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

Germany

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

Italy

Paolo Gambino

Canada

Sweden

Dag Gillberg

Steven Gottlieb

CERN

Per Grafstrom

Switzerland

Massimiliano Grazzini

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

In this paper, the existence of a massive dark photon, associated with a new gauge group is considered. The dark photon can be kinetically mixed with the photon. To study some applications, the thermo field dynamics formalism is used. Exploring the topological structure of this approach, the influence of dark photons on the Stefan–Boltzmann law and the Casimir effect at zero and finite temperature is calculated.

Brazil

V. Prata

A. Santos

Faqir Khanna

Stefan–Boltzmann law and Casimir effect for dark photons

The production of a heavy quark is accompanied by gluon bremsstrahlung, which is suppressed at small angles <math><mi>Θ</mi><mo>≲</mo><msub><mi>M</mi><mi>Q</mi></msub><mo>/</mo><mi>E</mi></math> for mass <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>Q</mi></mrow></msub></mrow></math> and high energy <math><mi>E</mi></math>, according to perturbative quantum chromodynamics (QCD) (“dead cone effect”). As particles at small angles typically have large momenta, the heavy quark mass also causes a suppression of high momentum particles. In this paper, we studied this effect in <math><mi>c</mi></math>- and <math><mi>b</mi></math>-quark events using data from Z boson decays in <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> annihilation. The heavy quark fragmentation function for charged particles is reconstructed in the momentum fraction variable <math><mi>x</mi></math> or <math><mi>ξ</mi><mo>=</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo></math> by removing the decays of the heavy quark hadrons. Indeed, we find an increasing suppression of particles with rising <math><mi>x</mi></math> down to a fraction of <math><mo>≲</mo><mn>1</mn><mo>/</mo><mn>10</mn></math> for particles with <math><mi>x</mi><mo>≳</mo><mn>0.2</mn></math> in <math><mi>b</mi></math>-quark and <math><mi>x</mi><mo>≳</mo><mn>0.4</mn></math> in <math><mi>c</mi></math>-quark jets in comparison to light quark fragmentation. The sensitivity to the dead cone effect in the present momentum analysis is considerably increased in comparison to the recently presented angular analysis. This amount of suppression and the differences between <math><mi>c</mi></math>- and <math><mi>b</mi></math>-quark fragmentation are in good quantitative agreement with the expectations based on perturbative QCD within the modified leading logarithmic approximation (MLLA) in the central kinematic region. The data also support a two parameter description in the MLLA of these phenomena (“limiting spectrum”). The sensitivity of these measurements to the heavy quark mass is investigated.

Wolfgang Ochs

Redamy Perez Ramos

Observation of the dead cone effect in charm and bottom quark jets and its QCD explanation

The Dirac field, spin 1/2 particles, is investigated in phase space. The Dirac propagator is defined. The Thermo Field Dynamics (TFD) formalism is used to introduce finite temperature. The energy-momentum tensor is calculated at finite temperature. The Stefan-Boltzmann law is established, and the Casimir effect is calculated for the Dirac field in phase space at zero and finite temperature. A comparative analysis with these results in standard quantum mechanics space is realized.

S.C. Ulhoa

R.G.G. Amorim

J.S. da Cruz Filho

A.F. Santos

F.C. Khanna

Spin-1/2 Particles in Phase Space: Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

We present state-of-the-art extractions of the strong coupling based on N3LO+NNLL accurate predictions for the two-jet rate in the Durham clustering algorithm at e + e − collisions, as well as a simultaneous fit of the two- and three-jet rates taking into account correlations between the two observables. The fits are performed on a large range of data sets collected at the LEP and PETRA colliders, with energies spanning from 35 GeV to 207 GeV. Owing to the high accuracy of the predictions used, the perturbative uncertainty is considerably smaller than that due to hadronization. Our best determination at the Z mass is α  s  (M Z) = 0.11881 ± 0.00063(exp.) ± 0.00101(hadr.) ± 0.00045(ren.) ± 0.00034(res.), which is in agreement with the latest world average and has a comparable total uncertainty.

Andrii Verbytskyi

Andrea Banfi

Hungary

Adam Kardos

Pier Monni

Gábor Somogyi

Zoltán Szőr

Zoltán Trócsányi

Zoltán Tulipánt

High precision determination of α  s  from a global fit of jet rates

Gravitons are described by the propagator in teleparallel gravity in nearly flat space-time. Finite temperature is introduced by using Thermofield Dynamics formalism. The gravitational Casimir effect and Stefan-Boltzmann law are calculated as a function of temperature. Then an equation of state for gravitons is determined.

T.F. Furtado

On Gravitational Casimir Effect and Stefan-Boltzmann Law at Finite Temperature

We use the scalar field constructed in phase space to analyze the analogous Stefan-Boltzmann law and Casimir effect, both of them at finite temperature. The temperature is introduced by Thermo Field Dynamics (TFD) formalism and the quantities are analyzed once projected in the space of coordinates. We show that using the framework of phase space it is possible to introduce a thermal energy which is related to temperature as it vanishes when the temperature tends to zero. In fact given such a correlation the formalism of TFD is equivalent when project is in momenta space when compared to coordinates space.

