<p>We perform a lattice calculation of the correlators of two chromoelectric fields in the adjoint representation connected by adjoint Wilson lines at nonzero temperature. These correlators arise in the study of quarkonium dynamics and of adjoint heavy quark diffusion in deconfined matter. We work in SU(3) gauge theory using either gradient flow or multilevel algorithms for noise reduction, and discuss the renormalization of the correlators on the lattice. We find that a Casimir factor rescaling relates the adjoint correlators corresponding to the diffusion of an adjoint heavy quark and the octet-octet quarkonium transitions to the chromoelectric correlator in the fundamental representation describing the diffusion of a heavy quark.</p>

Germany

Nora Brambilla

India

Saumen Datta

Marc Janer

Viljami Leino

Julian Mayer-Steudte

United States

Peter Petreczky

Antonio Vairo

Lattice study of correlators of chromoelectric fields for heavy quarkonium dynamics in the quark-gluon plasma

<p>Two decades ago the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mrow><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></mrow></math> state was discovered in the hadron spectrum with two heavy quarks. The discovery fueled a surge in experimental research, uncovering dozens of so called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mi>Y</mi><mi>Z</mi></mrow></math> exotics states lying outside the conventional quark model, as well as theoretical investigations into new forms of matter, such as quark-gluon hybrids, tetraquarks, and pentaquarks, with the potential of disclosing new information about the fundamental strong force. Among the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mi>Y</mi><mi>Z</mi><mi mathvariant="normal">s</mi></mrow></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mrow><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup><mrow><mo stretchy="false">(</mo><mn>3875</mn><mo stretchy="false">)</mo></mrow></math> states stand out for their striking characteristics and unleashed many discussions about their nature. Here, we address this question using the Born-Oppenheimer effective field theory (BOEFT) and show how QCD settles the issue of their composition. Not only do we describe well the main features of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mrow><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup><mrow><mo stretchy="false">(</mo><mn>3875</mn><mo stretchy="false">)</mo></mrow></math> states, but we also obtain predictions in the bottomonium sector. This opens the way to systematic applications of BOEFT to all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mi>Y</mi><mi>Z</mi><mi mathvariant="normal">s</mi></mrow></math>.</p>

Abhishek Mohapatra

Tommaso Scirpa

Nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>3872</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup><mo stretchy="false">(</mo><mn>3875</mn><mo stretchy="false">)</mo></math>

The physics of quarkonium created in heavy-ion collisions is intrinsically connected to the correlation functions of adjoint chromoelectric fields in quantum chromodynamics. We study such correlation functions in a weak-coupling expansion in a thermal medium. We identify three distinct gauge-invariant correlators, and evaluate them to next-to-leading order. Two of the resulting correlators turn out to be asymmetric. We pinpoint the source of this asymmetry to Matsubara zero modes associated with Wilson lines. The results are shown to agree well with recent lattice calculations at high temperatures.

Panayiotis Panayiotou

Spain

Saga Säppi

The chromoelectric adjoint correlators in Euclidean space at next-to-leading order

The discovery of XYZ exotic states in the hadronic sector with two heavy quarks represents a significant challenge in particle theory. Understanding and predicting their nature remains an open problem. In this work, we demonstrate how the Born-Oppenheimer (BO) effective field theory (BOEFT), derived from quantum chromodynamics (QCD) on the basis of scale separation and symmetries, can address XYZ exotics of any composition. We derive the Schrödinger coupled equations that describe hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia at leading order, incorporating nonadiabatic terms, and present the predicted multiplets. We define the static potentials in terms of the QCD static energies for all relevant cases. We provide the precise form of the nonperturbative low-energy gauge-invariant correlators required for the BOEFT: static energies, generalized Wilson loops, gluelumps, and adjoint mesons. These are to be calculated on the lattice, and we calculate here their short-distance behavior. Furthermore, we outline how spin-dependent corrections and mixing terms can be incorporated using matching computations. Lastly, we discuss how static energies with the same BO quantum numbers mix at large distances leading to the phenomenon of avoided level crossing. This effect is crucial to understand the emergence of exotics with molecular characteristics, such as the <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math>. With BOEFT both the tetraquark and the molecular picture appear as part of the same description.

