to-leading logarithmic order by determining the three-loop gluon Regge trajectory. This was based on computing multi-Reggeon exchanges using rapidity evolution in the shock-wave formalism, and disentangling between the Regge pole and Regge cut contributions. In the present paper we extend the relevant theoretical framework to 2 → 3 processes, and compute all multi-Reggeon exchanges necessary for extracting the two-loop Reggeon-gluon-Reggeon Lipatov vertex from 2 → 3 amplitudes. Then, specializing general amplitude methods to multi-Regge kinematics, we derive analytic expressions for non-planar two-loop $\textit{gg}$ → $\textit{ggg}$, $\textit{gq}$ → $\textit{ggq}$ and $\textit{qq}$ → $\textit{qgq}$ QCD amplitudes in that limit. Matching these to the multi-Reggeon computation, we determine the QCD Lipatov vertex in dimensional regularization at two loops through finite terms. We also determine the one-loop vertex through $$ \mathcal{O} $$ ($\textit{ϵ}$ ). All results are expressed in a compact form in terms of a basis of single-valued generalised polylogarithms, manifesting target-projectile symmetry and reality properties. Furthermore, our basis of functions is explicitly finite in the soft limit, featuring delicate cancellation of spurious rational poles by transcendental functions. Agreement between all three partonic channels, as well agreement of the maximal weight contributions with the super Yang-Mills Lipatov vertex provide robust checks of the result.

Giuseppe Laurentis

Italy

Giulio Falcioni

Einan Gardi

Calum Milloy

Leonardo Vernazza

The two-loop Lipatov vertex in QCD

We revisit the calculation of the soft gluon emission probability off a colour-singlet $$ q\overline{q} $$ system that evolves in a quark-gluon plasma. The $$ q\overline{q} $$ antenna is created in the presence of a medium and then emits a soft gluon outside. The gluon emission probability is modified with respect to the vacuum baseline due to interactions with the medium during the formation of the antenna and its propagation. Previous studies disregarded the former effect and found that the medium modification to the interference pattern of the antenna was controlled by the so-called critical angle , that exclusively depends on medium properties. We find that accounting for medium interactions during the antenna formation enhances the total rate of emissions off the $$ q\overline{q} $$ antenna. Interestingly, it also promotes the notion of a critical angle to a dynamic quantity, denoted $$ {\overset{\sim }{\theta}}_c $$ , that depends on both the medium and the antenna properties and is thus different for every splitting. As a consequence, depending on the region of parameter space, colour decoherence can either be delayed or accelerated with respect to previous estimates.

Spain

Xoán Mayo López

Portugal

Guilherme Milhano

Alba Soto-Ontoso

A generalized picture of colour decoherence in dense QCD media

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

France

Dmitry Chicherin

Vasily Sotnikov

Simone Zoia

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

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

United States

Wladimir Tschernow

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

We compute the complete set of two-loop beam functions for the transverse momentum distribution of the leading jet produced in association with an arbitrary colour-singlet system. Our results constitute the last missing ingredient for the calculation of the jet-vetoed cross section at small veto scales at the next-to-next-to-leading order, as well as an important ingredient for its resummation to next-to-next-to-next-to-leading logarithmic order. Our calculation is performed in the soft-collinear effective theory framework with a suitable regularisation of the rapidity divergences occurring in the phase-space integrals. We discuss the occurrence of soft-collinear mixing terms that might violate the factorisation theorem, and demonstrate that they are naturally absorbed into the beam functions at two loops in the exponential rapidity regularisation scheme when performing a multipole expansion of the measurement function. As in our recent computation of the two-loop soft function, we present the results as a Laurent expansion in the jet radius R. We provide analytic expressions for all flavour channels in x space with the exception of a set of R-independent non-logarithmic terms that are given as numerical grids. We also perform a fully numerical calculation with exact R dependence, and find that it agrees with our analytic expansion at the permyriad level or better. Our calculation allows us to define a next-to-next-to-leading order slicing method using the leading-jet p  T  as a slicing variable. As a check of our results, we carry out a calculation of the Higgs and Z boson total production cross sections at the next-to-next-to-leading order in QCD.

