# The ALICE experiment: a journey through QCD

S Acharya, D Adamová, A Adler, G Aglieri Rinella, M Agnello, N Agrawal, Z Ahammed, S Ahmad, S.U Ahn, I Ahuja, et al.

► To cite this version:

S Acharya, D Adamová, A Adler, G Aglieri Rinella, M Agnello, et al. The ALICE experiment: a journey through QCD. Eur.Phys.J.C, 2024, 84 (8), pp.813. 10.1140/epjc/s10052-024-12935-y. hal-04679289

HAL Id: hal-04679289

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# The ALICE experiment: a journey through QCD

## ALICE Collaboration*

CERN, 1211 Geneva 23, Switzerland

Received: 20 January 2023 / Accepted: 20 May 2024 / Published online: 14 August 2024
© CERN for the benefit of the ALICE Collaboration 2024

**Abstract** The ALICE experiment was proposed in 1993, to study strongly-interacting matter at extreme energy densities and temperatures. This proposal entailed a comprehensive investigation of nuclear collisions at the LHC. Its physics programme initially focused on the determination of the properties of the quark-gluon plasma (QGP), a deconfined state of quarks and gluons, created in such collisions. The ALICE physics programme has been extended to cover a broader ensemble of observables related to Quantum Chromodynamics (QCD), the theory of strong interactions. The experiment has studied Pb-Pb, Xe-Xe, p-Pb and pp collisions in the multi-TeV centre of mass energy range, during the Run 1-2 data-taking periods at the LHC (2009-2018). The aim of this review is to summarise the key ALICE physics results in this endeavor, and to discuss their implications on the current understanding of the macroscopic and microscopic properties of strongly-interacting matter at the highest temperatures reached in the laboratory. It will review the latest findings on the properties of the QGP created by heavy-ion collisions at LHC energies, and describe the surprising QGP-like effects in pp and p-Pb collisions. Measurements of few-body QCD interactions, and their impact in unraveling the structure of hadrons and hadronic interactions, will be discussed. ALICE results relevant for physics topics outside the realm of QCD will also be touched upon. Finally, prospects for future measurements with the ALICE detector in the context of its planned upgrades will also be briefly described.

The authors of this review are members of the ALICE Collaboration at CERN's Large Hadron Collider. ALICE's roughly 2000 members originate from 40 countries and 173 institutes - making it the largest-ever experimental collaboration dedicated to studying relativistic heavy-ion collisions. These members range from undergraduates to emeriti, scientists, and engineers, united in a global effort to move beyond the frontiers of science. They have, and continue to be, dedicated to unlocking nature's secrets of nuclear matter - with the ultimate goal of developing a profound understanding of the formation of the observable mass in the cosmos.

## Contents

Preface 3

1 Introduction 4

1.1 Emergent phenomena in QCD: the quark-gluon plasma 4

1.2 Experimental investigations of the QGP and QCD 6

1.2.1 The evolution of a heavy-ion collision 6

1.2.2 Observables in heavy-ion collisions 8

1.2.3 Small systems and searches for thresholds of QGP formation 10

1.2.4 Theoretical tools 11

1.3 Evolution of the field: from first studies to ALICE and beyond 14

1.4 ALICE: design considerations, implementation, operation 16

1.5 Overview of the key scientific questions addressed by ALICE 19

2 The quark-gluon plasma and its properties 23

2.1 Macroscopic system properties and QGP thermodynamics 23

2.1.1 Centrality of nucleus-nucleus collisions 24

2.1.2 Charged-particle multiplicity density at midrapidity 24

2.1.3 Determination of the initial energy density 26

2.1.4 Temperature of the system 27

2.1.5 Size and lifetime of the system 29

2.1.6 Conclusions 33

2.2 QGP evolution and its dynamical properties 34

2.2.1 First ALICE results of anisotropic flow 34

2.2.2 Identified hadron spectra, radial flow, and kinetic freeze-out temperatures 35

2.2.3 Hydrodynamic descriptions of global observables 36

2.2.4 Identified hadron anisotropic flow 38

2.2.5 Symmetric cumulants 40

2.2.6 Non-linear flow modes and flow vector fluctuations 41

<footnote>*e-mail: alice-publications@cern.ch</footnote>
<table>
  <tbody>
    <tr>
      <td>2.2.7</td>
      <td>Charge dependent and independent two-particle correlations</td>
      <td>41</td>
    </tr>
    <tr>
      <td>2.2.8</td>
      <td>Polarisation of hyperons and vector mesons</td>
      <td>43</td>
    </tr>
    <tr>
      <td>2.2.9</td>
      <td>Conclusions</td>
      <td>45</td>
    </tr>
    <tr>
      <td>2.3</td>
      <td>Hadronisation of the QGP</td>
      <td>46</td>
    </tr>
    <tr>
      <td>2.3.1</td>
      <td>Hadron gas composition and QCD thermodynamics</td>
      <td>47</td>
    </tr>
    <tr>
      <td>2.3.2</td>
      <td>Particle momentum and angular distributions: connection to microscopic hadronisation mechanisms</td>
      <td>52</td>
    </tr>
    <tr>
      <td>2.3.3</td>
      <td>The hadronic phase</td>
      <td>62</td>
    </tr>
    <tr>
      <td>2.3.4</td>
      <td>Conclusions</td>
      <td>64</td>
    </tr>
    <tr>
      <td>2.4</td>
      <td>Partonic interactions in matter</td>
      <td>66</td>
    </tr>
    <tr>
      <td>2.4.1</td>
      <td>Heavy-quark transport and diffusion measurements</td>
      <td>68</td>
    </tr>
    <tr>
      <td>2.4.2</td>
      <td>Jet quenching measurements</td>
      <td>71</td>
    </tr>
    <tr>
      <td>2.4.2.2</td>
      <td>Jet substructure modification</td>
      <td>77</td>
    </tr>
    <tr>
      <td>2.4.2.3</td>
      <td>Jet multiple scattering and deflection</td>
      <td>80</td>
    </tr>
    <tr>
      <td>2.4.3</td>
      <td>Conclusions</td>
      <td>81</td>
    </tr>
    <tr>
      <td>2.5</td>
      <td>Deconfinement and modification of the QCD force</td>
      <td>82</td>
    </tr>
    <tr>
      <td>2.5.1</td>
      <td>Study of the charmonium ground state: evidence for the (re)generation and demonstration of deconfinement at LHC energies</td>
      <td>83</td>
    </tr>
    <tr>
      <td>2.5.2</td>
      <td>Study of the bottomonium ground state: strong suppression and small (re)generation effects</td>
      <td>85</td>
    </tr>
    <tr>
      <td>2.5.3</td>
      <td>Study of non-QGP effects on ground state suppression</td>
      <td>86</td>
    </tr>
    <tr>
      <td>2.5.4</td>
      <td>Excited quarkonium states</td>
      <td>88</td>
    </tr>
    <tr>
      <td>2.5.5</td>
      <td>Discussion of further flow results for quarkonia</td>
      <td>90</td>
    </tr>
    <tr>
      <td>2.5.6</td>
      <td>Conclusions</td>
      <td>91</td>
    </tr>
    <tr>
      <td>2.6</td>
      <td>Early electromagnetic fields and novel QCD phenomena</td>
      <td>92</td>
    </tr>
    <tr>
      <td>2.6.1</td>
      <td>Searches for the early stage electromagnetic fields</td>
      <td>92</td>
    </tr>
    <tr>
      <td>2.6.2</td>
      <td>Searches for chiral anomalies</td>
      <td>93</td>
    </tr>
    <tr>
      <td>2.6.3</td>
      <td>Conclusions</td>
      <td>98</td>
    </tr>
    <tr>
      <td>2.7</td>
      <td>Quantitative characterisation of the QGP at the LHC</td>
      <td>98</td>
    </tr>
    <tr>
      <td>2.7.1</td>
      <td>Macroscopic properties</td>
      <td>98</td>
    </tr>
    <tr>
      <td>2.7.2</td>
      <td>Microscopic properties</td>
      <td>100</td>
    </tr>
    <tr>
      <td>3</td>
      <td>High-density QCD effects in proton-proton and proton-nucleus collisions</td>
      <td>104</td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Event classification for small collision systems</td>
      <td>105</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Dynamics and hadrochemistry of particle production</td>
      <td>107</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>Collective effects: anisotropic flow</td>
      <td>112</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Charmonium and bottomonium suppression in p-Pb collisions</td>
      <td>117</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Searches for jet quenching in small systems</td>
      <td>119</td>
    </tr>
    <tr>
      <td>3.6</td>
      <td>Conclusions</td>
      <td>121</td>
    </tr>
    <tr>
      <td>4</td>
      <td>The initial state of the collision</td>
      <td>122</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Electroweak-boson measurements</td>
      <td>123</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Photon-induced processes in heavy-ion collisions</td>
      <td>123</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Multiplicity and flow measurements</td>
      <td>127</td>
    </tr>
  </tbody>
</table>

<table>
  <tbody>
    <tr>
      <td>4.4</td>
      <td>Conclusions</td>
      <td>129</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Nuclear physics at the LHC: (anti)nuclei formation and hadron-hadron interactions</td>
      <td>130</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Production of nuclei from small to large collision-system size</td>
      <td>130</td>
    </tr>
    <tr>
      <td>5.1.1</td>
      <td>Light-nuclei yield measurements</td>
      <td>131</td>
    </tr>
    <tr>
      <td>5.1.2</td>
      <td>Flow of light nuclei in Pb-Pb collisions</td>
      <td>131</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Determination of a universal source for small colliding systems</td>
      <td>132</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Hyperon-nucleon and kaon-nucleon interactions</td>
      <td>133</td>
    </tr>
    <tr>
      <td>5.3.1</td>
      <td>Hypertriton lifetime</td>
      <td>133</td>
    </tr>
    <tr>
      <td>5.3.2</td>
      <td>Nucleon-hyperon and hyperon-hyperon interactions: p-Λ, Λ-Λ and p-Σ<sup>0</sup></td>
      <td>135</td>
    </tr>
    <tr>
      <td>5.3.3</td>
      <td>Multistrange hyperon-nucleon interactions</td>
      <td>136</td>
    </tr>
    <tr>
      <td>5.3.4</td>
      <td>K-p interaction: evidence of coupled-channel contributions</td>
      <td>137</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Baryon-antibaryon interactions in Pb-Pb collisions: possible existence of bound states</td>
      <td>138</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Conclusions</td>
      <td>140</td>
    </tr>
    <tr>
      <td>6</td>
      <td>QCD studies with high-Q<sup>2</sup> processes in pp collisions</td>
      <td>140</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Testing perturbative accuracy with jets</td>
      <td>141</td>
    </tr>
    <tr>
      <td>6.1.1</td>
      <td>Inclusive jet production</td>
      <td>141</td>
    </tr>
    <tr>
      <td>6.1.2</td>
      <td>Heavy-flavour jet production</td>
      <td>142</td>
    </tr>
    <tr>
      <td>6.1.3</td>
      <td>Jet substructure</td>
      <td>143</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Studying fragmentation with identified particles</td>
      <td>145</td>
    </tr>
    <tr>
      <td>6.2.1</td>
      <td>Light flavour hadrons</td>
      <td>145</td>
    </tr>
    <tr>
      <td>6.2.2</td>
      <td>Open heavy flavour hadrons</td>
      <td>146</td>
    </tr>
    <tr>
      <td>6.2.3</td>
      <td>Quarkonium</td>
      <td>150</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Direct experimental observation of the QCD dead cone</td>
      <td>150</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Conclusions</td>
      <td>151</td>
    </tr>
    <tr>
      <td>7</td>
      <td>ALICE contributions beyond QCD physics and synergies of heavy-ion physics with other fields</td>
      <td>152</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Neutron stars and the nuclear equation of state</td>
      <td>152</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Cosmogenic dark matter</td>
      <td>154</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Cosmic rays</td>
      <td>155</td>
    </tr>
    <tr>
      <td>7.3.1</td>
      <td>Data for cosmic-ray models</td>
      <td>155</td>
    </tr>
    <tr>
      <td>7.3.2</td>
      <td>ALICE cosmic muon measurements</td>
      <td>156</td>
    </tr>
    <tr>
      <td>7.4</td>
      <td>Atomic physics through lepton-pair production</td>
      <td>157</td>
    </tr>
    <tr>
      <td>7.5</td>
      <td>Searches for new particle states and new physics</td>
      <td>157</td>
    </tr>
    <tr>
      <td>7.6</td>
      <td>Tests of fundamental symmetries</td>
      <td>158</td>
    </tr>
    <tr>
      <td>7.7</td>
      <td>AdS/CFT correspondence and heavy-ion collisions</td>
      <td>159</td>
    </tr>
    <tr>
      <td>7.8</td>
      <td>Ultracold gases and Bose-Einstein condensates</td>
      <td>159</td>
    </tr>
    <tr>
      <td>7.9</td>
      <td>Chiral magnetic effect and new materials</td>
      <td>160</td>
    </tr>
    <tr>
      <td>7.10</td>
      <td>Machine learning</td>
      <td>160</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Summary</td>
      <td>161</td>
    </tr>
    <tr>
      <td>9</td>
      <td>Outlook: ALICE detector and physics for the next two decades</td>
      <td>165</td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>ALICE upgrades and data-taking campaigns for the LHC Runs 3 and 4</td>
      <td>165</td>
    </tr>
    <tr>
      <td>9.1.1</td>
      <td>Major ALICE 2 detector upgrade</td>
      <td>165</td>
    </tr>
    <tr>
      <td>9.1.2</td>
      <td>Proposed upgrades for Run 4</td>
      <td>166</td>
    </tr>
  </tbody>
</table>

<figure><img src="image_3.png" /></figure>
<table><tr><td>9.1.3</td><td>Target data samples and integrated luminosities for LHC Runs 3 and 4</td><td>166</td></tr><tr><td>9.2</td><td>ALICE 3 detector proposal for the LHC Run 5 and beyond</td><td>167</td></tr><tr><td>9.3</td><td>Physics prospects with the ALICE 2 and ALICE 3 detectors</td><td>167</td></tr><tr><td>References</td><td></td><td>172</td></tr></table>

<figure><img src="image_4.png" /></figure>

## Preface

Relativistic heavy-ion collisions at the LHC create the quark-gluon plasma (QGP): the hottest and densest fluid ever studied in the laboratory on Earth. In contrast to normal nuclear matter, the QGP is a state where quarks and gluons are not confined inside hadrons. It is speculated that the early Universe existed in such a state around ~ $10^{-6}$ s after the Big Bang. The strong force, which principally governs behaviour of the quarks and gluons in such a plasma, and all nuclear matter, is responsible for the vast majority of the visible mass in the Universe. The ALICE detector (Fig. 1) was specifically designed to study the QGP created at LHC energies, and this review is dedicated to summarising what has been learnt in

the last decade. In particular, we will address the following questions. What does the Standard Model have to say about the plausibility of the existence of the QGP? Are there emergent phenomena that arise from high density QCD? Does the QGP behave as a gas or a liquid? How quickly does it expand? What happens to the QGP when it is excited by the presence of large momentum/mass quarks or gluons? How do QGP constituents react to the enormous electromagnetic fields created by heavy-ion collisions? Are the extremely high energies at the LHC sufficient for the formation of the QGP in proton-proton collisions? How do the hadrons that emerge in the dense medium after the transition from QGP to normal matter interact between themselves? And what can we learn from these studies regarding the properties of the extremi-
ties in the Universe, such as the core of neutron stars? This review intends to answer these questions in the context of the ALICE measurements performed in the last decade. We will begin by an introduction discussing the theoretical ground- ings in Quantum Chromodynamics (QCD) for the existence of the QGP, and what heavy-ion collisions can teach us (and have already taught us) about QGP properties. At the end of the introduction, the above questions will be reformulated in the context of what we can learn about QGP formation at LHC energies, and what advantages the ALICE detec- tor brings to this endeavour. We will then discuss how our experimentally based conclusions constrain QGP properties from heavy-ion collisions, and after that we will review mea- surements of QGP-like signatures in proton–proton (pp) and proton–lead (p–Pb) collisions. We will also discuss the con- tributions of ALICE measurements in pushing QCD to its limits in the context of few-body interactions. We will finish by discussing connections to other fields of physics from our studies, and conclude by summarising what we have learnt so far, and what we look forward to for the future phases of ALICE in the 2020s and 2030s.

