We report on the complete computation of the mixed QCD-electroweak corrections to the neutral-current Drell-Yan process. Our calculation holds in the entire range of dilepton invariant masses. We present phenomenological results for several kinematical distributions in the case of bare muons both in the resonant region and for high invariant masses. We also consider the forward-backward asymmetry, which is a key observable to measure the weak mixing angle. We finally extend our calculation to dressed leptons and compare our results in the massless limit to those available in the literature.

Belgium

Italy

Tommaso Armadillo

Roberto Bonciani

Switzerland

Luca Buonocore

Simone Devoto

Massimiliano Grazzini

Stefan Kallweit

India

Narayan Rana

Alessandro Vicini

Mixed QCD-EW corrections to the neutral-current Drell-Yan process

We present the first fully differential next-to-next-to-leading order (NNLO) QCD calculation for the production of a top-antitop quark pair in association with a Higgs boson $$ \left(t\overline{t}H\right) $$ at hadron colliders. The computation is exact, except for the finite part of the two-loop virtual contribution, which we estimate using two different methods that yield consistent results within their respective uncertainties. The first method relies on a soft-Higgs factorisation formula that we develop up to the three-loop order. The second is based on a high-energy expansion in the small top-mass limit. Combining the newly computed corrections with the complete set of next-to-leading order (NLO) QCD+EW results provides the most advanced perturbative prediction currently available at the LHC for both inclusive and differential $$ t\overline{t}H $$ cross sections. The uncertainties due to the missing exact two-loop contribution are conservatively estimated to be at the percent level, both for the total cross section and for most of the differential distributions, and therefore subleading compared to the residual perturbative uncertainties.

Javier Mazzitelli

Germany

Chiara Savoini

Precise predictions for

We consider subleading power corrections to event shape variables in  $$e^+e^-$$  <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> </mrow> </math>   collisions at the first order in the QCD coupling  $$\alpha _{\textrm{S}}$$  <math> <msub> <mi>α</mi> <mtext>S</mtext> </msub> </math>  . We start from the jettiness variable  $$\tau _2$$  <math> <msub> <mi>τ</mi> <mn>2</mn> </msub> </math>   and the  $$y_{23}$$  <math> <msub> <mi>y</mi> <mn>23</mn> </msub> </math>   resolution variable for the  $$k_T$$  <math> <msub> <mi>k</mi> <mi>T</mi> </msub> </math>   jet clustering algorithm and we analytically compute the corresponding cumulative cross section. We investigate the origin of the different power suppressed contributions in the two-jet limit and trace it back to their different coverage of the phase space. We extend our analysis to the case of thrust and of the C-parameter, and we finally discuss a class of observables that depend on a continuous parameter giving different weight to central and forward emissions and we evaluate the corresponding subleading power corrections.

CERN

Flavio Guadagni

Luca Rottoli

Subleading power corrections for event shape variables in  $$e^+ e^-$$  <math> <mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msup> <mi>e</mi> <mo>-</mo> </msup> </mrow> </math>   annihilation

We consider the class of inclusive hadron collider processes in which one or more energetic jets are produced, possibly accompanied by colourless particles. We provide a general formulation of a slicing scheme for this class of processes, by identifying the various contributions that need to be computed up to next-to-leading order (NLO) in QCD perturbation theory. We focus on two novel observables, the one-jet resolution variable ∆E  t  and the n-jet resolution variable   <math> <msubsup> <mi>k</mi> <mi>T</mi> <mtext>ness</mtext> </msubsup> </math>  $$ {k}_T^{\textrm{ness}} $$ , and explicitly compute all the ingredients needed to carry out NLO computations using these variables. We contrast the behaviour of these variables when the slicing parameter becomes small. In the case of   <math> <msubsup> <mi>k</mi> <mi>T</mi> <mtext>ness</mtext> </msubsup> </math>  $$ {k}_T^{\textrm{ness}} $$  we also present results for the hadroproduction of multiple jets.