Jonathan Gaunt

Luca Rottoli

Robert Szafron

Quark and gluon two-loop beam functions for leading-jet p  T  and slicing at NNLO

We present the analytic expressions for the two-loop form factors for the production or decay of pseudo-scalar quarkonia, in a scheme where the quarks are produced at threshold. We consider the two-loop amplitude for the process   <math> <mi>γγ</mi> <mo>↔</mo> <mmultiscripts> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>1</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>1</mn> </mmultiscripts> </math>  $$ \gamma \gamma \leftrightarrow {}^1{S}_0^{\left[1\right]} $$ , that was previously known only numerically, as well as for the processes   <math> <mi>gg</mi> <mo>↔</mo> <mmultiscripts> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>1</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>1</mn> </mmultiscripts> </math>  $$ gg\leftrightarrow {}^1{S}_0^{\left[1\right]} $$ ,   <math> <mi>γg</mi> <mo>↔</mo> <mmultiscripts> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>8</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>1</mn> </mmultiscripts> </math>  $$ \gamma g\leftrightarrow {}^1{S}_0^{\left[8\right]} $$  and   <math> <mi>gg</mi> <mo>↔</mo> <mmultiscripts> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>8</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>1</mn> </mmultiscripts> </math>  $$ gg\leftrightarrow {}^1{S}_0^{\left[8\right]} $$ , which have not been computed before. The two-loop corrections to   <math> <mi>gg</mi> <mo>↔</mo> <mmultiscripts> <msubsup> <mi>S</mi> <mn>0</mn> <mfenced> <mn>1</mn> </mfenced> </msubsup> <mprescripts></mprescripts> <none></none> <mn>1</mn> </mmultiscripts> </math>  $$ gg\leftrightarrow {}^1{S}_0^{\left[1\right]} $$  are the last missing ingredients for a full NNLO calculation of η  Q  hadro-production. We discuss how the singularity structure of the amplitudes is affected by the threshold kinematics, which in particular introduces Coulomb singularities. In this context, we first show how the usual structure of the infrared singularities degenerates at threshold kinematics, and then extract the anomalous dimensions governing the Coulomb singularities for colour-singlet and octet channels, the latter being presented here for the first time. We give high-precision numerical results for the hard functions, which can be used for phenomenological studies of η  Q  production and decay at NNLO.

Matteo Becchetti

Claude Duhr

Melih Ozcelik

Two-loop form factors for pseudo-scalar quarkonium production and decay

We compute the master integrals relevant for the two-loop corrections to pseudo-scalar quarkonium and leptonium production and decay. We present both analytic and high-precision numerical results. The analytic expressions are given in terms of multiple polylogarithms (MPLs), elliptic multiple polylogarithms (eMPLs) and iterated integrals of Eisenstein series. As an application of our results, we obtain for the first time an analytic expression for the two-loop amplitude for para-positronium decay to two photons at two loops.

Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay

We present the calculation of the two-loop soft function for the transverse momentum distribution of the leading jet produced in association with any colour-singlet system (e.g. a Higgs or a Z boson). This constitutes a central ingredient for the resummation of the above distribution as well as the jet-vetoed cross section at the next-to-next-to-next-to-leading logarithmic order, both of which play an important role in the precision physics programme at the Large Hadron Collider. The calculation is performed in soft-collinear effective theory with an appropriate regularisation of the rapidity divergences that occur in the phase-space integrals. We obtain analytic results by employing an exponential regulator and by taking a Laurent expansion in the jet radius R. All expressions are presented as supplementary material attached to this article.

The analytic two-loop soft function for leading-jet p  T

We present the calculation of the three distinct non-planar hexa-box topologies for five-point one-mass processes. These three topologies are required to obtain the two-loop virtual QCD corrections for two-jet-associated W, Z or Higgs-boson production. Each topology is solved by obtaining a pure basis of master integrals and efficiently constructing the associated differential equation with numerical sampling and unitarity-cut techniques. We present compact expressions for the alphabet of these non-planar integrals, and discuss some properties of their symbol. Notably, we observe that the extended Steinmann relations are in general not satisfied. Finally, we solve the differential equations in terms of generalized power series and provide high-precision values in different regions of phase space which can be used as boundary conditions for subsequent evaluations.

Two-loop hexa-box integrals for non-planar five-point one-mass processes