# 1 Introduction

## 1.1 Emergent phenomena in QCD: the quark–gluon plasma

Understanding nature is one of the main goals of science. In the reductionist approach, one tries to infer all mani- festations of reality using a compact set of relations [1,2]. Such an approach has proved to be very fruitful in pinning down fundamental interactions, and discovering the basic building blocks of the universe. It has led to a framework, the Standard Model of particle physics [3–6], that describes the electromagnetic, weak and strong forces. It also allows for quantitative predictions of phenomena involving elemen- tary particles, using the mathematical language of quantum field theory [7]. The strong interaction, the major topic of ALICE’s research, is responsible for the very existence of atomic nuclei. Broadly speaking, it is also responsible for ~95% of the visible mass of the Universe.

Quantum Chromodynamics (QCD) is one of the pillars of the Standard Model of particle physics. It is a gauge field the- ory describing the strong interaction [8]. One of its distinctive features is asymptotic freedom, a consequence of the non- Abelian nature of the SU(3) group on which QCD is based. Asymptotic freedom results in the interaction between quarks and gluons (partons), the elementary particles that experience strong interaction, becoming weaker when their momentum exchange increases. Therefore, the coupling constant $\alpha_S$ of the strong interaction depends on the momentum scale of the interaction. Processes corresponding to large momentum transfers between partons [9], where $\alpha_S$ is small, can be inves-

tigated via a perturbative approach (pQCD). This involves an expansion to higher orders in $\alpha_S$ [10], with only lower orders contributing significantly. For calculations concern- ing low-momentum processes, the perturbative approach of QCD breaks down, as higher orders become dominant [11]. Such a non-perturbative regime, where the coupling constant becomes large, is of the utmost interest for the description of several essential features of the strong interaction. In par- ticular, in this regime confinement is observed [12]. This leads to the fact that quarks and gluons, the elementary par- ticles carrying the “charge” of the strong interaction, known as colour, cannot be isolated, and therefore not directly observed. Only composite objects, hadrons, which do not possess a net colour charge, can be detected. Another key fea- ture of strong interaction in the non-perturbative regime is the spontaneous breaking of chiral symmetry [13–15]. This gives rise to the predominant fraction of the mass of hadrons, for example protons and neutrons. Theoretical approaches such as Lattice QCD [16–18] and Effective Field Theories [19– 21] represent viable solutions for the description of the non- perturbative regime of strong interactions. In this domain, guidance provided by experimental findings considerably helps the progress of the theory.

The behaviour of extended systems subject to the strong interaction provides a unique avenue to understand the strong interaction further. For the electromagnetic interaction, con- densed matter physics [22] studies phenomena as magnetism and superconductivity. These are prime examples of emer- gent behaviour [23], in the sense that their manifestations do not directly arise from the laws governing microscopic interactions. The study of “QCD condensed matter” can be performed by producing a many body system of quarks and gluons, under the conditions of large energy density. Heating such a system, with zero (or very small) net baryon den- sity, to a temperature exceeding 150–160 MeV (equivalent to more than $10^{12}$ K), leads to the creation of the quark– gluon plasma (QGP). This is a state of matter where two of the basic features of low-temperature QCD, confinement and chiral symmetry breaking, are no longer present [24–26].

A primary role for the study of the QGP is played by Latt- ice QCD. Lattice QCD uses continuous space-time discre- tised on a lattice of finite size, with quarks defined at lattice sites, and gluons for the links connecting those sites. This approach can be used for QCD regarding static problems in the non-perturbative domain. Therefore, it is well suited for the study of the thermodynamic properties of a partonic sys- tem, such as the QGP. The thermodynamic properties can be estimated more precisely with increasing computational power. In this respect, reliable estimates of the properties of strongly interacting matter via Lattice QCD calculations require a number of important aspects. These include the use of realistic masses [27] for the dominant quark flavours (the so-called (2+1)-flavour QCD, corresponding to up, down and
<figure><img src="image_5.png" /><figcaption>**Fig. 1** An ALICE event display of detected particles created in a Pb–Pb $\sqrt{s_{NN}}$ = 5.02 TeV collision</figcaption></figure>

strange quarks), and a controlled extrapolation to the contin-
uum limit [28]. Indeed the charm, bottom and top quarks are
too heavy to significantly add to the dynamics of the sys-
tem, and it is the dynamics of the strange and, most impor-
tantly, of the light up and down quarks to determine the rich
phase structure of QCD. Once the equation of state of QCD
matter is evaluated with this technique, it is then possible
to extract the temperature dependence of various thermody-
namic quantities, such as the pressure, the energy and entropy
density. The main results of such calculations demonstrate
that a strongly interacting system with zero net baryon den-
sity evolves smoothly from a confined (hadronic) towards
a deconfined (quarks and gluons) state, when its tempera-
ture is increased up to $T \sim 155$ MeV [26,28]. Since there
is no discontinuity in thermodynamic variables, a crossover
transition occurs, where de-confined and confined hadronic
matter can co-exist and no latent heat is involved. The cor-
responding temperature is commonly indicated as “pseudo-
critical temperature” $T_{pc}$. Moreover, it is found that such a
cross-over occurs is in coincidence with the restoration of the
chiral symmetry. The liberation of many new degrees of free-
dom is indicated by a strong increase in the energy density
($\epsilon$) normalised to the fourth power of temperature (Stefan–
Boltzmann law) around the deconfinement temperature. This
can be clearly seen in Fig. 2, where the extrapolation to the
continuum of lattice simulations of (2+1)-flavour QCD is
shown [26]. The temperature dependence of other thermo-
dynamical quantities as pressure ($p$) and entropy density ($s$)
are also reported.

Figure 2 also shows that even at $T \sim 400$ MeV, the energy
density is still ~ 20% lower than the non-interacting ideal-
gas limit. It is expected that this limit will be reached at very
large $T$, due to the asymptotic freedom of QCD [9], i.e. the
vanishing strength of the strong interaction for increasingly
large momentum transfers. This residual difference has pro-
found consequences on the properties of the QGP, with the
remaining coupling responsible for various QGP features.
In particular, below the ideal gas limit, the QGP manifests
itself as a strongly interacting system of quarks and gluons.
It behaves as an almost perfect liquid [29], with the pres-
ence of bound states of quasi-particles (qq, gg, qg) [30].
These properties have furnished in the literature to the defi-
nition of sQGP, with the letter “s” underlining its strong cou-
pling [31]. Strongly coupled systems like the QGP also occur
in the domain of electromagnetic interactions. For example,
ultracold atomic Fermi gases in the extremely small pico-
eV temperature range are the subject of an intense research
activity [32].

It is worth stressing that the result shown in Fig. 2 is
obtained for a QCD medium with zero net baryon number.
This is a particularly relevant configuration as it corresponds
to that of the early universe, where deconfined quarks and
gluons also hadronised around $T_c$. A complementary situa-
tion, corresponding to large baryon density and relatively low
temperature, may also lead to the creation of the QGP. This
could be present in the core of neutron stars [33]. In the labo-
ratory, a situation corresponding to that of the early universe
can be obtained by colliding heavy ions in the energy range
accessible to hadron colliders. The CERN Large Hadron Col-
<figure><img src="image_6.png" /><figcaption>**Fig. 2** Pressure, energy density and entropy density normalised to the 4th (3rd for the latter) power of the temperature, from the Lattice QCD calculations of the HotQCD Collaboration, see Ref. [26]. The dark lines show the prediction of the Hadron Resonance Gas model, the horizontal line corresponds to the ideal gas limit for the energy density. The vertical band indicates the cross-over transition region. Corresponding results from the Wuppertal–Budapest Collaboration can be found in Ref. [28]</figcaption></figure>

lider (LHC) [34] is currently the facility where, starting from 2010, the highest collision energies are attained. The values of the center-of-mass energies per nucleon pair have reached $\sqrt{s_{NN}} \sim 5$ TeV. The main scope of this review is to analyse and discuss the results obtained by ALICE (A Large Ion Collider Experiment) [35] in order to study QGP properties.

In the following sections of this Introduction, a discussion of the general aspects of experimental and theoretical investigations of high temperature and energy QCD using the ALICE detector will occur (Sect. 1.2). Then, a brief review of the history and the main results obtained from lower energy facilities regarding QGP formation will be given (Sect. 1.3). A concise introduction of the ALICE detector, with key details on its design, realisation and performances will follow (Sect. 1.4). Finally, an overview of the key scientific questions addressed by ALICE, in light of QGP studies, but also exploring other areas of QCD, will be described (Sect. 1.5). In that last Section the organisation of the following chapters will also be discussed.

## 1.2 Experimental investigations of the QGP and QCD

The purpose of this section is three-fold. It will start by summarising the conceptual framework that has arisen in the field of heavy-ion physics over a number of decades. It will then introduce commonly measured observables, which will be shown and elaborated on in subsequent chapters. It will then finish by a brief description of the main theoretical approaches that attempt to use such observables to further investigate the QGP, and more generally QCD. Additionally

theoretical approaches in the context of comparisons to ALICE data will also be explored in subsequent chapters.

### 1.2.1 The evolution of a heavy-ion collision

Collisions of heavy ions with ultra-relativistic energies are used to create the QGP in the laboratory. The evolution of a heavy-ion collision is commonly described in terms of a series of stages, which can in principle be factorised. They include: (i) an initial state, defined by the wave-functions of the projectiles, which are universal and independent of any specific scattering process; (ii) large-$Q^2$ interactions of partons drawn from the projectiles<fnref n="1" />; (iii) smaller-$Q^2$ interactions generating a pre-equilibrated parton gas; (iv) equilibration and expansion of the QGP; (v) hadron formation; (vi) chemical freeze-out of hadrons; (vii) hadronic interactions that subsequently freeze-out kinetically; (viii) free-streaming of stable particles to the detector.<fnref n="2" /> This evolution is illustrated schematically in Fig. 3.<fnref n="3" /> As the heavy ions collide, an extremely dense region of partons is excited and deposits energy and entropy in the overlap region of the collision. Before they interact, the nuclei at the LHC will be highly Lorentz contracted, as indicated in Fig. 3. The impact parameter $b$ is the distance between the centres of the colliding nuclei. It very closely related to the number of nucleons in the nuclei that participate in an inelastic interaction (at least once), referred to as $N_{part}$. It also controls the volume of the collision region. The total number of inelastic nucleon-nucleon collisions is referred to as $N_{coll}$. When $b$ is small, $N_{part}$ and $N_{coll}$ will be large, and vice-versa. The largest number $N_{part}$ can reach is $2A$ (if each colliding nuclei have the same number $A$ of nucleons) and for $N_{coll}$ the respective number is $\sim A^{4/3}$ (which can reach values of $\sim 2000$ for Pb–Pb at LHC energies). Nucleons not participating in the collision are defined as spectators, and continue travelling approximately along the beam direction after the collision. In the inelastic nucleon–nucleon interactions, a variety of QCD processes occur, which, as mentioned, involve a range of $Q^2$ momentum transfers, and each range plays a distinct role.

The partons within the nuclei that are involved in the smaller-$Q^2$ interactions determine the overall energy density and entropy deposition in the initial state, and their interaction rate is largely driven by $N_{part}$. Such parton interactions lead to a “lumpiness” of the initial density profile, which is the result of fluctuations in the distribution of nuclear matter, and is depicted in Fig. 3. Immediately after the collision, the smaller-$Q^2$ interactions occur in the context of a weakly coupled pre-equilibrium phase. This is followed by the creation

<footnote n="1">¹ $Q^2$ being the 4 momentum transfer squared.</footnote>

<footnote n="2">² For a more detailed conceptual description of heavy-ion collisions, see Ref. [36].</footnote>

<footnote n="3">³ This figure was inspired by illustrations from Chun Shen.</footnote>
ation of even softer partons in these processes, which enable the formation of a strongly coupled QGP phase. The hard processes from large-$Q^2$ interactions, with their rate driven by $N_{\text{coll}}$, enable the creation of high momentum gluons and high momentum/mass quarks, as indicated in Fig. 3 via the gluon and charm quark trajectories. As they have short wavelengths, they will interact with other quarks and gluons on a microscopic level, leading to energy loss effects (the energy being transferred to the medium), and therefore they offer information on the opaqueness of the QGP. The interactions of high-momentum partons with the QGP can be radiative as indicated in Fig. 3 for a gluon, as well as elastic, as indicated by the change in direction of the charm quark. The amount of energy loss will depend on the colour charge, momentum, mass, type of process (inelastic or elastic), the distance traversed (path length) of the hard scattered parton, and is subject to stochastic processes. The heavy quarks produced via hard processes can also form quarkonia (bound heavy quark-antiquark states), with their production rate being suppressed because the binding force between the quark and anti-quark is weakened (screened) by the presence of the colour charge of quarks and gluons. That suppression is closely related to the temperature of the QGP, and can be counterbalanced by a regeneration process that recombines heavy quarks participating in the medium interactions, depending on the abundance of heavy quarks. In addition, the parton fragmentation processes (indicated by the yellow cone) lead to jets, partonic showers that arise from these high energy partons, and that fragment into experimentally observable hadrons once the shower components reach low virtuality. That fragmentation pattern in the medium can be altered compared to vacuum-like conditions, e.g. $e^+e^-$ collisions.