Matthias Berwein

Hybrids, tetraquarks, pentaquarks, doubly heavy baryons, and quarkonia in Born-Oppenheimer effective theory

The static QCD force from the lattice can be used to extract <math><msub><mi>Λ</mi><mover><mi>MS</mi><mo>¯</mo></mover></msub></math>, which determines the running of the strong coupling. Usually, this is done with a numerical derivative of the static potential. However, this introduces additional systematic uncertainties; thus, we use another observable to measure the static force directly. This observable consists of a Wilson loop with a chromoelectric field insertion. We work in the pure SU(3) gauge theory. We use gradient flow to improve the signal-to-noise ratio and to address the field insertion. We extract <math><mrow><msubsup><mrow><mi>Λ</mi></mrow><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msubsup></mrow></math> from the data by exploring different methods to perform the zero-flow-time limit. We obtain the value <math><mrow><msqrt><mrow><mn>8</mn><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msqrt><msubsup><mrow><mi>Λ</mi></mrow><mrow><mover><mrow><mi>MS</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></msubsup><mo>=</mo><mn>0.62</mn><msubsup><mrow><mn>9</mn></mrow><mrow><mo>−</mo><mn>26</mn></mrow><mrow><mo>+</mo><mn>22</mn></mrow></msubsup></mrow></math>, where <math><msub><mi>t</mi><mn>0</mn></msub></math> is a flow-time reference scale. We also obtain precise determinations of several scales: <math><msub><mi>r</mi><mn>0</mn></msub><mo>/</mo><msub><mi>r</mi><mn>1</mn></msub></math>, <math><msqrt><mrow><mn>8</mn><msub><mi>t</mi><mn>0</mn></msub></mrow></msqrt><mo>/</mo><msub><mi>r</mi><mn>0</mn></msub></math>, <math><msqrt><mrow><mn>8</mn><msub><mi>t</mi><mn>0</mn></msub></mrow></msqrt><mo>/</mo><msub><mi>r</mi><mn>1</mn></msub></math>, and we compare these to the literature. The gradient flow appears to be a promising method for calculations of Wilson loops with chromoelectric and chromomagnetic insertions in quenched and unquenched configurations.

Static force from generalized Wilson loops on the lattice using the gradient flow

We compute the suppression of bottomonium in the quark-gluon plasma using the three-loop QCD static potential. The potential describes the spin-averaged bottomonium spectrum below threshold with a less than 1% error. Within potential nonrelativistic quantum chromodynamics and an open quantum systems framework, we compute the evolution of the bottomonium density matrix. The values of the quarkonium transport coefficients are obtained from lattice QCD measurements of the bottomonium in-medium width and thermal mass shift; we additionally include for the first time a vacuum contribution to the dispersive coefficient <math><mi>γ</mi></math>. Using the three-loop potential and the values of the heavy quarkonium transport coefficients, we find that the resulting bottomonium nuclear modification factor is consistent with experimental observations, while at the same time reproducing the lattice measurements of the in-medium width.

Tom Magorsch

Michael Strickland

Peter Vander Griend

Bottomonium suppression from the three-loop QCD potential

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!

Franz Gross

Eberhard Klempt

Stanley Brodsky

Andrzej Buras

Volker Burkert

Gudrun Heinrich

Karl Jakobs

Curtis Meyer

Kostas Orginos

Johanna Stachel

Giulia Zanderighi

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

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

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

We demonstrate the importance of quantum jumps in the nonequilibrium evolution of bottomonium states in the quark-gluon plasma. Based on nonrelativistic effective field theory and the open quantum system framework, we evolve the density matrix of color singlet and octet pairs. We show that quantum regeneration of singlet states from octet configurations is necessary to understand experimental results for the suppression of both bottomonium ground and excited states. The values of the heavy-quarkonium transport coefficients used are consistent with recent lattice QCD determinations.

