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

We formulate a collinear partonic shower algorithm that achieves next-to-single-logarithmic (NSL, $$ {\alpha}_s^n{L}^{n-1} $$ ) accuracy for collinear-sensitive non-singlet fragmentation observables. This entails the development of an algorithm for nesting triple-collinear splitting functions. It also involves the inclusion of the one-loop double-collinear corrections, through a -dependent NLO-accurate effective 1 → 2 branching probability, using a formula that can be applied more generally also to future full showers with 1 → 3 splitting kernels. The specific NLO branching probability is calculated in two ways, one based on slicing, the other using a subtraction approach based on recent analytical calculations. We close with demonstrations of the shower’s accuracy for non-singlet partonic fragmentation functions and the energy spectrum of small- quark jets. This work represents an important conceptual step towards general NNLL accuracy in parton showers.

Netherlands

Melissa Beekveld

United Kingdom

Mrinal Dasgupta

Australia

Basem El-Menoufi

Jack Helliwell

Gavin Salam

A collinear shower algorithm for NSL non-singlet fragmentation

We consider the production of a pair of heavy quarks $$Q\overline{Q }$$ in association with a generic colour singlet system at lepton colliders, and present the first analytic calculation of the two-loop soft function differential in the total momentum of the real radiation. The calculation is performed by reducing the relevant Feynman integrals into a canonical basis of master integrals by means of integration-by-parts identities. The resulting integrals are then evaluated by solving a system of differential equations in the kinematic invariants, whose boundary conditions are determined analytically with some care due to the presence of Coulomb singularities. The fully differential soft function is expressed in terms of Goncharov polylogarithms. This result is an essential ingredient for a range of N LL resummations for key collider observables at lepton colliders, such as the $$Q\overline{Q }V$$ production cross section at threshold and observables sensitive to the total transverse momentum of the radiation in heavy-quark final states. Moreover, it constitutes the complete final-final dipole contribution to the fully differential soft function needed for the description of $$Q\overline{Q }V$$ production at hadron colliders, which plays an important role in the LHC physics programme.

China

Ze Liu

The two-loop fully differential soft function for

We study the collinear fragmentation of highly energetic jets defined with a small jet radius. In particular, we investigate how the corresponding fragmentation functions differ from their hadronic counterpart defined in the common   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  scheme. We find that the anomalous dimensions governing the perturbative evolution of the two fragmentation functions differ starting at the two loop order. We compute for the first time the new anomalous dimensions at two loops and confirm our predictions by comparing the inclusive small-R jet spectrum against a fixed order perturbative calculation at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>s</mi> <mn>2</mn> </msubsup> </math>  $$ {\alpha}_s^2 $$ ). To investigate the dependence of the anomalous dimension on the kinematic cutoff variable, we study the fragmentation functions of Cambridge jets defined with a transverse momentum cutoff as opposed to an angular cutoff R. We further study the evolution of the small-R fragmentation function with an alternative cutoff scale, proportional to zR, representing the maximum possible transverse momentum of emissions within a jet. In these cases we find that the two-loop anomalous dimensions coincide with the   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  DGLAP ones, highlighting a correspondence between the   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  scheme and a transverse-momentum cutoff.

CERN

Alexander Karlberg

Two-loop anomalous dimensions for small-R jet versus hadronic fragmentation functions

Jet calculus offers a unique mathematical technique to bridge the area of QCD resummation with Monte Carlo parton showers. With the ultimate goal of constructing next-to-next-to-leading logarithmic (NNLL) parton showers we study, using the language of generating functionals, the collinear fragmentation of final-state partons. In particular, we focus on the definition and calculation of the Sudakov form factor, which physically describes the no-emission probability in an ordered branching process. We review recent results for quark jets and compute the Sudakov form factor for the collinear fragmentation of gluon jets at NNLL. The NNLL corrections are encoded in a z dependent two-loop anomalous dimension   <math> <mi>B</mi> </math>  $$ \mathcal{B} $$  2(z), with z being a suitably defined longitudinal momentum fraction. This is obtained from the integration of the relevant 1 → 3 collinear splitting kernels combined with the one-loop corrections to the 1 → 2 counterpart. This work provides the missing ingredients to extend the methods of jet calculus in the collinear limit to NNLL and gives an important element of the next generation of NNLL parton shower algorithms. As an application we derive new NNLL results for both the fractional moments of energy-energy correlation FC  x  and the angularities λ  x  measured on mMDT/Soft-Drop (β = 0) groomed jets.

Collinear fragmentation at NNLL: generating functionals, groomed correlators and angularities

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

United States

Robert Szafron

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

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

A significant challenge in the tagging of boosted objects via machine-learning technology is the prohibitive computational cost associated with training sophisticated models. Nevertheless, the universality of QCD suggests that a large amount of the information learnt in the training is common to different physical signals and experimental setups. In this article, we explore the use of transfer learning techniques to develop fast and data-efficient jet taggers that leverage such universality. We consider the graph neural networks LundNet and ParticleNet, and introduce two prescriptions to transfer an existing tagger into a new signal based either on fine-tuning all the weights of a model or alternatively on freezing a fraction of them. In the case of W-boson and top-quark tagging, we find that one can obtain reliable taggers using an order of magnitude less data with a corresponding speed-up of the training process. Moreover, while keeping the size of the training data set fixed, we observe a speed-up of the training by up to a factor of three. This offers a promising avenue to facilitate the use of such tools in collider physics experiments.

Frédéric Dreyer

Radosław Grabarczyk

Leveraging universality of jet taggers through transfer learning

We consider the production of a pair of heavy quarks and illustrate the derivation of the MiNNLOPS method to match next-to-next-to-leading order calculations with parton showers (NNLO+PS) for this class of processes. As a first application, we construct an event generator for the fully differential simulation of hadronic top-quark pair production at NNLO+PS and discuss all details of its implementation in a parton shower Monte Carlo framework. We present new phenomenological results for the Large Hadron Collider obtained by including the tree-level decays of the top quarks, while accounting for spin-correlation effects. A comprehensive comparison to LHC measurements shows an excellent description of experimental data across multiple hadronic and leptonic particle-level observables. The computer code is available for download within the Powheg-Box.

Javier Mazzitelli

Italy

Paolo Nason

France

Emanuele Re

Marius Wiesemann

Giulia Zanderighi

Top-pair production at the LHC with MINNLOPS

Non-global QCD observables are characterised by a sensitivity to the full angular distribution of soft radiation emitted coherently in hard scattering processes. This complexity poses a challenge to their all-order resummation, that was formulated at the leading-logarithmic order about two decades ago. In this article we present a solution to the long-standing problem of their resummation beyond this order, and carry out the first complete next-to-leading logarithmic calculation for non-global observables. This is achieved by solving numerically the recently derived set of non-linear differential equations which describe the evolution of soft radiation in the planar, large-N  c  limit. As a case study we address the calculation of the transverse energy distribution in the interjet rapidity region in e + e  −  → dijet production. The calculation is performed by means of an algorithm that we formulate in the language of jet-calculus generating functionals, which also makes the resummation technique applicable to more general non-global problems, such as those that arise in hadronic collisions. We find that NLL corrections are substantial and their inclusion leads to a significant reduction of the perturbative scale uncertainties for these observables. The computer code used in the calculations is made publicly available.

Andrea Banfi

Higher-order non-global logarithms from jet calculus