The evolution of the QGP for most processes involved in soft interactions after ~1 fm/c can be understood as follows. Since the mean free path of the vast majority of QGP constituents is expected to be much smaller than the size of the QGP formed (assuming these constituents are strongly coupled), multiple interactions drive the expansion. This expansion is highly influenced by the non-uniform energy distribution in the initial state as a function of space, which creates pressure gradients in the QGP stage, that act to cool the system and to smooth out the lumps as time progresses. If the length scales of these gradients are larger than the mean free path, such an evolution exhibits long-wavelength behaviour, which can be described as a liquid in the context of hydrodynamics. A radial flow occurs due to a greater pressure at the centre of the QGP compared to the outskirts, and this leads to a common velocity field outwards. The rate of the hydrodynamic expansion is influenced by the QGP’s bulk viscosity, which is its resistance to volume growth. Anisotropic flow is the result of a directional dependence to these pressure gradients. This occurs due to spatial anisotropies in the initial state. These arise if the collision zone is almond shaped (at

$b > 0$), or due to the lumpiness of the initial state. Such spatial anisotropies are converted to momentum anisotropies via the hydrodynamic response. This is influenced by the QGP’s shear viscosity, which quantifies the resistance to fluid deformation.

Since the nuclei are charged, the movement of the incoming beams sets up a very large magnetic field (from the protons in the nuclei with relativistic energies), which can also have an influence on motion of quarks during the QGP stage. The Chiral Magnetic Effect (CME), which is purported to result from strong parity violation, could lead to a splitting among positive and negative quarks in the QGP along the direction of the extremely large magnetic fields produced from the colliding ions. The decaying magnetic field will also induce an electric field, which can move the electrically charged quarks accordingly. Such an effect prolongs the decay time of the magnetic field (Lenz’s law). The high-temperature QGP also produces thermal radiation, in the form of both photons and lepton-antilepton pairs. The electromagnetic radiation does not interact via the strong force in the QGP, and can therefore be used to gain information about the temperature from early to late times in the collision processes. In addition, strange quarks, which have masses below the deconfinement temperature (therefore the QGP temperature) can also be produced easily in the QGP stage, with the dominant mechanism coming from gluon interactions.

During the evolution, the parts of the QGP that cool below the transition temperature $T_{\text{pc}}$ will hadronise. Since it is likely that the temperature and energy density of the medium will diminish with increasing distance from the collision centre, and, as indicated by Lattice QCD, the transition to normal nuclear matter is a smooth cross-over, it is possible that hadronisation will happen at different times at different places in the phase-space. Nonetheless, the formation of hadrons from the QGP can proceed as follows. Hard partons in a jet will fragment and hadronise in the similar manner as in elementary collisions. For partons at lower momenta, if they share a similar space and momenta as other partons, they can combine into hadrons via coalescence. Heavy (charm or beauty) quarks can also combine with heavy antiquarks, thus forming quarkonia and give rise to an additional production mechanism for closed heavy-flavour hadrons, which can compensate the deficit due to melting in the QGP.

After the hadrons materialise out of quarks and gluons, primarily pions ($\pi$), kaons (K) or protons (p), the energy density may be large enough to allow for inelastic interactions, with a consequent evolution of their “chemical” composition, in terms of particle species. A loosely bound state such as the deuteron (d) shown in Fig. 3 is particularly sensitive to such interactions, as it can be easily formed or destroyed. Such interactions cease at the chemical freeze-out temperature $T_{\text{chem}}$, fixing the particle composition. Elastic interactions can still continue, demonstrated by the $\pi$-K interaction
<figure><img src="image_7.png" /><figcaption>Fig. 3 The evolution of a heavy-ion collision at LHC energies</figcaption></figure>

in Fig. 3, and halt at the kinetic freeze-out temperature $T_{kin}$, which is achieved at the time of ~ 10 fm/c. At this point, the particle momenta are fixed. These particles travel towards the ALICE detector, where they will be measured ~ 10$^{15}$ fm/c after the initial collision. The high-energy beams of the LHC provide an unprecedented opportunity to study the QGP in the laboratory. The highest centre of mass energy per nucleon pair ($\sqrt{s_{NN}}$) achieved at the LHC in Pb-Pb collisions has been 5.02 TeV, which is ~ 25 times higher than the top energy available at the Relativistic Heavy-Ion Collider (which began taking data in 2000). This in principle allows for the hottest, densest, and longest ever lived QGP formed in the laboratory to be probed using the ALICE detector.

### 1.2.2 Observables in heavy-ion collisions

There are various experimental probes used to investigate all phases of heavy-ion collisions: the initial state, the QGP phase, and the final hadronic phase. Each of these probes has varying sensitivities to each phase. To start, a fundamental quantity for many of these probes is the Lorentz-invariant differential yield of final state particles, given by:

$$E \frac{d^3 N}{dp^3} = \frac{1}{2\pi p_T} \frac{d^2 N}{dp_T dy}, \qquad (1)$$

which is the number density of the particle three-momentum scaled by the particle energy (E). This will depend on the measured particle species in question, the transverse momentum, $p_T$, and rapidity, $y$. When the particle species is not

known, pseudorapidity $\eta$ is used instead,<fnref n="4" /> and both are equivalent when the particle energy is much greater than its mass. Different $p_T$ ranges will probe different physical processes, and for clarity in this review, one can define low-$p_T$ as $p_T \lesssim 2$ GeV/c, intermediate-$p_T$ as $2 \lesssim p_T \lesssim 8$ GeV/c, and high-$p_T$ as $p_T \gtrsim 8$ GeV/c. This classification is not intended to provide a rigid distinction regarding these processes, but to aid the reader regarding references to momentum ranges.

**Initial state.** Regarding the initial state, for a given collision, the multiplicity can be determined, which is an addition of the number of charged hadrons in a broad momentum range. It plays a critical role in providing a selection on a range of impact parameters, $b$, for heavy-ion collisions. It will be large when $b$ is small, which leads to large numbers of $N_{part}$ that will correspondingly produce large numbers of particles. Such small-$b$ collisions are referred to as central i.e. head-on, whereas collisions with large impact parameters and small numbers of $N_{part}$ (and fewer produced particles) are referred to as peripheral. The multiplicity therefore provides an experimental handle on the centrality of a collision. Such a handle is extremely useful, as many of the system properties such as the energy density or lifetime depend upon the centrality. The multiplicity can also provide a measure of the initial state entropy. If the system hydrodynamically evolves without internal resistance i.e. viscous effects are minimal, the multiplicity in the final state can be used to determine the

<footnote n="4">⁴ Defined as $\eta = -\ln[\tan(\theta/2)]$, $\theta$ being the polar emission angle of the particle. A more complete description of kinematic variables can be found in a Review of Particle Physics by the Particle Data Group [37].</footnote>
initial state entropy, as entropy is conserved in this case. If viscous effects become appreciable, entropy is created during the evolution, and the multiplicity then serves as an upper limit to the initial state entropy directly.

Ultra peripheral collisions (UPCs) provide another handle on the initial state. These are heavy-ion collisions that typically have very large b values (greater than the nuclear diameter e.g. ~ 15 fm for Pb). The nuclei in these collisions are sufficiently separated such that short-range strong interactions are highly suppressed (and therefore no QGP formation), yet the huge electromagnetic fields generated in the vicinity of the colliding nuclei lead to photon mediated interactions. Measurements of the differential cross sections of light (e.g. $\rho^0$) and heavy (e.g. J/ψ) vector mesons from these collisions are especially sensitive to the momentum distribution of partons (i.e. quarks and gluons) inside the nucleus, parameterised as nuclear parton distribution functions (nPDFs). These distributions are modified compared with proton-proton collisions parton distribution functions (PDFs), and nPDFs play a critical role for understanding the production rate of hard processes in the initial stages of heavy-ion collisions.

**QGP phase.** The amount of the hydrodynamic collective motion developed in the QGP phase can be explored using measurements sensitive to radial and anisotropic flow [38], which develop transverse to the beam direction. In the low-$p_T$ region, radial flow causes an increase in the transverse momentum ($p_T$) of higher-mass hadrons. Measurements of the mean transverse momentum $\langle p_T \rangle$ of identified particles can therefore be used to extract information on the amount of radial flow that has developed. In order to investigate anisotropic flow, Eq. 1 can be expanded into the azimuthal angular dependence of the transverse momentum vector direction. The most widely used decomposition expresses the $\varphi$ (the angular component of this vector) dependence of the produced particle density as a Fourier series as follows [39]:

$$ \frac{dN}{d\varphi} \propto 1 + 2 \sum_{n=1}^{\infty} v_n \cos[n(\varphi - \Psi_n)]. \qquad (2) $$

The $v_n$ terms, which can be determined experimentally, are referred to as anisotropic flow coefficients. They depend on the particle species, $p_T$ and $\gamma$. The term $n$ corresponds to the order of anisotropic flow, and $\Psi_n$ is the corresponding symmetry plane angle, which is the angular direction of anisotropic flow for the order $n$ of interest. For the vast majority of produced hadrons, non-zero $v_n$ coefficients arise mainly from the QGPs hydrodynamic response (with a relatively small contribution from the hadronic state). Regarding the coupling of the created electromagnetic fields with quarks, the CME effect introduces charge-dependent sine terms in Eq. 2 for the produced hadrons with corresponding

coefficients. These are small relative to the $v_n$ coefficients, but can also be explored experimentally. Any motion due to an electric field in the QGP will lead to charge-dependent $v_1$ coefficients.

Hard probes, as mentioned, are produced in the earliest times of the collision. One of the ways they can be explored is by extracting their nuclear modification factor $R_{AA}$. This observable is constructed to be sensitive to changes of the dynamics of hard processes in heavy-ion collisions with respect to expectations from elementary pp collisions. Within a $\gamma$ or $\eta$ interval, $R_{AA}$ as a function of transverse momentum is defined as follows:

$$ R_{AA}(p_T) = \frac{1}{\langle T_{AA} \rangle} \frac{dN_{AA}(p_T)/d p_T}{d\sigma_{pp}(p_T)/d p_T}. \qquad (3) $$

The average nuclear overlap function $\langle T_{AA} \rangle$ is obtained from the average number of $N_{coll}$ divided by the inelastic nucleon-nucleon cross section for the centrality range of interest, and its estimation is discussed in Sect. 1.2.4. For hard processes, the yield $N_{AA}$ in heavy-ion collisions is expected to scale with the average nuclear overlap function $\langle T_{AA} \rangle$ when compared to the production cross section $\sigma_{pp}$ in pp collisions, in the absence of any QGP or initial state nuclear effects. That being the case, if $R_{AA}(p_T) = 1$, production from heavy-ion collisions can be considered as a superposition of nucleon-nucleon collisions. Any departure from unity reveals how these processes are modified in heavy-ion collisions. In particular, $R_{AA}$ is expected to be below unity at high-$p_T$ for inclusive hadrons from partons undergoing in-medium energy loss. Jet-finding algorithms can also be applied in heavy-ion collisions. The internal structure of jet showers is governed by quantum interference effects, resulting in the phenomenon of “angle-ordering”, whereby the highest $p_T$ hadrons in the shower are on average most closely aligned with the nominal jet axis. A key parameter in jet measurements is the “jet radius” or “resolution parameter” $R$, which is effectively the size of the aperture through which the jet shower is viewed. The measured jet yields can be used to determine $R_{AA}$, while the jet radius dependence of the $R_{AA}$ or jet substructure measurements provide information about the medium modifications of the quark and gluon radiation patterns. The heavy-ion jets can be compared to jets in pp collisions, where no such modifications are expected. Finally, measurements of $v_n$ at high-$p_T$ can be used to explore the path length dependence of energy loss. Rather than resulting from the hydrodynamic response, finite values can arise as large momentum partons lose less energy in the $\Psi_n$ direction as the path length is smaller, compared to the perpendicular direction where this is larger, within the elongated nuclear overlap region. This leads to a larger abundance of high-$p_T$ hadrons in the $\Psi_n$ direction, which, in turn, results in an observed non-zero $v_n$.
A specific class of hard probes, heavy quarks, gives access to several QGP features that can be accessed via differential measurements of $R_{AA}$ and $v_n$. A key difference is that the corresponding hadrons are associated with early-stage probes across their entire $p_T$ range. D mesons, which carry the vast majority of the charm quarks produced, can be investigated differentially as function of $p_T$ via measurements of $v_n$ and $R_{AA}$. Measurements of $v_n$ at low-$p_T$ are sensitive to degree to which heavy quarks take part in the collective expansion of the QGP and approach thermalisation. Measurements of $R_{AA}$ at high-$p_T$ provide insights on the energy loss processes of heavy quarks, which maybe smaller than for light quarks or gluons, due to the dead cone effect [40]. Measurements of quarkonia (e.g. J/ψ or Υ(1S)) $R_{AA}$ and $v_n$ are also carried out as a function of $p_T$. The $R_{AA}$ studies allow the investigation of the aforementioned suppression and regeneration processes, with the latter expected to be dominant in the low-$p_T$ region due to larger heavy quark multiplicity in that kinematic range. Anisotropic flow measurements give complementary information to those from open heavy flavours, with the observed effect not involving the contribution of the flow of light quarks.

Electroweak probes, e.g. photons, leptons or Z/W bosons, provide another class of measurements. If they arise from hard processes in the initial stages, measurements of $R_{AA}$ at high-$p_T$ are expected to be unity, and any deviations from this reveal the influences of non-QGP processes that affect this measurement e.g. isospin effects or differences in nPDFs and PDFs. On the other hand, measurements of direct photon (i.e. not from hadron decays) spectra at low-$p_T$ are expected to arise mainly from the softer processes in the QGP involving charged quarks, and their slope can be related to the temperature when they are created. Direct photons can also be produced from hadronic interactions, so that the observed yield is a convolution of their emission along the whole collision history. Since photons are produced in an expanding QGP medium, they will be blue-shifted, which has to be taken into account regarding their connection to the temperature of the QGP.