Jürg Haag

Exploring slicing variables for jet processes

Quantum Chromodynamics, the theory of quarks and gluons, whose interactions can be described by a local SU(3) gauge symmetry with charges called “color quantum numbers”, is reviewed; the goal of this review is to provide advanced Ph.D. students a comprehensive handbook, helpful for their research. When QCD was “discovered” 50 years ago, the idea that quarks could exist, but not be observed, left most physicists unconvinced. Then, with the discovery of charmonium in 1974 and the explanation of its excited states using the Cornell potential, consisting of the sum of a Coulomb-like attraction and a long range linear confining potential, the theory was suddenly widely accepted. This paradigm shift is now referred to as the November revolution. It had been anticipated by the observation of scaling in deep inelastic scattering, and was followed by the discovery of gluons in three-jet events. The parameters of QCD include the running coupling constant,  $$\alpha _s(Q^2)$$  <math> <mrow> <msub> <mi>α</mi> <mi>s</mi> </msub> <mrow> <mo>(</mo> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </math>  , that varies with the energy scale  $$Q^2$$  <math> <msup> <mi>Q</mi> <mn>2</mn> </msup> </math>   characterising the interaction, and six quark masses. QCD cannot be solved analytically, at least not yet, and the large value of  $$\alpha _s$$  <math> <msub> <mi>α</mi> <mi>s</mi> </msub> </math>   at low momentum transfers limits perturbative calculations to the high-energy region where  $$Q^2\gg \varLambda _{{\textrm{QCD}}} ^2\simeq $$  <math> <mrow> <msup> <mi>Q</mi> <mn>2</mn> </msup> <mo>≫</mo> <msubsup> <mi>Λ</mi> <mrow> <mtext>QCD</mtext> </mrow> <mn>2</mn> </msubsup> <mo>≃</mo> </mrow> </math>   (250 MeV) $$^2$$  <math> <msup> <mrow></mrow> <mn>2</mn> </msup> </math>  . Lattice QCD (LQCD), numerical calculations on a discretized space-time lattice, is discussed in detail, the dynamics of the QCD vacuum is visualized, and the expected spectra of mesons and baryons are displayed. Progress in lattice calculations of the structure of nucleons and of quantities related to the phase diagram of dense and hot (or cold) hadronic matter are reviewed. Methods and examples of how to calculate hadronic corrections to weak matrix elements on a lattice are outlined. The wide variety of analytical approximations currently in use, and the accuracy of these approximations, are reviewed. These methods range from the Bethe–Salpeter, Dyson–Schwinger coupled relativistic equations, which are formulated in both Minkowski or Euclidean spaces, to expansions of multi-quark states in a set of basis functions using light-front coordinates, to the AdS/QCD method that imbeds 4-dimensional QCD in a 5-dimensional deSitter space, allowing confinement and spontaneous chiral symmetry breaking to be described in a novel way. Models that assume the number of colors is very large, i.e. make use of the large  $$N_c$$  <math> <msub> <mi>N</mi> <mi>c</mi> </msub> </math>  -limit, give unique insights. Many other techniques that are tailored to specific problems, such as perturbative expansions for high energy scattering or approximate calculations using the operator product expansion are discussed. The very powerful effective field theory techniques that are successful for low energy nuclear systems (chiral effective theory), or for non-relativistic systems involving heavy quarks, or the treatment of gluon exchanges between energetic, collinear partons encountered in jets, are discussed. The spectroscopy of mesons and baryons has played an important historical role in the development of QCD. The famous X,Y,Z states – and the discovery of pentaquarks – have revolutionized hadron spectroscopy; their status and interpretation are reviewed as well as recent progress in the identification of glueballs and hybrids in light-meson spectroscopy. These exotic states add to the spectrum of expected  $$q{{\bar{q}}}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   mesons and qqq baryons. The progress in understanding excitations of light and heavy baryons is discussed. The nucleon as the lightest baryon is discussed extensively, its form factors, its partonic structure and the status of the attempt to determine a three-dimensional picture of the parton distribution. An experimental program to study the phase diagram of QCD at high temperature and density started with fixed target experiments in various laboratories in the second half of the 1980s, and then, in this century, with colliders. QCD thermodynamics at high temperature became accessible to LQCD, and numerical results on chiral and deconfinement transitions and properties of the deconfined and chirally restored form of strongly interacting matter, called the Quark–Gluon Plasma (QGP), have become very precise by now. These results can now be confronted with experimental data that are sensitive to the nature of the phase transition. There is clear evidence that the QGP phase is created. This phase of QCD matter can already be characterized by some properties that indicate, within a temperature range of a few times the pseudocritical temperature, the medium behaves like a near ideal liquid. Experimental observables are presented that demonstrate deconfinement. High and ultrahigh density QCD matter at moderate and low temperatures shows interesting features and new phases that are of astrophysical relevance. They are reviewed here and some of the astrophysical implications are discussed. Perturbative QCD and methods to describe the different aspects of scattering processes are discussed. The primary parton–parton scattering in a collision is calculated in perturbative QCD with increasing complexity. The radiation of soft gluons can spoil the perturbative convergence, this can be cured by resummation techniques, which are also described here. Realistic descriptions of QCD scattering events need to model the cascade of quark and gluon splittings until hadron formation sets in, which is done by parton showers. The full event simulation can be performed with Monte Carlo event generators, which simulate the full chain from the hard interaction to the hadronic final states, including the modelling of non-perturbative components. The contribution of the LEP experiments (and of earlier collider experiments) to the study of jets is reviewed. Correlations between jets and the shape of jets had allowed the collaborations to determine the “color factors” – invariants of the SU(3) color group governing the strength of quark–gluon and gluon–gluon interactions. The calculated jet production rates (using perturbative QCD) are shown to agree precisely with data, for jet energies spanning more than five orders of magnitude. The production of jets recoiling against a vector boson,  $$W^\pm $$  <math> <msup> <mi>W</mi> <mo>±</mo> </msup> </math>   or Z, is shown to be well understood. The discovery of the Higgs boson was certainly an important milestone in the development of high-energy physics. The couplings of the Higgs boson to massive vector bosons and fermions that have been measured so far support its interpretation as mass-generating boson as predicted by the Standard Model. The study of the Higgs boson recoiling against hadronic jets (without or with heavy flavors) or against vector bosons is also highlighted. Apart from the description of hard interactions taking place at high energies, the understanding of “soft QCD” is also very important. In this respect, Pomeron – and Odderon – exchange, soft and hard diffraction are discussed. Weak decays of quarks and leptons, the quark mixing matrix and the anomalous magnetic moment of the muon are processes which are governed by weak interactions. However, corrections by strong interactions are important, and these are reviewed. As the measured values are incompatible with (most of) the predictions, the question arises: are these discrepancies first hints for New Physics beyond the Standard Model? This volume concludes with a description of future facilities or important upgrades of existing facilities which improve their luminosity by orders of magnitude. The best is yet to come!