Miguel Ángel Escobedo

Ajaharul Islam

Anurag Tiwari

Regeneration of bottomonia in an open quantum systems approach

We present results for the static energy in (<math><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math>)-flavor QCD over a wide range of lattice spacings and several quark masses, including the physical quark mass, with ensembles of lattice-gauge-field configurations made available by the MILC Collaboration. We obtain results for the static energy out to distances of nearly 1 fm, allowing us to perform a simultaneous determination of the scales <math><msub><mi>r</mi><mn>1</mn></msub></math> and <math><msub><mi>r</mi><mn>0</mn></msub></math>, as well as the string tension <math><mi>σ</mi></math>. For the smallest three lattice spacings we also determine the scale <math><msub><mi>r</mi><mn>2</mn></msub></math>. Our results for <math><msub><mi>r</mi><mn>0</mn></msub><mo>/</mo><msub><mi>r</mi><mn>1</mn></msub></math> and <math><msub><mi>r</mi><mn>0</mn></msub><msqrt><mi>σ</mi></msqrt></math> agree with published (<math><mn>2</mn><mo>+</mo><mn>1</mn></math>)-flavor results. However, our result for <math><msub><mi>r</mi><mn>1</mn></msub><mo>/</mo><msub><mi>r</mi><mn>2</mn></msub></math> differs significantly from the value obtained in the (<math><mn>2</mn><mo>+</mo><mn>1</mn></math>)-flavor case, which is most likely due to the effect of the charm quark. We also report results for <math><msub><mi>r</mi><mn>0</mn></msub></math>, <math><msub><mi>r</mi><mn>1</mn></msub></math>, and <math><msub><mi>r</mi><mn>2</mn></msub></math> in fm, with the former two being slightly lower than published (<math><mn>2</mn><mo>+</mo><mn>1</mn></math>)-flavor results. We study in detail the effect of the charm quark on the static energy by comparing our results on the finest two lattices with the previously published (<math><mn>2</mn><mo>+</mo><mn>1</mn></math>)-flavor QCD results at similar lattice spacing. We find that for <math><mrow><mi>r</mi><mo>&gt;</mo><mn>0.2</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> our results on the static energy agree with the (<math><mn>2</mn><mo>+</mo><mn>1</mn></math>)-flavor result, implying the decoupling of the charm quark for these distances. For smaller distances, on the other hand, we find that the effect of the dynamical charm quark is noticeable. The lattice results agree well with the two-loop perturbative expression of the static energy incorporating finite charm mass effects. This is the first time that the decoupling of the charm quark is observed and quantitatively analyzed on lattice data of the static energy.

Rafael L. Delgado

Andreas S. Kronfeld

Sebastian Steinbeißer

Johannes H. Weber

Static energy in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD: Scale setting and charm effects

Under some assumptions on the hierarchy of relevant energy scales, we compute the nonrelativistic QCD (NRQCD) long-distance matrix elements (LDMEs) for inclusive production of J/ψ, ψ(2S), and Υ states based on the potential NRQCD (pNRQCD) effective field theory. Based on the pNRQCD formalism, we obtain expressions for the LDMEs in terms of the quarkonium wavefunctions at the origin and universal gluonic correlators, which do not depend on the heavy quark flavor or the radial excitation. This greatly reduces the number of nonperturbative unknowns and substantially enhances the predictive power of the nonrelativistic effective field theory formalism. We obtain improved determinations of the LDMEs for J/ψ, ψ(2S), and Υ states thanks to the universality of the gluonic correlators, and obtain phenomenological results for cross sections and polarizations at large transverse momentum that agree well with measurements at the LHC.

South Korea

Hee Chung

Xiang-Peng Wang

Inclusive production of J/ψ, ψ(2S), and Υ states in pNRQCD