**Hadronic phase.** A variety of methods are available to investigate the hadronic phase of the heavy-ion collisions, and will be outlined in the following. Measurements of identified hadron spectra and $v_n$ at intermediate-$p_T$ are sensitive to the coalescence of quarks upon hadronisation from the QGP phase, which will compete with hadronisation from the well known fragmentation processes observed in QCD interactions. Such a coalescence mechanism enhances baryon production rates and $v_n$ values compared to mesons in this $p_T$ range. The chemical freeze-out temperature $T_{chem}$ can be explored by measuring the total yields of specific particles (dN/dy) via an integration over $p_T$ in Eq. 1 – for hadrons that do not decay quickly via the strong or electromagnetic interactions (e.g. protons, kaons, charged pions, hyperons etc.). Measurements of the production of strongly decaying resonances can offer insight into the duration of the hadronic phase. Their decay products for temperatures both above and below $T_{chem}$ are subject to elastic scattering. If these effects are strong enough, they will lead to a depletion in the resonance yields, as the kinematics of the particles resulting from a resonance decay are altered in such a way that the resonance cannot be reconstructed anymore. On the other hand, regeneration processes from hadron–hadron interactions may form resonances during this phase. Final state hadron–hadron interactions can also be studied via the pair-wise femtoscopic correlation function, which can be extracted experimentally as:

$$C(k^*) = \frac{N_{\text{pairs (same event)}}}{N_{\text{pairs (background)}}}, \qquad (4)$$

where $k^*$ is the invariant momentum difference between the pairs. The numerator is extracted from pairs produced in the same collision, and the denominator is obtained from a background hypothesis that assumes no correlated pairs, and is normalised such that $C(k^*) = 1$ in the absence of same-event correlations. Femtoscopic correlations arise from quantum correlations which are sensitive to the size of the system at freeze-out, or final state hadron–hadron interactions, both of which predominately occur at low-$p_T$. Finally, the slope of hadron spectra can be used to infer the kinetic freeze-out temperature $T_{kin}$. Such an approach relies on modelling to disentangle the contribution from radial flow, and will be discussed in the next chapter.

### 1.2.3 Small systems and searches for thresholds of QGP formation

Proton–proton collisions, where initial state nuclear effects are not relevant, were envisioned to provide a crucial reference to ultrarelativistic heavy-ion collisions. Collisions of p–Pb also have a further complementary role, by providing a reference where cold nuclear matter (CNM) effects are expected, but in principle, no QGP effects. CNM effects include modifications of the parton distribution functions in the nucleus compared to the proton, multiple scatterings in nucleons that collide with more than one other nucleon, the Cronin Effect [41], parton energy loss in cold nuclear matter, and absorption of the produced hadrons by the nucleus. One way to study CNM effects independently of the formation of the QGP (in principle), is by measuring $R_{pPb}$, which is an analogous application of Eq. 3 for p–Pb collisions. As CNM effects are also expected to occur in heavy-ion collisions, such measurements are very important for disentangling CNM effects from QGP effects, and are of interest in their own right. Proton–proton collisions, and to some extent proton–nucleus collisions, have provided opportunities for
studying QCD interactions in few-body systems. For example, inclusive and heavy-flavour jet cross sections provide stringent tests on perturbative QCD (pQCD), while measurements of the production of identified hadrons can be used to test the universality (collision-system independence) of parton-to-hadron fragmentation functions (FFs). The use of femtoscopic measurements to study final state hadron-hadron interactions, which occur for any collision system producing hadrons, is another example.

All things considered, pp and p-Pb collisions therefore provide an opportunity to study QCD at the LHC in a more dilute environment. Studying the multiplicity dependence of observables described in the previous section for pp and p-Pb collisions provides a means to explore the thresholds for QGP formation. Unlike the case for heavy-ion collisions, high multiplicity events in pp and p-Pb collisions are not expected to result from a trivial increase in the amount of colliding matter: this is obviously the case for pp collisions as $N_{\text{part}}$ is always 2. Rather, given the initial system volume is somewhat fixed, high multiplicity events may be associated with collisions that have energy densities exceeding the values required for QGP formation. Indeed, the highest number of particles produced in such collisions are comparable to peripheral heavy-ion collisions at lower energies, where QGP formation is established. Even if the energy densities in pp and p-Pb collisions reach values needed for QGP formation, it is not clear a priori if and which typical QGP effects can be observed in their study. A detailed investigation in this sense, using measurements developed in heavy-ion collisions to explore QGP-like behaviour - such as anisotropic flow coefficients, the production rate of hadrons from hard processes, or changes in hadronisation mechanisms as a function of multiplicity, represents a fascinating avenue of study.

### 1.2.4 Theoretical tools

A wide variety of theoretical tools are employed to investigate the aforementioned processes in heavy-ion and light system collisions. They range from fundamental QCD calculations to effective theories and phenomenological models. Some of the main goals of comparing the output of theory to the previously described measurements are as follows:

(i) Map the spatial and momentum distributions of the initial state.

(ii) Provide quantitative constraints on QGP properties and probe QGP behaviour.

(iii) Understand the contribution of hadronic interactions to final state measurements and explore the global properties of the hadronic phase.

(iv) Assess the degree QGP effects or few body QCD interactions describe the data in heavy and small systems.

**Initial state.** The use of initial state modeling is an essential prerequisite to determine QGP properties, and is of vast interest in and of itself. The Monte Carlo (MC) Glauber approach can be used to infer $\langle N_{\text{part}} \rangle$, $\langle N_{\text{coll}} \rangle$, and $\langle T_{AA} \rangle$ within a particular centrality/multiplicity range for Pb-Pb and p-Pb collisions [42,43]. This approach assumes the nucleons in the nucleus are positioned according to the Woods-Saxon density distribution. The nucleons travel in an unperturbed trajectory irrespective of whether they interact with other nucleons. The criterion for an inelastic nucleon-nucleon interaction then depends on the cross section of a single interaction, which can be inferred from experimental measures in pp collisions. These assumptions are particularly relevant for heavy-ion collisions at the LHC that are in transparency regime, which implies the nuclear remnants after a collision travel in a straight line.

Another output of such an approach is the initial state spatial eccentricity, $\epsilon_n$, in the (transverse) $xy$ plane. This depends on how the matter in the initial state is distributed. It is an essential input for the hydrodynamic model chains (which will be described shortly), as non zero values of $\epsilon_n$ generate anisotropic flow. Other initial state models go further. For example, the IP-Glasma model uses MC Glauber assumptions [44], but then determines the spatial gluon distributions in a nucleon using the QCD based Colour Glass Condensate ansatz [45]. It attempts to model these distributions and how they overlap in a heavy-ion collision in the context of nuclear saturation effects, which control the high number of low-$x$ gluons populating the transverse extent of nucleons in a high-energy collision. Saturation effects dictate the population of the vast majority of partons at low-$x$, and their corresponding spatial distributions. This additional feature influences the value $\epsilon_n$ compared to the MC Glauber approach, and IP-Glasma generally predicts higher values of $\epsilon_2$ in a particular centrality range. A comparison of the energy densities from MC-Glauber (derived from nucleons) and IP-Glasma (gluons) is shown in Fig. 4 in the $xy$ plane. For the study of hard processes in both elementary and nuclear collisions, knowledge of the initial momentum distributions of partons also is an essential prerequisite. These distributions can be constrained using measurements from UPC collisions with the appropriate theoretical comparisons. Parameterisations of PDFs in protons, expressed as a function of the fractional momentum $x$ of the parton and the momentum transfer $Q^2$ are modeled generally starting from the results of deep inelastic scattering experiments. In the nucleus, typical depletion effects (i.e. a reduction in the nPDF) such as shadowing are prominent at low-$x$ ($\ll 0.1$), and can have notable consequences on the production of hard processes. Anti-shadowing effects lead to parton excesses relative to pp collisions at high-$x$ ($\sim 0.1$). Recent parameterisations
<figure><img src="image_8.png" /><figcaption>**Fig. 4** The xy distributions of the initial energy density (arbitrary units) from the MC-Glauber and IP-Glasma models for a heavy-ion collision [44]</figcaption></figure>

of nPDFs including these effects via fits the data have been investigated elsewhere [46,47].

**QGP properties and behaviour.** Viscous hydrodynamic theory is one of the main methods employed to investigate the dynamical evolution of the QGP. Lattice QCD, being a static theory in space and time, cannot be used for this purpose. Viscous hydrodynamic theory is used in the context of a hydrodynamic model chain e.g. IP-Glasma+MUSIC [48] or TRENTO-VISHNU [49], with components that attempt to describe each stage of a collision. These components include the energy/entropy density from an initial state model at the moment of the collision, a pre-equilibrium phase that describes the weakly coupled dynamics prior to the hydrodynamic phase, an implementation of the viscous hydrodynamics to describe the evolution of the QGP when it is formed, a partclisation scheme to create hadrons when the deconfinement temperature is reached, and finally a model to describe hadronic interactions. The models used for hadronic interactions, often called hadronic afterburners, will be described later in this subsection. The viscous hydrodynamic stage itself uses a Lattice QCD equation of state to evolve a strongly interacting system via energy and momentum conservation equations. As the final output of these hydrodynamic model chains are the momentum distributions of hadrons, this facilitates direct comparisons to data; in particular measurements sensitive to radial and anisotropic flow. An example of an intermediate output of such a model is shown in Fig. 5, which demonstrates the evolution of the energy density as function of space (in the transverse plane) and time.

For anisotropic flow orders of $n = 2$, or 3, hydrodynamic models exhibit the following factorisation relation:

$$v_n \approx \kappa_n \epsilon_n \qquad (5)$$

where the variable $\kappa_n$ encodes the efficiency of the QGP in converting the spatial anisotropy $\epsilon_n$ to a momentum anisotropy $v_n$. That efficiency is dictated by two prominent QGP transport properties, which are needed as inputs to hydrodynamic models, and are $\eta/s$ and $\zeta/s$. The terms $\eta$ and $\zeta$ are the shear and bulk viscosities respectively, and as mentioned, they influence the amount of anisotropic and radial flow respectively, while $s$ is the entropy density. The $\eta/s$ and

$\zeta/s$ terms can be deduced from different coupling regimes that can be explored for the QGP, and are ultimately constrained by measurements sensitive to radial and anisotropic flow. These regimes range from beyond Standard Model approaches that include infinite coupling regimes from string theory techniques (AdS), to Standard Model QCD descriptions in the strong and weak coupling limits [50–54].

The production of hard probes and their interactions with the QGP offers distinct information regarding QGP properties. The description of the hadron production from these processes relies on the application of the QCD Collinear Factorisation theorem [55] to heavy-ion collisions. As an example, it can be applied as follows to determine the differential hadron ($h$) production cross section [56]:

$$d\sigma_{AB \to hX} \approx \sum_{abjj'd} f_{a/A}(x_a) \otimes f_{b/B}(x_b) \otimes d\sigma_{ab \to jd} \otimes P_{j \to j'} \otimes D_{h/j'}(z_{j'}). \qquad (6)$$

The A and B terms correspond to the colliding nuclei. The f terms represent the parton distribution functions of those nuclei at longitudinal momentum fractions $x_a, x_b$ for incoming partons a and b. The term $d\sigma_{ab \to jd}$ is the cross section of a hard scattered parton j, while $P_{j \to j'}$ describes the effect of the parton j interacting with the coloured medium before fragmenting into hadrons. The term D is the fragmentation function in terms of the fractional momentum z of the hadron from the modified parton j'. As mentioned, the hard-scattered partons interact with the colour-charged constituents of the QGP through elastic collisions [57] and inelastic scatterings, with the latter giving rise to medium-induced gluon radiation [58,59]. Such interactions cause a modification of the parton shower, usually denoted as “jet quenching” [60]. The simplest manifestation of jet quenching is a reduction of the energy/momentum of the jet (and of the leading hadron emerging from the jet fragmentation) due to the transport of the initial parton energy to large angles from the jet direction (“energy loss” [61]). In the high-momentum regime, the influence and the effects of the QGP, mainly the collisional and radiative energy loss, can be calculated in the context of pQCD utilising either (semi)analytical calculations of in-medium energy loss [62,63] or Monte Carlo simulations of the microscopic interactions that occur in an expanding QGP [64,65]. The main energy loss mechanism
<figure><img src="image_9.png" /><figcaption>**Fig. 5** A mapping of the energy density in the QGP phases vs time and space for a mid-central ($b = 7.5$ fm) Pb-$\sqrt{s_{NN}}$ = 5.02 TeV collision using the TRENTo-VISHNU model chain [49]</figcaption></figure>

for high momentum partons is expected to be radiative gluon emission [66]. In the framework of the Factorisation theorem, the medium-induced energy degradation of the leading parton can be described through medium-modified parton fragmentation functions [67]. These models for QGP interactions explore the energy loss of gluons (which are the majority of hard partons produced at LHC energies), light quarks, and heavy quarks, which are influenced differently in the QGP due to the different colour charges and masses [40]. The parameter that is typically used to characterise the medium-induced energy loss in such models is the transport coefficient $\hat{q}$, which can be considered a quantification of the opacity of the QGP. It is defined as the average of the squared transverse momentum transfer per mean free path of a parton due to energy loss in the QGP, and like $\eta/s$, is directly linked to the coupling regime of the QGP. Amongst other things, these models provide as an output $R_{AA}$, which can be directly compared to experimental measurements of jets, or hadrons at high-$p_T$. They can also be compared to $v_n$ for hadrons from hard processes, in order to test the assumptions of the path-length dependent energy loss used in these approaches. In addition, Monte Carlo based models provide predictions for medium induced modifications of jet properties (such as radial energy profile, jet substructure, and fragmentation functions) in heavy-ion collisions compared to pp interactions.

The study of the transport of heavy quarks (charm and bottom) represents another key topic for the characterisation of QGP properties [68]. In this respect transport models, which attempt to model the heavy quark-parton interactions on a microscopic level as a function of space and time, provide valuable information. The production and transport of heavy quarks themselves in heavy-ion collisions can be modelled under a couple of advantageous assumptions. Firstly, the heavy quark production rates in the initial stages can also be inferred from pQCD calculations, or direct measurements from pp collisions, and secondly, since the heavy-quark masses are considerably higher than the expected QGP temperatures at the LHC, further production of heavy quarks is expected to be small in the QGP stage. In addition, as the temperature of the medium is smaller than the heavy quark mass, the typical momentum exchanges of heavy quarks with the medium are small. The propagation of heavy quarks in

the QGP can therefore be described as a “Brownian motion” with many small-momentum kicks. A key variable to characterise how heavy quarks interact in the QGP is the heavy-quark spatial diffusion coefficient, $D_s$, which is related to the relaxation time of heavy quarks in the QGP [69]. This time defines the rate at which the initial momentum distribution of the heavy quarks (which are produced far out of equilibrium with the QGP) approaches a thermal spectrum. If this is small in relation to the medium lifetime and the expansion rate of the system, heavy quarks participate in the collective motion of the QGP. It can also be determined for the infinite, strong, and weak coupling regimes [70], like $\hat{q}$ and $\eta/s$. Measurements of heavy quark hadron $v_n$ and $R_{AA}$ at low and intermediate-$p_T$ provides constraints for the values of $D_s$ used in these approaches. Heavy quark transport models can also investigate the production of heavy quarkonium states in the QGP, which is strongly affected by a medium with free colour charges. Typically, one solves a rate equation including suppression and regeneration effects. The former are evaluated starting from the modifications of the quarkonium spectral functions in the QGP, constrained by potential models validated by Lattice QCD inputs, while the latter are tuned from the measured heavy quark multiplicity, which is very large ($> 10^2$) for charm quarks in central Pb–Pb collisions at LHC energies.