United States

Franz Gross

Eberhard Klempt

Stanley Brodsky

Andrzej Buras

Volker Burkert

Gudrun Heinrich

Karl Jakobs

Curtis Meyer

Kostas Orginos

Michael Strickland

Johanna Stachel

Giulia Zanderighi

Nora Brambilla

Peter Braun-Munzinger

Daniel Britzger

Simon Capstick

Tom Cohen

Volker Crede

Martha Constantinou

United Kingdom

Christine Davies

Luigi Debbio

Achim Denig

Carleton DeTar

Alexandre Deur

Russia

Latvia

Yuri Dokshitzer

Hans Dosch

Jozef Dudek

Monica Dunford

Evgeny Epelbaum

Spain

Miguel Escobedo

Harald Fritzsch

Japan

Kenji Fukushima

Paolo Gambino

Canada

Sweden

Dag Gillberg

Steven Gottlieb

Per Grafstrom

Boris Grube

JINR

Alexey Guskov

Toru Iijima

Xiangdong Ji

Frithjof Karsch

Stefan Kluth

John Kogut

Frank Krauss

Shunzo Kumano

Australia

Derek Leinweber

Heinrich Leutwyler

China

Hai-Bo Li

Yang Li

France

Bogdan Malaescu

Chiara Mariotti

Pieter Maris

Simone Marzani

Wally Melnitchouk

Johan Messchendorp

Harvey Meyer

Ryan Mitchell

Chandan Mondal

Frank Nerling

Sebastian Neubert

Marco Pappagallo

Saori Pastore

José Peláez

Andrew Puckett

Jianwei Qiu

Klaus Rabbertz

Alberto Ramos

Patrizia Rossi

Azerbaijan

Anar Rustamov

Andreas Schäfer

Stefan Scherer

Matthias Schindler

Steven Schramm

Mikhail Shifman

Edward Shuryak

Torbjörn Sjöstrand

George Sterman

Iain Stewart

Joachim Stroth

Eric Swanson

Costa Rica

Guy Téramond

Ulrike Thoma

Antonio Vairo

Danny Dyk

James Vary

Javier Virto

Marcel Vos

Christian Weiss

Markus Wobisch

Sau Wu

Christopher Young

Feng Yuan

Xingbo Zhao

Xiaorong Zhou

50 Years of quantum chromodynamics

The production of a top-antitop quark pair in association with a <math><mrow><mi>W</mi></mrow></math> boson (<math><mi>t</mi><mover><mi>t</mi><mo>¯</mo></mover><mi>W</mi></math>) is one of the heaviest signatures currently probed at the Large Hadron Collider. Since the first observation reported in 2015, the corresponding rates have been found to be consistently higher than the standard model predictions, which are based on next-to-leading order calculations in the QCD and electroweak interactions. We present the first next-to-next-to-leading order QCD computation of <math><mi>t</mi><mover><mi>t</mi><mo>¯</mo></mover><mi>W</mi></math> production at hadron colliders. The calculation is exact, except for the finite part of the two-loop virtual corrections, which is estimated using two different approaches that lead to consistent results within their uncertainties. We combine the newly computed next-to-next-to-leading order QCD corrections with the complete next-to-leading order QCD plus electroweak results, thus obtaining the most advanced perturbative prediction available to date for the <math><mi>t</mi><mover><mi>t</mi><mo>¯</mo></mover><mi>W</mi></math> inclusive cross section. The tension with the latest ATLAS and CMS results remains at the <math><mrow><mn>1</mn><mi>σ</mi><mtext>−</mtext><mn>2</mn><mi>σ</mi></mrow></math> level.