Stefan-Boltzmann Law and Casimir Effect for the Scalar Field in Phase Space at Finite Temperature

We present a comparison of the computation of energy–energy correlation in $$e^{+}e^{-}$$ <math><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>-</mo></msup></mrow></math>  collisions in the back-to-back region at next-to-next-to-leading logarithmic accuracy matched with the next-to-next-to-leading order perturbative prediction to LEP, PEP, PETRA, SLC and TRISTAN data. With these predictions we perform an extraction of the strong coupling constant taking into account non-perturbative effects modelled with Monte Carlo event generators. The final result at NNLO+NNLL precision is $$\alpha _{S}(M_{Z})= 0.11750\pm 0.00018 {\text{( } exp.)}\pm 0.00102{\text{( }hadr.)}\pm 0.00257{\text{( }ren.)}\pm 0.00078{\text{( }res.)}$$ <math><mrow><msub><mi>α</mi><mi>S</mi></msub><mrow><mo>(</mo><msub><mi>M</mi><mi>Z</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0.11750</mn><mo>±</mo><mn>0.00018</mn><mrow><mtext>(</mtext><mspace width="0.333333em"></mspace><mi>e</mi><mi>x</mi><mi>p</mi><mo>.</mo><mo>)</mo></mrow><mo>±</mo><mn>0.00102</mn><mrow><mtext>(</mtext><mspace width="0.333333em"></mspace><mi>h</mi><mi>a</mi><mi>d</mi><mi>r</mi><mo>.</mo><mo>)</mo></mrow><mo>±</mo><mn>0.00257</mn><mrow><mtext>(</mtext><mspace width="0.333333em"></mspace><mi>r</mi><mi>e</mi><mi>n</mi><mo>.</mo><mo>)</mo></mrow><mo>±</mo><mn>0.00078</mn><mrow><mtext>(</mtext><mspace width="0.333333em"></mspace><mi>r</mi><mi>e</mi><mi>s</mi><mo>.</mo><mo>)</mo></mrow></mrow></math> .

Precise determination of $$\alpha _{S}(M_Z)$$ <math><mrow><msub><mi>α</mi><mi>S</mi></msub><mrow><mo>(</mo><msub><mi>M</mi><mi>Z</mi></msub><mo>)</mo></mrow></mrow></math>  from a global fit of energy–energy correlation to NNLO+NNLL predictions

It has recently been suggested (S.B. Giddings (2016) [2] ) that the Hawking black-hole radiation spectrum originates from an effective quantum “atmosphere' which extends well outside the black-hole horizon. In particular, comparing the Hawking radiation power of a (3+1) -dimensional Schwarzschild black hole of horizon radius rH with the familiar Stefan–Boltzmann radiation power of a (3+1) -dimensional flat space perfect blackbody emitter, Giddings concluded that the source of the Hawking semi-classical black-hole radiation is a quantum region outside the Schwarzschild black-hole horizon whose effective radius rA is characterized by the relation Δr≡rA−rH∼rH . It is of considerable physical interest to test the general validity of Giddings's intriguing conclusion. To this end, we study the Hawking radiation of (D+1) -dimensional Schwarzschild black holes. We find that the dimensionless radii rA/rH which characterize the black-hole quantum atmospheres, as determined from the Hawking black-hole radiation power and the (D+1) -dimensional Stefan–Boltzmann radiation law, are a decreasing function of the number D+1 of spacetime dimensions. In particular, it is shown that radiating (D+1) -dimensional Schwarzschild black holes are characterized by the relation (rA−rH)/rH≪1 in the large D≫1 regime. Our results therefore suggest that, at least in some physical cases, the Hawking emission spectrum originates from quantum excitations very near the black-hole horizon.

Israel

Shahar Hod

Hawking radiation and the Stefan–Boltzmann law: The effective radius of the black-hole quantum atmosphere

Where does Hawking radiation originate? A common picture is that it arises from excitations very near or at the horizon, and this viewpoint has supported the “firewall” argument and arguments for a key role for the UV-dependent entanglement entropy in describing the quantum mechanics of black holes. However, closer investigation of both the total emission rate and the stress tensor of Hawking radiation supports the statement that its source is a near-horizon quantum region, or “atmosphere,” whose radial extent is set by the horizon radius scale. This is potentially important, since Hawking radiation needs to be modified to restore unitarity, and a natural assumption is that the scales relevant to such modifications are comparable to those governing the Hawking radiation. Moreover, related discussion suggests a resolution to questions regarding extra energy flux in “nonviolent” scenarios, that does not spoil black hole thermodynamics as governed by the Bekenstein–Hawking entropy.

Steven B. Giddings

Hawking radiation, the Stefan–Boltzmann law, and unitarization