**Hadronic interactions and the hadronic phase.** Hadronic transport and statistical models play a crucial role for the description of this phase. Regarding the transport models (e.g. UrQMD [71]), they attempt to describe microscopic hadron–hadron interactions in space and time. They require hadronic cross sections as inputs, and as mentioned are employed as a hadronic afterburner for hydrodynamic model chains, in order to account for final state interactions. The cross sections required for various hadron species interactions are subject to experimental uncertainties, and in some cases, in particular for rarely produced hadrons, have no experimental constraints and therefore require a theoretical input. The application of statistical models on the other hand assumes that the system is thermalised at the hadronic stage, which allows for macroscopic thermodynamic properties to be determined based on fits to measurements of hadron abundances [72]. The chemical freeze-out temperature is a prime
example, as these hadron abundances are fixed at this tem-
perature. The baryochemical potential $\mu_B$, which is related
to the net baryon density i.e. the matter-antimatter asym-
metry of the system, is another example. Statistical models
are also particularly useful as Lattice QCD calculations pre-
dict the deconfinement temperature, therefore the chemical
freeze-out temperature obtained from statistical models pro-
vides a lower limit for such a prediction. Lattice QCD or chil-
ral effective theories can also be used to explore the strong
potential between produced hadrons that undergo low $Q^2$
interactions, in both heavy-ion and light system collisions.
It can prescribe potentials between the copiously produced
short-lived hadrons (such as $\Lambda$ or $\Xi$) and protons, which can
be tested via femtoscopic measurements.

QGP effects and few body QCD interactions in heavy and small systems. A number of event generators are used in the modelling of heavy-ion or light system collisions, which incorporate QGP effects to a varying degree. The AMPT model uses phenomenological strings to determine the QCD fields created in the initial collision [73]. These strings break to form quarks, which subsequently interact in accordance with transport equations, and then hadronise to produce particle momentum distributions, which can be compared to data. It has an afterburner, similar to UrQMD, called ART, that operates in a similar way. The EPOS model also attempts to model QGP behaviour [74], and like AMPT it utilises QCD strings, but differs in its use of a core-corona approach. In the core, the energy density of the strings is sufficient to invoke the QGP description, which is subject to a hydrodynamic evolution. In the corona nucleon-nucleon collisions are treated as independent entities regarding hadron production from such collisions. The HIJING model is somewhat simpler [75]: it assumes nucleon-nucleon interactions produce hadrons independently, and is therefore used as a non-QGP reference with respect to comparisons to heavy-ion data (although it has a jet-quenching option). As mentioned, QCD can be tested directly in the more dilute environment of pp, p-Pb and ultra peripheral Pb-Pb collisions. Such tests take advantage of the more standard tools used in high energy physics. An event generator that does so is PYTHIA [76], which provides a well established model for pp collisions. PYTHIA, like HIJING, AMPT and EPOS, describes the QCD field using phenomenological strings which break to produce low-$p_T$ hadrons, and also incorporates hard scatterings coupled with event by event parton showers and hadronisation models to describe the production of high-$p_T$ hadrons. Other Monte Carlo generators such as MC@NLO [77,78] and POWHEG [79] are available to study heavy flavour production. They couple the modeling of the hadronic final state with a perturbative description of the hard scattering process, and are useful to model complex observables involving correlations or heavy-flavour jets. PYTHIA has also been

recently extended in the PYTHIA/Angantyr format to model
heavy-ion collisions [80], without including a hydrodynamic
evolution.

1.3 Evolution of the field: from first studies to ALICE and beyond

Before the LHC start-up, the search for the existence of the
quark-gluon plasma and the study of its properties were
already the quintessential reason for the existence of a broad
community of physicists, with a background ranging from
traditional nuclear physics to high-energy particle physics.
Such a community had a relevant fraction of its roots in exper-
iments performed with ion beams in the few GeV energy
range, starting at BEVALAC in the 1970s [81], with strong
theory groups devoted to the study of nuclear matter under
extreme conditions being also formed in those years [82,83].
A decisive boost for the field occurred with the availability
of nuclear collisions at the BNL AGS ($\sqrt{s_{NN}}$ ~ 5 GeV) [84]
and CERN SPS ($\sqrt{s_{NN}}$ ~ 20 GeV) [85], both starting in
1986. On the theory side a whole series of observables that
could be considered as signatures of the formation of the QGP
were proposed. Prime examples of these signals included the
suppression of the $J/\psi$ yield induced by colour screening in
the QGP [86] and the enhancement of strangeness production
due to the reduction from constituent to current quark mass
in a chirally symmetric and deconfined medium [87], lead-
ing to a strong excess of multi-strange baryons relative to pp
collisions. Enhancements of the dilepton spectrum due to the
modification of the $\rho$ spectral function [88,89] and the pro-
duction of a thermal signal from the deconfined state [90,91]
were also among the predicted signatures. All of these sig-
nals were observed to various extents in Pb-Pb collision
studies performed by several experiments at the SPS in the
1990s [92–96]. Although alternative explanations not involv-
ing deconfinement could be formulated for some of them, the
multiple simultaneous observations of the various proposed
signatures was considered by the community as a compelling
evidence for the production of a new state of matter with the
characteristics of the QGP and a press release in that sense
was issued by CERN in 2000 [85].

A few months later (June 2000) the first heavy-ion collider,
RHIC, entered operation at Brookhaven National Laboratory,
increasing the available collision energy by about one order
of magnitude. With this step, the study of the production of
hard probes of the QGP received an extraordinary boost and
only a few months later some new discoveries came to light
in this sector, with the first observation of the suppression
in Au–Au collisions at $\sqrt{s_{NN}}$ = 130 GeV of high-$p_T$ parti-
cle production [97,98]. This signal, much awaited for, was
immediately connected to the energy loss of hard partons in
the QGP [99, 100] and initiated an entire field of investigation,
extended through the years to the study of the modification of
jet properties in the medium [101–103]. At the same time, the increase of the hadron multiplicity up to pseudorapidity densities $dN_{ch}/d\eta \sim 650$ for central Au–Au collisions [104] made accurate studies of the global characteristics of the events possible. It was shown, from the measurements of $v_2$ at low-$p_T$ and $R_{AA}$ at high-$p_T$ from charged hadrons that the medium created in nuclear collisions at RHIC behaves like a nearly ideal fluid [105]. Such a medium implies a thermal equilibration at early times and with constituent interactions that have a very short mean free path, somewhat superseding the early notion of a weakly-coupled, gas-like QGP [106]. The values of $\eta/s$ implied from these measurements were close to those expected from the creation of an infinitely coupled fluid prescribed from AdS/CFT: $1/4\pi$ [107]. In the later RHIC years (circa 2009), a claim of the observation of a CME-like signal was announced [108].

The first round of RHIC results [109–112] was an important input for the shaping of the physics program for heavy-ion collisions at the LHC. Studies were carried out mainly in the environment of the ALICE Collaboration, but also by groups participating in the ATLAS and CMS experiments. The further jump in energy by a factor > 20 ($\sqrt{s_{NN}} = 5.5$ TeV for Pb–Pb collisions for the nominal LHC running conditions) creates ideal conditions for the study of hard processes, thanks to the increase in the corresponding production cross sections, both in the strong (charm, beauty) and electroweak (W, Z) sector. At the same time, the further logarithmic but significant increase in the hadronic multiplicity created an obvious interest in the possibility of precise studies of the chemical composition and of collective effects of the medium, also addressing such quantities and their fluctuations on an event-by-event basis. In this situation, and considering the fact that the lifetime of the QGP was now expected to be much larger than that of the hadronic phase, the LHC was clearly considered as the ultimate machine for the quantitative study of the deconfined phase [113,114]. The ALICE experiment and its physics program were designed to address the study of particle production in the widest possible momentum range, from the soft to the hard sector, and to have the capability of answering a number of fundamental questions on the behaviour of the QGP [115,116]. Corresponding physics programs for the ATLAS [117] and CMS [118] experiments were also developed in a later phase, with a stronger emphasis on the detection of high-$p_T$ probes, due to the specific design of those experiments, mainly conceived for the study of high-luminosity pp collisions. During the first decade of LHC operation, also the LHCb experiment developed a program to study proton–nucleus and peripheral nucleus–nucleus collisions.

As mentioned in Sect. 1.1, the QGP created at LHC energies is expected to have a vanishing net baryon-number density. Indeed, when the collision energy becomes very high a “transparency” regime sets in, with the baryon number

<figure><img src="image_10.png" /><figcaption>**Fig. 6** A schematic representation of the QCD phase diagram. The green line and band shows the $\mu_B$ region accessible to Lattice QCD calculations [119]. The line shows the pseudocritical temperature, whereas the band represents the half-width of the crossover transition i.e. the temperatures where the QGP and hadrons can co-exist. The open points show experimental results for the determination of the chemical freeze-out parameters [120–123]. The location of atomic nuclei is also shown, as well as conjectured regions for the presence of a first order phase transition and of a critical point</figcaption></figure>

carried out by the colliding nuclei being located outside the centre-of-mass rapidity region ($y_{CM} = 0$). The latter is the hottest region, where particle production is maximal and the highest temperatures are reached [124,125]. In these kinematic conditions the transition between deconfined and hadronic phase is characterised by a rapid cross-over at the pseudocritical temperature $T_{pc}$ [26,28].

In a wider perspective, by studying heavy-ion collisions over extended energy ranges, it is possible to obtain information on the QCD phase diagram, usually shown in terms of temperature and baryochemical potential $\mu_B$. The latter quantity corresponds to the energy needed to increase the baryon number by one unit at fixed volume and entropy of the system and is proportional to the net baryon density of the system. A conceptual representation of the QCD phase diagram is illustrated in Fig. 6, where the region explored by LHC experiments is approximately shown. Moving towards lower collision energies implies a lower initial temperature of the QGP phase and an increase in the net baryon density of the system due to an increased “stopping” of the baryonic number in a region closer to $y_{CM} = 0$. The open points depicted in Fig. 6 show the values $T_{chem}$ and $\mu_{B,ch}$ corresponding to the chemical freeze-out of the system at LHC energy (blue) and for lower collision energies (red). In the region where reliable Lattice QCD calculations exist, indicated as a green band, the confinement temperature coincides, within uncertainties, with $T_{chem}$, showing that the chemical equilibration of the system occurs at, or shortly after, hadronisation. At
<figure><img src="image_11.png" /><figcaption>**Fig. 7** The ALICE detector. A short description of the various subdetectors, as well as information on their kinematic coverage, is given in the text</figcaption></figure>

large $\mu_B$, a well known technical issue of Lattice QCD calculations, the sign problem [126], forces to rely on approximate methods for the calculation of the thermodynamical quantities [127,128]. These methods predict a first-order phase transition at large baryon density that should end in a critical point when decreasing $\mu_B$ [129,130]. Studies of this region of the QCD phase diagram are extremely relevant also for our understanding of astrophysical objects as compact neutron stars [131]. They represent a field of investigation which is being developed in these years with experiments at existing (RHIC low beam-energy scans, SPS) and forthcoming (FAIR, NICA) facilities [132–137]. It is clearly complementary to the one accessible at top RHIC and LHC collider energies where, as already discussed in Sect. 1.1, a hot and long-lived QGP phase, similar to that of the early Universe, is produced. The program at the highest collider energies will continue at RHIC in the 2020s with the sPHENIX [138] detector and at the LHC in the 2020s and 2030s with the participation of all four large experiments [139].

## 1.4 ALICE: design considerations, implementation, operation

ALICE (A Large Ion Collider Experiment) was proposed in March 1993 as “a dedicated heavy-ion experiment” aimed at studying nuclear collisions at the LHC, in order to analyse the existence of QCD bulk matter and the QGP [140,141]. Being the only detector specifically devoted to QGP studies, it was designed to access a large number of specific observables in a wide transverse momentum range, in order to shed light on the various stages of the evolution of the heavy-ion collisions,

from the initial state to the QGP phase and to the transition to hadronic matter. At the same time, specific aspects of pp physics were part of the ALICE program from the beginning, and gained progressively more importance.

The ALICE detector is situated at the interaction point IP2 of the LHC. The apparatus that was operated in the LHC Runs 1 and 2 (‘ALICE 1’, shown in Fig. 7) is described in detail in Ref. [35] and its performance is discussed in Ref. [142] and will be shortly reported hereafter. The detector is based on a central barrel, covering full azimuth and the pseudorapidity region $|\eta| < 0.9$. It provides a robust particle identification up to $p_T \sim 20$ GeV/c, together with a very low momentum cut-off ($p_T \sim 0.15$ GeV/c, and even lower for specific reconstruction settings and collision systems) and excellent capabilities for primary and secondary vertex reconstruction (i.e. $\sim 100$ µm resolution for the study of the 2- and 3-prong decays of D mesons). The main charged-particle tracking detectors of ALICE are the Inner Tracking System (ITS) [143], composed of six tracking layers, two Silicon Pixel Detectors (SPD), two Silicon Drift Detectors (SDD), and two Silicon Strip Detectors (SSD), and a large Time Projection Chamber (TPC) [144]. The latter, together with the SSD and SDD, provide charged-particle identification via measurement of the specific ionisation energy loss $dE/dx$. External to the TPC, the tracking is complemented by a Transition Radiation Detector (TRD) [145] and a Time Of Flight system (TOF) [146]. These two detectors also provide electron identification and charged particle identification at intermediate-momenta, respectively. Outside the TOF, the azimuthal region is shared by two electromagnetic calorimeters, with a thickness of about 20 radiation lengths: (i) the
high-resolution PHOton Spectrometer (PHOS) [147], based on PbWO₄ crystals, (ii) the EMCal [148, 149], based on a layered Pb-scintillator sampling technique. The former is used for photon and neutral meson detection, while the latter due to its significantly larger acceptances is also employed for electron and (di)jet measurements. Finally a High Momentum Particle Identification Detector (HMPID) [150], provides, thanks to the RICH technique, further PID capability at intermediate-$p_T$. The central barrel detectors are embedded in the L3 solenoid magnet which delivers a magnetic field up to $B = 0.5$ T. Figure 8 illustrates, as an example, the PID performance of the TPC which allows excellent separation for various hadrons and light nuclei at low-$p_T$. A good separation between protons, kaons and pions is also achieved in the region of the relativistic rise of $dE/dx$, up to $p_T \sim 20$ GeV/c. The PID performance of the TOF system is shown in Fig. 8. Furthermore, an overview of the ALICE capabilities for the measurement of various hadrons is shown in both Figs. 8 and 9, where the approximate coverage, extending to very low-$p_T$, for different meson and baryon species is shown, together with the corresponding detection techniques.