Precise Predictions for the Associated Production of a <math><mi>W</mi></math> Boson with a Top-Antitop Quark Pair at the LHC

We consider extensions of the soft-gluon effective coupling that generalize the Catani-Marchesini-Webber (CMW) coupling in the context of soft-gluon resummation beyond the next-to-leading logarithmic accuracy. Starting from the probability density of correlated soft emission in d dimensions we introduce a class of soft couplings relevant for resummed QCD calculations of hard-scattering observables. We show that at the conformal point, where the d-dimensional QCD β function vanishes, all these effective couplings are equal and they are also equal to the cusp anomalous dimension. We present explicit results in d dimensions for the soft-emission probability density and the soft couplings at the second-order in the QCD coupling α S. In d = 4 dimensions we obtain the explicit relation between the soft couplings at   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (  <math> <msubsup> <mi>α</mi> <mi>S</mi> <mn>3</mn> </msubsup> </math>  $$ {\alpha}_{\textrm{S}}^3 $$ ). Finally, we study the structure of the soft coupling in the large-n  F  limit and we present explicit expressions to all orders in perturbation theory. We also check that, at the conformal point, our large-n  F  results agree with the known result of the cusp anomalous dimension.

Stefano Catani

Argentina

Daniel Florian

Soft-gluon effective coupling: perturbative results and the large-n  F  limit to all orders

We consider QCD radiative corrections to the production of a heavy-quark pair in hadronic collisions. We present the computation of the soft-parton contributions at low transverse momentum of the heavy-quark pair up to second order in the QCD coupling α S. These results complete the evaluation at the next-to-next-to-leading order (NNLO) of the transverse-momentum resummation formula for this process. Moreover, they give all the ingredients that are needed for the NNLO implementation of the q  T  subtraction formalism for heavy-quark production. We discuss the details of the computation and we provide a code that can be used to obtain the relevant results in numerical form.

Soft-parton contributions to heavy-quark production at low transverse momentum

The associated production of a Higgs boson with a top-antitop quark pair is a crucial process at the LHC since it allows for a direct measurement of the top-quark Yukawa coupling. We present the computation of the radiative corrections to this process at the next-to-next-to-leading order (NNLO) in QCD perturbation theory. This is the very first computation for a <math><mrow><mn>2</mn><mo>→</mo><mn>3</mn></mrow></math> process with massive colored particles at this perturbative order. We develop a soft Higgs boson approximation for loop amplitudes, which enables us to reliably quantify the impact of the yet unknown two-loop contribution. At the center-of-mass energy <math><mrow><msqrt><mrow><mi>s</mi></mrow></msqrt><mo>=</mo><mn>13</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></mrow></math>, the NNLO corrections increase the next-to-leading order result for the total cross section by about 4% and lead to a significant reduction of perturbative uncertainties.

Higgs Boson Production in Association with a Top-Antitop Quark Pair in Next-to-Next-to-Leading Order QCD

We consider the class of inclusive hadron collider processes in which several energetic jets are produced, possibly accompanied by colorless particles [such as Higgs boson(s), vector boson(s) with their leptonic decays, and so forth]. We propose a new variable that smoothly captures the <math><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></math> to <math><mi>N</mi></math>-jet transition. This variable, that we dub <math><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>ness</mi></mrow></msubsup></mrow></math>, represents an effective transverse momentum controlling the singularities of the <math><mi>N</mi><mo>+</mo><mn>1</mn></math>-jet cross section when the additional jet is unresolved. The <math><msubsup><mi>k</mi><mi>T</mi><mrow><mi>ness</mi></mrow></msubsup></math> variable offers novel opportunities to perform higher-order calculations in quantum chromodynamics by using nonlocal subtraction schemes. We study the singular behavior of the <math><mi>N</mi><mo>+</mo><mn>1</mn></math>-jet cross section as <math><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>ness</mi></mrow></msubsup><mo>→</mo><mn>0</mn></mrow></math> and, as a phenomenological application, we use the ensuing results to evaluate next-to-leading-order corrections to <math><mrow><mi>H</mi><mo>+</mo><mi>jet</mi></mrow></math> and <math><mi>Z</mi><mo>+</mo><mn>2</mn></math>-jet production at the LHC. We show that <math><mrow><msubsup><mrow><mi>k</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>ness</mi></mrow></msubsup></mrow></math> performs extremely well as a resolution variable and appears to be very stable with respect to hadronization and multiple-parton interactions.

Effective transverse momentum in multiple jet production at hadron colliders