The ALICE coverage extends outside the midrapidity region by means of a muon spectrometer and of various sets of smaller forward detectors. The muon spectrometer [151] covers the pseudorapidity region $-4 < \eta < -2.5$, it detects charmonia and bottomonia, via their dimuon decay, with a mass resolution of $\sim 70$ and $\sim 150$ MeV/c², respectively, Z bosons and low-mass hadrons ($\omega, \phi$). It also measures single-muon production from decays of heavy-flavour hadrons and W± bosons. It is based on a dipole magnet with a 3 Tm bending power, and on sets of tracking (Cathode Pad Chambers, CPC) and triggering (Resistive Plate Chambers, RPC) detectors. A hadron absorber, with a thickness of 10 interaction lengths ($\lambda_i$) separates the muon spectrometer from the interaction point (IP), and a second absorber ($7.2 \lambda_i$-thick) is positioned between the tracking and the trigger chambers. The other forward detectors include a silicon-strip Forward Multiplicity Detector [152] (FMD, on both sides of the IP at $-3.4 < \eta < -1.7$ and $1.7 < \eta < 5$) and a preshower/gas counter Photon Multiplicity Detector [153] (PMD, at $2.3 < \eta < 3.9$) for the measurement of charged particles and photons. Furthermore, the V0 [154], composed of two sets of scintillator detectors covering $-3.7 < \eta < -1.7$ (V0C) and $2.8 < \eta < 5.1$ (V0A), defines the trigger for minimum-bias collisions and is also used to classify events based on multiplicity or centrality. The T0 [152], two quartz Cerenkov detectors ($-3.3 < \eta < -3$ and $4.6 < \eta < 4.9$), are used to determine the timing and longitudinal position of the interaction. Finally two sets of Zero-Degree Calorimeters [155] (ZDC) detect spectator neutrons and protons emitted at very small angle and are used mainly for the determination of the

<figure><img src="image_12.png" /><figcaption>**Fig. 8 (Top)** The $dE/dx$ signal in the ALICE TPC as a function of magnetic rigidity. The expected curves for various particle species are also shown, with the inset panel showing the TOF mass measurement providing additional separation for helium isotopes when $p/Z > 2.3$ GeV/c. (Bottom) The Time-of-Flight measured in the TOF system as a function of the particle momentum. Tracks are selected with standard cuts inside the pseudorapidity region $|\eta| < 0.5$</figcaption></figure>

centrality of the collisions and to reject out-of-bunch interactions.

For a large experiment as ALICE, the triggering scheme is based on various decision levels [156, 157]. The Level 0 trigger decision (L0) is made $\sim 0.9$ µs after the collision using V0, T0, EMCal, PHOS, and the muon spectrometer trigger chambers. The events accepted at L0 are further evaluated by the Level 1 (L1) trigger algorithm, which also involves TRD and ZDC, and the corresponding decision is made $\sim 6.5$ µs after L0. The Level 2 (L2) decision, taken after about 100 µs corresponding to the drift time of the TPC, triggers the sending of the event data to DAQ and, in parallel, to the High Level Trigger system (HLT), which also plays an important role in data compression. In physics data taking a mixture of so-called “minimum bias” and rare triggers is used. In Pb-Pb collisions, a typical definition of the minimum bias
<figure><img src="image_13.png" /><figcaption>**Fig. 9** ALICE particle identification and reconstruction capabilities, with the $p_T$ coverage corresponding to the published measurements based on pp or Pb–Pb data samples. For W (Z) bosons, the selection $p_T > 10$ (20) GeV/c is performed on the transverse momentum of the decay muon, without defining a sharp $p_T$ range</figcaption></figure>

trigger is a coincidence of signals on the two V0 detectors, that can be complemented by the requirement of signals in both sets of ZDCs, to suppress electromagnetic interactions. Additional selections on the centrality of the event, based on the V0 signal, can also be included at the trigger level. Several rare triggers, designed to enhance the statistical significance for specific physics processes, are also implemented. As an example, the experiment can trigger on the production of single muon and muon pairs in the forward spectrometer [158], with a programmable $p_T$ threshold which can be as low as 0.5 GeV/c, and on the deposition of a certain amount of

energy in a number of adjacent cells in the EMCal and/or PHOS. Similar trigger conditions are implemented for the pp data taking, with the addition of a specific set of triggers involving the detection of one or more electrons in the TRD, and with the definition of a high-multiplicity trigger based on appropriate thresholds on the V0 signals or on the number of hits on the SPD. Details on the triggering scheme of the experiment can be found in Ref. [142].

ALICE has started physics data taking in 2009 with the first LHC pp collisions at $\sqrt{s} = 0.9$ TeV, and since then has collected data with all the collision systems and energies that became available during Run 1 (2009–2013) and Run 2 (2015–2018). In particular, Pb–Pb data taking occurred in 2010 and 2011 at $\sqrt{s_{NN}} = 2.76$ TeV, and in 2015 and 2018 at $\sqrt{s_{NN}} = 5.02$ TeV. A short run with Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV was also performed in 2017. Proton–proton data taking at the same energies as Pb–Pb, serving as a reference for nucleus–nucleus reactions but also for specific QCD studies, were also performed along the years. For the study of cold nuclear matter effects, after a short pilot run in 2012, p–Pb collisions were studied in 2013 and 2016, at $\sqrt{s_{NN}} = 5.02$ and 8.16 TeV. In addition to these combinations of collision system and energy, ALICE has taken pp data at all the other collision energies that became available along the years, up to $\sqrt{s} = 13$ TeV.

The data samples collected by ALICE can be characterised by two quantities. The first is the number of recorded minimum bias events $N_{MB}$, which is relevant for all the measurements which involve low-$p_T$ particle production. The minimum-bias event rate is limited essentially by the read-out time of the detectors and their occupancy, and when necessary, in particular for pp and p–Pb collisions, was tuned by acting on the beam optics (up to values of ~200 kHz for pp data taking at $\sqrt{s} = 13$ TeV). A typical feature of ALICE data taking is the possibility of operating with a low pile-up rate in pp collisions, allowing as an example a detailed study of the dependence of particle production on the charged hadron multiplicity. Typical values for $N_{MB}$ in the LHC Run 2 are ~4 × 10⁹ for pp collisions at $\sqrt{s} = 13$ TeV, complemented by ~2×10⁹ high-multiplicity events. For p–Pb, about ~8 × 10⁸ minimum-bias events were collected, mainly at $\sqrt{s_{NN}} = 5.02$ TeV. The corresponding number for Pb–Pb collisions at the same energy was about 3 × 10⁸ minimum-bias events, plus a similar number of recorded events with a further selection on centrality. The second relevant quantity is the recorded integrated luminosity $L_{int}$ for rare triggers, with the largest values corresponding typically to (di)muon triggers. For this specific class of triggers, in the LHC Run 2, about 36 pb⁻¹ were collected for $\sqrt{s} = 13$ TeV pp collisions and 25 nb⁻¹ for p–Pb collisions at $\sqrt{s_{NN}} = 8.16$ TeV. The corresponding number for Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV was about 800 µb⁻¹. In Table 1 the collision systems, center-of-mass energies, inte-
<table><caption>**Table 1** Collision systems studied by ALICE during LHC run 1 and 2. The center of mass energy per nucleon–nucleon collision, the integrated luminosity for rare triggers and the number of collected minimum bias</caption><thead><tr><th>System</th><th>Year(s)</th><th>√s<sub>NN</sub> (TeV)</th><th>L<sub>int</sub></th><th>N<sub>MB</sub></th></tr></thead><tbody><tr><td>Pb–Pb</td><td>2010, 2011</td><td>2.76</td><td>75 μb<sup>-1</sup></td><td>1.3 × 10<sup>8</sup></td></tr><tr><td>Pb–Pb</td><td>2015, 2018</td><td>5.02</td><td>800 μb<sup>-1</sup></td><td>6 × 10<sup>8</sup></td></tr><tr><td>Xe–Xe</td><td>2017</td><td>5.44</td><td>0.3 μb<sup>-1</sup></td><td>1.1 × 10<sup>6</sup></td></tr><tr><td>p–Pb</td><td>2013, 2016</td><td>5.02</td><td>18 nb<sup>-1</sup></td><td>8 × 10<sup>8</sup></td></tr><tr><td>p–Pb</td><td>2016</td><td>8.16</td><td>25 nb<sup>-1</sup></td><td>1.3 × 10<sup>8</sup></td></tr><tr><td>pp</td><td>2009</td><td>0.9</td><td>200 μb<sup>-1</sup></td><td>0.5 × 10<sup>6</sup></td></tr><tr><td>pp</td><td>2011</td><td>2.76</td><td>100 nb<sup>-1</sup></td><td>1.3 × 10<sup>8</sup></td></tr><tr><td>pp</td><td>2010, 2011</td><td>7</td><td>1.5 pb<sup>-1</sup></td><td>1.6 × 10<sup>9</sup></td></tr><tr><td>pp</td><td>2012</td><td>8</td><td>2.5 pb<sup>-1</sup></td><td>3.1 × 10<sup>8</sup></td></tr><tr><td>pp</td><td>2015, 2017</td><td>5.02</td><td>1.3 pb<sup>-1</sup></td><td>10<sup>9</sup></td></tr><tr><td>pp</td><td>2015–2018</td><td>13</td><td>36 pb<sup>-1</sup></td><td>6 × 10<sup>9</sup></td></tr></tbody></table>

events are shown. The latter quantity includes samples where selections on hadronic multiplicity (pp) or collision centrality (Pb–Pb) were performed

grated luminosity and number of minimum-bias events are listed.

The data size corresponding to event samples collected in each year of data taking can easily exceed tens of PetaBytes. The large computing and storage power required to process such amounts in order to perform the reconstruction of the physics objects from the raw data is ensured by the Worldwide LHC Computing Grid (WLCG) infrastructure, based on about 200 computing centres located all around the world [159].

In ALICE, the track reconstruction scheme is based on the Kalman filter [142, 160, 161]. In the central barrel, after a preliminary interaction vertex finding based on SPD information, track finding in the TPC is performed and track candidates are then matched to clusters in the ITS layers. Tracks are extrapolated to their point of closest approach to the preliminary interaction vertex and are then refitted in the outward direction, including the more external detectors and evaluating quantities relevant for PID. A final refit, starting from the external radius of the TPC is then performed and a final primary interaction vertex is computed. Secondary vertices from particle decays and photon conversions are subsequently searched, pairing tracks exceeding a defined distance of closest approach to the primary vertex and applying several quality cuts. More complex secondary vertex reconstruction (for example for heavy-flavour decays) is performed later at the analysis stage. In the muon spectrometer a Kalman-based algorithm is also adopted, and the track candidates are extrapolated to the SPD-based vertex position and finally corrected for the non-negligible effect of muon energy loss and multiple scattering in the hadron absorber [162]. Photon identification in the calorimeters is based on the detection of clusters in those detectors. The electron/positron contribution is rejected

by checking the absence of reconstructed charged tracks in the vicinity [116, 142]. Cuts on various shower shape parameters allow rejection of hadronic showers. A complementary detection technique via study of photon conversions to $e^+e^-$ pairs in the TPC is also employed [163]. Finally, jet measurements, particularly delicate in the complex heavy-ion collision environment, are performed combining information from charged particle tracking in ITS, TPC and EMCal information for the neutral energy carried by photons from hadronic decays [149, 164, 165].

The ALICE apparatus has now undergone a major upgrade (see Sect. 9.1) and the ‘ALICE 2’ version has recently started data taking in the LHC Run 3 [166, 167]. A completely new detector ‘ALICE 3’ [168] is proposed for the LHC Run 5 and beyond (see Sect. 9.2).

## 1.5 Overview of the key scientific questions addressed by ALICE

High-energy nuclear collisions in the multi-TeV energy range at the LHC provide ideal conditions to produce a long-lived QGP as well as abundant hard probes of such a state up to high $p_T$. An accurate and quantitative characterisation of the QGP state and its related phenomenology, via the study of the observables discussed in Sect. 1.2, is the subject of the ALICE experimental program and has inspired the design of its detector. In addition to this, owing to its high-precision tracking and excellent PID performance down to very low $p_T$ ALICE has contributed to addressing further fundamental aspects of the strong interaction and QCD. In particular, more general insights on the manifestations of hadronic interactions can be carried out by studying observables like the production of light anti- and hyper-nuclei, the kaon-nucleon
and the hyperon-nucleon interaction. Due to the difficulty in obtaining hyperon beams, such studies can be efficiently performed in the ALICE environment, either with Pb-Pb or high-multiplicity pp collisions. Another set of physics topics related to the strong interaction and not directly connected to QGP, where ALICE gives an important contribution, is the study of specific high-$Q^2$ QCD processes in pp collisions. These include the measurement of inclusive and heavy-flavour jet production, and the study of their substructure. Studies of open heavy flavour, quarkonia and high-$p_T$ hadrons also represent an area of interest for the experiment.

In the following, we briefly introduce the main topics investigated by ALICE, in the form of physics questions. The corresponding answers will be the object of the following chapters of this review.

1. What are the thermodynamic and global properties of the QGP produced at the LHC?

the QGP can be considered as an extended system of strongly interacting quarks and gluons in thermal equilibrium and as such can be characterised by a set of intensive quantities like temperature and energy density. Also, its size and lifetime are essential parameters for its characterisation. Various observables, such as hadronic multiplicity, transverse energy distributions, and direct photon observables are known to be related to these quantities and have been thoroughly investigated in the past. At LHC energies, which provide the highest temperatures to explore the QGP, they fully retain their importance for the evaluation of the thermodynamic properties of the QGP produced in Pb-Pb collisions and, once the temperature of the system decreases below the pseudocritical temperature, also of the subsequent hadronic phase. The analysis of the evolution of these quantities from small to large collision systems is a field of investigation that has received much attention at the LHC and is crucial for the understanding of the emergence of QGP-related features. Results on these studies are reported in Sect. 2.1 and Sect. 3.1 for Pb-Pb and smaller collision systems, respectively.

2. What are the hydrodynamic and transport properties of the QGP?

As the highest QGP temperatures are expected to be achieved at the LHC, that positions heavy-ion collisions at the LHC as paramount when investigating how hydrodynamic collective motion develops in such an extreme environment. In particular, the temperature dependence of $\eta/s$ and $\zeta/s$ can be studied to the fullest extent, via measurements sensitive to anisotropic and radial flow respectively, as the largest possible ranges of temperatures of the QGP can be probed. The temperature dependence of these parameters were not established prior to first heavy-ion collisions at the LHC. Given the discovery

of a perfect liquid at RHIC, constraining the temperature dependence of $\eta/s$ and $\zeta/s$ is crucial in assessing how “perfect” the QGP liquid created at the LHC is, as a weak coupling picture is expected at higher temperatures, due to the scale dependence of the QCD coupling constant. Finally, we will investigate, via measurements of charmed hadrons, whether low-momentum charm quarks produced out of equilibrium in the initial stages are able to participate in the QGP’s collective motion. Such an observation was not established prior to the LHC, and will help to constrain the temperature dependence of the diffusion coefficient $D_s$ via comparisons to transport models. The ALICE detector is particularly well suited to this task at the LHC due to its large $p_T$ and $\eta$ acceptance, given that hydrodynamic and transport models provide predictions over the full phase space. These topics will be addressed in Sects. 2.2 and 2.4.

3. How does the QGP affect the formation of hadrons?

As mentioned, the hadronisation of the QGP can be studied via measurements of hadron production, which provides information on how hadronic states emerge from the bulk of thermalised partons, when the expanding medium approaches the pseudocritical temperature. The abundances of different hadron species were observed at AGS, SPS and RHIC energies to closely follow the expected equilibrium populations of a hadron-resonance gas at the temperature at which the chemical freeze-out of the system occurs. This thermal origin of particle production allows for a macroscopic description of the properties of the hadron gas in terms of thermodynamic variables via statistical-hadronisation models and provides access to the chemical freeze-out parameters relevant for the QCD phase diagram. At the microscopic level, hadron formation from the QGP medium can be influenced by the presence of the bulk of thermalised partons. Recombination models were proposed after the first measurements of hadron production at RHIC and provided a natural explanation for some unexpected results of baryon and meson production and $v_2$ measurements in the intermediate-$p_T$ region. In these models, the quarks of the bulk, which are close to each other in phase space, can combine into hadrons. The recombination process is expected to dominate for partons at low and intermediate $p_T$, whereas energetic partons generally escape from the QGP and hadronise outside the medium via fragmentation. High precision measurements of production of different hadron species, including strange and charm mesons and baryons, as well as light nuclei and hypernuclei, are possible with the ALICE detector at the LHC in different collision systems with improved precision, especially at intermediate- and high-$p_T$, compared to results at RHIC energies. This can provide new insights
into the hadronisation mechanisms and the properties of the hadronic phase of the system evolution. The results of studies on QGP hadronisation will be discussed in Sect. 2.3.

4. *How does the QGP affect the propagation of energetic partons?*

The evidence of strong energy loss of high-momentum partons, via single-hadron $R_{AA}$ measurements, was one of the main discoveries at RHIC energies. This led to the first quantitative investigations for key energy loss parameters, such as $\hat{q}$. At high $p_T$, jet production is the main observable regarding the propagation of energetic quarks/gluons, both in vacuum and in a medium. The LHC, thanks to the abundant production of such probes, represents a unique laboratory for the study of their coupling to the QGP. Jet energy loss, jet substructure modification, jet deflection and the emission of large-angle radiation are powerful handles for a more detailed phenomenological understanding and for constraining the key theoretical parameters. Specific studies as the colour-charge (quarks vs gluons) and the quark-mass dependence (beauty vs charm vs light partons) of energy loss offer the possibility of a detailed testing on QCD-based predictions. Also, jets probe the QGP at short distances, as the QGP cannot be strongly coupled at all scales. When probed at short distances, its quasi-particle structure, i.e., the free quarks and gluons, should emerge, and their signature is expected to be a large momentum transfer to the jet or its constituents. These topics will be addressed in Sect. 2.4.

5. *How does deconfinement in the QGP affect the QCD force?*

The strong interaction binding a heavy quark-antiquark pair is expected to be strongly modified in the presence of a medium like the QGP, characterised by a high density of (deconfined) colour charges. Results at SPS and RHIC energies showed a significant suppression of charmonium states, depending on their binding energy and clearly related to the formation of a strongly interacting system. At the same time they also demonstrated that a quantitative understanding of the observation needs to consider the modification of spectral properties of quarkonia in the medium as they can be inferred from Lattice QCD and implemented in effective theory calculations. The LHC is ideally placed for a decisive step forward in these studies, thanks to the availability of high-statistics results in both the charmonium and, for the first time, bottomonium sector. Specific open points concern the existence and the characteristics of the various bound states in the QGP and the possibility of a (re)generation of quarkonia at the confinement threshold, which constitutes in itself an evidence for the production of a deconfined state.

The results on quarkonium-related observables will be described and discussed in Sect. 2.5.

6. *Can the QGP lead to discovery of novel QCD effects?*

In addition to forming the necessary conditions to study the deconfined QCD matter in a controlled way, heavy-ion collisions provide also the possibility to probe novel QCD phenomena that are associated with a chirality imbalance, and thus to parity violation in strong interactions. This imbalance is a consequence of topological transitions between the different vacuum states of QCD, yet another reflection of the non-Abelian nature of the theory, via a mechanism similar to the one responsible for the creation of baryon asymmetry in the early universe. The coupling of these effects with the extremely large magnetic field produced in peripheral heavy-ion collisions is expected to lead to an electric dipole moment of QCD matter, imprinted in the motion of final state particles. This is known as the Chiral Magnetic Effect (CME) which is believed to have a signal within experimental reach. The associated motion of deconfined quarks can only occur in the QGP. The CME is of particular interest at the LHC as heavy-ion collisions at these energies are expected to provide the largest ever magnetic field created in the laboratory. The ALICE results will be discussed in Sect. 2.6, also in relation with the general status of the field in studies of the early stage electromagnetic fields and on probing anomalous QCD effects.

7. *What are the minimal conditions of QGP formation?*

Several signatures of a strongly interacting medium, such as a relatively large strangeness production and anisotropic flow, appear prominently in mid-central and central nucleus-nucleus collisions. In this context, a natural question is if these phenomena also appear in very peripheral nucleus-nucleus collisions or, alternatively, in high-multiplicity proton-proton or proton-nucleus collisions. The ALICE detector is able to conduct a comprehensive study regarding the limits of QGP formation by studying the multiplicity dependence of several observables in pp and p-Pb collisions, which was made possible due to the unprecedented amount of data from these smaller collision systems delivered by the LHC. Advances in several key topics, including strangeness production, the onset of collective behaviour and energy loss phenomena, were achieved, leading to a better understanding of the microscopic origin of many of these phenomena and bringing forth new theoretical challenges. These results will be discussed in Sect. 3.

8. *What is the nature of the initial state of heavy-ion collisions?*

An accurate description of the initial state of colliding nuclei is crucial for the interpretation of many of the measurements carried out in heavy-ion collisions
at the LHC, and ultimately for the extraction of the relevant QGP parameters. In particular, the characterisation of the spatial distribution of nuclear matter in terms of energy and density profiles before the QGP formation has an important influence on the quantitative evaluation of its fundamental parameters (e.g. viscosity). Various approaches were developed along the years, considering either nucleons, constituent quarks, or gluons as the fundamental degrees of freedom. The Colour Glass Condensate framework implies the spatial extent of low-$x$ gluons at LHC energies is determined by the saturation scale. This should have a discernible impact on anisotropic flow coefficients. The corresponding measurements therefore provide an opportunity to test nuclear gluon saturation approaches using the ALICE detector. At a complementary level, a quantitative description of hard processes requires the knowledge of the modifications of the parton longitudinal momentum distribution in the nuclei. Here, a strong suppression (nuclear shadowing) was previously observed at low parton momentum fractions ($x < 10^{-2}$). The measurement of such modifications to the parton distribution functions (PDF), which can be seen as a consequence of saturation effects in the low-$x$ region, suffers from significant uncertainties. The availability of results with high statistical precision on electroweak boson production in p-Pb and Pb-Pb, as well as on vector-meson photoproduction in ultraperipheral Pb-Pb collisions, represent the main source of improvement that becomes available at LHC energies. The corresponding results will be described in Sect. 4.

## 9. What is the nature of hadron-hadron interactions?

The residual strong interaction among stable and unstable hadrons can be studied using femtoscopic correlation techniques at the LHC in pp and p-Pb collisions, investigated by ALICE with unprecedented precision. This is due to the large acceptance provided by ALICE, and the large integrated luminosities for light systems provided by the LHC. These measured observables can be directly connected to the relative wave function among the hadrons of interest. The small size of the source created in these light collision systems (about 1 fm) provides a unique environment to test the short-range strong interactions among hadron pairs. As mentioned, the measurements sensitive to these interactions can be compared with Lattice QCD or chiral effective theory predictions of hadron-hadron potentials, and the success of these comparisons, in particular for the rarely produced hadrons, may have implications for the modelling of neutron star cores. Hadron-hadron interactions can also be studied by measuring the binding energies of light antinuclei and hypernuclei produced in pp, p-Pb and Pb-Pb collisions. A microscopic understanding of

the formation process of light nuclei can be achieved by employing the measurement of the source size extracted from femtoscopy and testing whether hadron interactions can explain nuclei formation via the coalescence of nucleons. These topics are treated in Sect. 5.

10. Can ALICE elucidate specific aspects of perturbative QCD and of related "long distance" QCD interactions?
Precise perturbative QCD (pQCD) calculations serve as a baseline for searches for physics Beyond the Standard Model. Moreover, observables with a well understood production cross section in proton-proton collisions are essential to interpret the additional complexity of nonzero temperature QCD in heavy-ion collisions. Measurements of high-$Q^2$ processes typically involve both perturbatively calculable cross sections, as well as long-distance QCD interactions described by non-perturbative objects such as parton distribution functions and fragmentation functions which are commonly assumed to be universal, i.e. independent of the collision system, energy, and observable. Jet measurements, for example, test the importance of higher order terms in state-of-the-art pQCD calculations, and identified particle measurements constrain nonperturbative objects and test the universality of their fragmentation functions. The ALICE high-precision tracking and particle identification capabilities, with a focus on low to moderate transverse momentum, are unique at the LHC and complementary to the other LHC detectors. The moderate collision rates at the ALICE interaction point provide a clean, low-pileup environment for a variety of high-precision QCD studies. Results of QCD studies with high-$Q^2$ processes in pp collisions will be discussed in Sect. 6.

The next chapters of this review are organised as follows: Sect. 2 contains an extended description of the quantitative studies of the QGP properties, as investigated by ALICE mainly in Pb-Pb collisions. Sect. 3 aims at connecting observations obtained for large and small collision systems with the scope of analyzing the emergence of collective effects related to the strong interaction and possibly to QGP formation. Section 4 will describe the contributions of ALICE to the determination of the properties of the initial state of the high-energy collision systems, which are relevant both for the interpretation of final state quantities and for the study of specific QCD aspects (parton saturation). Section 5 will address studies of the interaction between hadrons, manifested in the formation of nuclear systems also including strange quarks. Section 6 deals with selected studies of high-$Q^2$ processes in pp collisions where ALICE has given significant contributions in the frame of LHC experimental results. Section 7 presents ALICE contributions to fields not directly related with LHC physics. Section 8 is devoted to the conclu-
sions. Section 9 discusses future prospects with the upgraded ‘ALICE 2’ (Run 3–4), the proposed new detector ‘ALICE 3’ for Run 5 and beyond, and the open questions that will be addressed by future measurements.

## 2 The quark–gluon plasma and its properties

The quantitative study of the properties of the quark–gluon plasma produced in nuclear collisions at the LHC represents the core of the ALICE physics program. The possibility of colliding Pb ions at the highest energy presently available at a particle accelerator, with significant further increase only foreseen in a relatively distant future (HE-LHC, FCC), implies that many of the results that will be described in this chapter will represent for a long time the state-of-the-art of our comprehension of low(zero)-$\mu_B$ deconfined matter. Due to the large initial temperature, the conditions are favourable for the formation of a long-lived system which quickly reaches thermalisation, allowing a precise estimate of its intensive properties. At the same time, the abundant production of hard partons at LHC energy, constitutes an intense source of probes of the QGP phase, allowing a test of its transport properties and micro-structure.

This chapter is organized in sections, each one dealing with specific physics aspects of the QGP. After a short introduction to the corresponding topics, the main results obtained by ALICE are described and discussed, with reference to theoretical models that are found to be relevant for the understanding of the results and the extraction of a physics message.

Specifically, characterisation of the Pb–Pb events in terms of centrality selection will be discussed first, followed by estimates of the attained energy density and temperature of the QGP, and by a discussion on the spatial extension of the strongly interacting system (Sect. 2.1). Then, a discussion on the dynamical properties of the QGP will be presented, mainly dealing with the study of its collective motion that offers insight on the coupling strength of the system (Sect. 2.2). In the next step, a discussion of the phenomena related to the (pseudo)critical transition of the QGP towards the hadronic phase is carried out, by describing results on the chemical equilibration and kinetic thermalisation of the system and investigating the duration of the hadronic phase (Sect. 2.3). Then, Sect. 2.4 deals with the use of high-momentum transfer processes as a probe of the QGP, their in-medium interactions being used to quantitatively understand dynamical features such as energy transport and equilibration. A specific set of hard probes, quarkonia, exhibits a strong sensitivity to deconfinement and is more generally sensitive to the modification of the QCD force in the medium, with the corresponding results shown in Sect. 2.5. Then, Sect. 2.6 deals with novel QCD phenomena that can

be catalysed by the presence of very strong magnetic fields generated in heavy-ion collisions, as local P and CP violating effects in the strong interaction.

A brief and schematic summary of the main findings completes each section, and the chapter is then concluded in Sect. 2.7, which presents a discussion of the extracted values of QGP-related quantities.

### 2.1 Macroscopic system properties and QGP thermodynamics

The first question to answer is whether a quark–gluon plasma is formed in Pb–Pb collisions at the LHC and how we can characterise it based on its thermodynamic properties. As discussed in Sect. 1, crossing the QCD phase transition boundary and forming the QGP require that the initial energy density and temperature reached in the collision are larger than the critical values. For these, the most recent Lattice QCD calculations report for the energy density $\varepsilon_c = (0.42 \pm 0.06)$ GeV/fm$^3$ [169], and two slightly different values of pseudocritical temperature, $T_{pc} = (156.5 \pm 1.5)$ MeV [169] and $T_{pc} = (158 \pm 0.6)$ MeV [119]. As the system expands and cools down, a second transition from a deconfined quark and gluon medium to hadrons takes place. Therefore, the first quantities we are interested in determining to characterise the system are the initial energy density and the temperature. The total transverse energy of the particles produced in the final state is a measure of the energy density, whereas multiplicity in the final state is closely related to entropy production in the collision. As in the rapid expansion of the fireball both the total energy and the entropy are expected to be approximately conserved, the use of final state quantities to estimate the initial energy and entropy density is justified. This is strictly valid in a non-viscous hydrodynamics scenario. The amount of energy concentrated in the collision zone can be controlled experimentally by selecting events based on centrality, as discussed in Sect. 2.1.1. Results on the charged-particle multiplicity density measured in Pb–Pb collisions at the LHC are summarised and presented in comparison to those at lower energies as well as in p–Pb and pp collisions in Sect. 2.1.2. In Sect. 2.1.3, we provide an estimate of the energy density as a function of centrality and discuss its relation to hydrodynamic calculations. While hadrons are produced in the freeze-out phase of the collision, photons and dileptons are emitted during the entire evolution of the fireball and provide an independent measure of the temperature. Therefore, hadrons and electromagnetic radiation measure different temperatures, corresponding to different stages of the system evolution. This is discussed in Sect. 2.1.4. Besides these thermodynamic properties, the system produced in the collision can be further characterised in terms of its spatial extension and duration by using femtoscopic techniques. The measurement of momentum correlations among parti-
cles produced in the final state (i.e. at the freeze-out of the system) provides important information on the particle emitting source and on the underlying dynamics. The size and lifetime of the system are discussed in Sect. 2.1.5.

### 2.1.1 Centrality of nucleus-nucleus collisions

Nuclei are extended objects and the degree of geometrical overlap between them in the collision, expressed in terms of the impact parameter ($b$), varies. Since $b$ is not directly measurable, an experimental proxy, centrality, is used to characterise the amount of nuclear overlap in the collisions, as anticipated in Sect. 1.2.4. Centrality is commonly expressed in percentiles of the total nucleus-nucleus cross section ($\sigma_{PbPb} = (7.67 \pm 0.16)^{\text{syst}}$) b for Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [170]) and, conventionally, small (large) percentiles as for instance 0–5% (80–90%) correspond to central (peripheral) collisions [171].

In ALICE, centrality is determined by measuring the signal in the V0 scintillator arrays, which is proportional to the number of particles that strike them. The signal amplitude measured at forward rapidity in the V0 is observed to be strongly correlated with the multiplicity of charged particles measured at midrapidity. This is shown in the top panel of Fig. 10, which illustrates the correlation between the summed V0A and V0C (the sum is referred to as V0M) signal amplitudes, and the number of charged-particle track segments reconstructed using the ITS at midrapidity ($|\eta| < 1.4$). The observed relation of proportionality between the particle multiplicity at midrapidity and the particle multiplicity at forward rapidity, which are two causally-distant observables, constitutes a purely data-driven justification to our centrality classification of events based on the V0 information. Alternatively, centrality can be measured using the particle multiplicity in the ITS or the energy deposited by the spectator nucleons in the ZDC, as discussed in detail in Ref. [171]. The V0 estimator remains the most common choice for centrality classification for the purpose of studying particle production at midrapidity, due to the rapidity separation. As illustrated in the bottom panel of Fig. 10 for a sample of 5.02 TeV Pb–Pb collision events, the centrality classes are defined by slicing the V0M signal amplitude distribution. The latter is fitted with a Glauber model [172] (see also Sect. 1.2.4) coupled to a model for particle production based on a negative binomial distribution (NBD), to obtain $\langle N_{\text{coll}} \rangle$ and $\langle N_{\text{part}} \rangle$ for each centrality class, as detailed in [170,171]. The NBD is used to reproduce the amplitude for peripheral events where the multiplicity approaches the one observed in pp collisions [171]. In peripheral nucleus–nucleus collisions, the selection of the centrality classes on the basis of charged-particle multiplicity, leads to a bias in the determination of the collision geometry parameters [173], which has to be taken into account when interpreting mea-

<figure><img src="image_14.png" /><figcaption>**Fig. 10** Summed signal amplitude in the V0 scintillators in triggered Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. (Top) Correlation between the V0M amplitude, at forward rapidity, and the number of charged-particle track segments (tracklets) reconstructed in the ITS at midrapidity ($|\eta| < 1.4$). (Bottom) Fit (red line) to the V0M distribution with a Glauber model coupled to a negative binomial distribution (NBD). Gray and white bands indicate the classification of the events into centrality classes, with centrality expressed in percentile of the hadronic cross section</figcaption></figure>

surements that use these parameters, like the nuclear modification factor (see Sect. 2.4.2). The event classification based on multiplicity in pp as well as p–Pb collisions is discussed later, in Sect. 3.1.

### 2.1.2 Charged-particle multiplicity density at midrapidity

In heavy-ion collisions, the charged-particle multiplicity per unit of (pseudo)rapidity ($dN_{ch}/d\eta$) is studied as a function of centrality as well as in comparison to different collision systems and at various collision energies. This allows one to investigate the role of the initial energy density and the mechanisms responsible for particle production.

The primary charged-particle multiplicity density at midrapidity ($|\eta| < 2$) can be obtained from the number of short track segments formed using the position of the primary vertex and all possible combinations of hits between the
two SPD layers (the two innermost layers of the ITS). The deposited energy signal in the FMD is used to measure the charged-particle pseudorapidity density in the forward regions ($-3.5 < \eta < -1.8$ and $1.8 < \eta < 5$). In both cases, the multiplicity is corrected for the background from secondary particles as well as for efficiency and detector acceptance. Details on the methods can be found in [174], for instance. The number of particles produced at midrapidity increases from peripheral to central collisions, from few tens of particles per unit of rapidity to few thousands. In central (0–5%) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV and in the range $|\eta| < 0.5$, $\langle dN_{ch}/d\eta \rangle = 1943 \pm 54$ and $\langle N_{part} \rangle = 385 \pm 3$, compared to $\langle dN_{ch}/d\eta \rangle = 17.52 \pm 1.84$ and $\langle N_{part} \rangle = 7.3 \pm 0.1$ in peripheral (80–90%) collisions [174,175].

To compare the bulk particle production in different collision systems, the charged-particle multiplicity measured at midrapidity ($|\eta| < 0.5$) is scaled by the number of nucleon pairs participating in the collision, $\langle N_{part} \rangle/2$. Figure 11 shows the midrapidity charged-particle multiplicity normalised by the number of the participating nucleon pairs, $\frac{2}{\langle N_{part} \rangle} \langle dN_{ch}/d\eta \rangle$, in pp, p$\bar{p}$, p(d)A and in central heavy-ion collisions as a function of the centre-of-mass energy per nucleon pair, $\sqrt{s_{NN}}$. In particular, ALICE central (0–5%) heavy-ion data include Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV and 5.02 TeV as well as Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV. The dependence of $\frac{2}{\langle N_{part} \rangle} \langle dN_{ch}/d\eta \rangle$ on the centre-of-mass energy can be fitted with a power-law function of the form of $\alpha \times s^\beta$, resulting in an exponent $\beta = 0.152 \pm 0.003$ for central Å collisions, under the assumption of uncorrelated uncertainties. It is a much stronger s-dependence than for inelastic (INEL) pp and non-single-diffractive (NSD) p(d)A collisions, where a value of $\beta = 0.103 \pm 0.002$ is obtained. This indicates that heavy-ion collisions are more efficient in converting the initial beam energy into particle production at midrapidity than pp or p–Pb collisions.

Figure 12 presents the charged-particle multiplicity density $dN_{ch}/d\eta_{lab}$ as a function of pseudorapidity for central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. For symmetric collision systems, such as Pb–Pb and pp, the laboratory reference frame coincides with the centre-of-mass one and the $dN_{ch}/d\eta_{lab}$ distributions are symmetric (hence hereafter indicated as $dN_{ch}/d\eta$). When available, the data are extended into the non-measured region $-5 < \eta < -3.5$ by reflecting the $3.5 < \eta < 5$ values around $\eta = 0$. An asymmetry between the proton and the lead hemispheres is observed in the p–Pb distribution, in which the number of charged particles is higher in the Pb-going side (positive $\eta_{lab}$). The results are compared to the EPOS 3.4 model for Pb–Pb and pp collisions and to PYTHIA 8.3 for all the three collision systems (see Sect. 1.2.4 for an introduction to these models). For Pb–Pb collisions, both models reproduce quantitatively the $dN_{ch}/d\eta$ in the midrapidity region. The comparison

<figure><img src="image_15.png" /><figcaption>**Fig. 11** Collision energy dependence of the charged-particle pseudorapidity density at midrapidity ($|\eta| < 0.5$) normalised to the average number of participants, $\frac{2}{\langle N_{part} \rangle} \langle dN_{ch}/d\eta \rangle$. Data from central Å collisions [175–185] are compared to measurements in non-single diffractive p(d)A collisions [186–189] and inelastic (INEL) pp and p$\bar{p}$ collisions [174,190–195]. ALICE heavy-ion data include Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV and 5.02 TeV as well as Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV. All values of $\langle N_{part} \rangle$ used for the normalisation of the data are the results of Glauber model calculations. The lines are power law fits to the data and the bands represent the uncertainties on the extracted dependencies</figcaption></figure>

at forward rapidities is not as good as at midrapidity. For p–Pb collisions, PYTHIA 8.3 overpredicts the data in the Pb-fragmentation region, while it reproduces well the trend for midrapidity and for the p-fragmentation side. For pp collisions, EPOS 3.4 is overestimating the data by around 20% in the full pseudorapidity range.

Particle production has been measured in small to medium-sized and large colliding systems such as Xe–Xe and Pb–Pb with unprecedentedly high precision. The $dN_{ch}/d\eta$ uncertainties range from around 3% for central Å collisions, at midrapidity, to around 10% for peripheral results, in the forward region.

Figure 13 shows the centrality dependence of $\frac{2}{\langle N_{part} \rangle} \langle dN_{ch}/d\eta \rangle$, with the centrality being expressed in terms of $\langle N_{part} \rangle$. Data from 5.02 TeV Pb–Pb and 5.44 TeV Xe–Xe collisions are compared to lower energy results in Au–Au and Cu–Cu collisions by the PHOBOS experiment at RHIC.

To disentangle energy dependence from $N_{part}$ dependence, the Au–Au, Cu–Cu and Xe–Xe data are scaled by factors calculated using the fit function of Fig. 11 for the top 5% most central collisions. The values of $\frac{2}{\langle N_{part} \rangle} \langle dN_{ch}/d\eta \rangle$ for Pb–Pb and Xe–Xe collisions decrease by a factor of two from
<figure><img src="image_16.png" /><figcaption>Fig. 12 (Left) Pseudorapidity density of charged particles in 0–5% central Pb–Pb collisions [174], NSD p–Pb collisions [188], and INEL pp collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [196] in ALICE. Dashed lines represent EPOS 3.4 model calculations [197–199], available only for pp and Pb–Pb collisions at this energy and PYTHIA 8.3 [80], available</figcaption></figure>

<figure><img src="image_17.png" /></figure>

for all collision systems. Error bars represent systematic uncertainties on data, statistical uncertainties are negligible. (Right) Ratio between EPOS 3.4 and PYTHIA 8.3 model calculations relative to Pb–Pb (top), p–Pb (middle), and pp (bottom) data. Shaded bands represent the relative systematic uncertainties on data

the most central to the most peripheral collisions, where they agree with the values measured in minimum bias pp and p–Pb collisions. At the same $N_{part}$, Pb–Pb undergoes less binary collisions than Xe–Xe. This causes the separation between the shapes of Xe–Xe and Pb–Pb for the $\langle \frac{2}{N_{part}} \rangle \langle dN_{ch}/d\eta \rangle$ as a function of $\langle N_{part} \rangle$. Hints for a deviation between the lower energy Au–Au and Cu–Cu results from the LHC data for $\langle N_{part} \rangle < 100$ are visible. This deviation is not significant given the large uncertainties and it could be attributed to the different Glauber model implementation used to estimate $N_{part}$ in ALICE [175] and PHOBOS [184]. For the 5% most central Xe–Xe and Pb–Pb collisions the $\langle \frac{2}{N_{part}} \rangle \langle dN_{ch}/d\eta \rangle$ increases steeply. A qualitatively similar feature is observed also in Cu–Cu collisions.

In Fig. 13 a comparison to PYTHIA 8.3 [200] calculations is also shown with black solid lines for the same selection in centrality as done in data ($N_{ch}$-selected). Dotted lines represent, instead, a selection performed on the $N_{part}$. The calculations for the most central events exhibit a steep increase for the $N_{ch}$-selected results only, indicating that the sources of the rise are fluctuations of the charged-particle multiplicity at a fixed number of particle sources, $N_{sources}$. Selection biases and auto-correlation effects are covered more in detail in Sect. 3.1. The $N_{ch}/N_{sources}$ fluctuation bias can be avoided by selecting centrality and calculating the $N_{part}$ using the energy deposited by spectator nucleons in the Zero Degree Calorimeters. A methodology to derive these geometrical quantities from spectator detection near beam rapidity is under study in the Collaboration. Measurements in light-

nucleus collisions at the LHC Run 3 could allow us to constrain further the underlying particle production mechanisms, and to describe the increase with energy, see Fig. 11, and centrality, Fig. 13, bridging the gap between the trends observed in pp and pA collisions and those in the mid-sized Xe–Xe and the large Pb–Pb systems.

### 2.1.3 Determination of the initial energy density

The energy density and temperature of the early collision stages are key physical quantities since they determine whether the critical conditions for the QCD phase transition are reached in the collision. In nuclear collisions, both can be, in principle, controlled by selecting events based on their centrality. The energy density in the collision can be estimated from the total produced transverse momentum using the ‘Bjorken-estimate’ [201]:

$$ \varepsilon_{Bj}(\tau) = \frac{1}{S_T \tau} \frac{dE_T}{dy}, \qquad (7) $$

where $dE_T/dy$ is the total produced transverse energy $E_T = \sqrt{p_T^2 + m^2}$ per unit of rapidity, and $S_T$ is the transverse size of the interaction region at proper time $\tau$. This estimate is valid for a free-streaming system that undergoes boost-invariant longitudinal expansion and no transverse expansion.

From the measurements of the charged-particle pseudorapidity density and assuming that the rapidity of the charged particles produced in the collisions is normally distributed [174,202], we extract a lower-bound estimate of the
<figure><img src="image_18.png" /><figcaption>Fig. 13 $\frac{2}{\langle N_{\text{part}} \rangle} \langle dN_{\text{ch}}/d\eta \rangle$ in Pb–Pb collisions [175], p–Pb collisions [188] and pp collisions [196] at $\sqrt{s_{NN}} = 5.02$ TeV, Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV [176] scaled by a factor of 0.98, Au–Au and Cu–Cu collisions at $\sqrt{s_{NN}} = 0.2$ TeV [184] scaled by 2.55 as a function of $\langle N_{\text{part}} \rangle$. PYTHIA 8.3 calculations [200] for $N_{\text{ch}}$-selection are shown as black lines, dotted lines are $N_{\text{part}}$-selected calculations</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 14 Lower-bound estimate of the energy density times the formation time $\tau$ in pp, p–Pb, and Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV as a function of the number of participating nucleons [203]. The transverse area is calculated as the total area of overlap between participating nucleons using Eq. 8. The open circles are from Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV and calculated via direct measurements of $E_T$ [204]. A fit to a power law $aN_{\text{part}}^p$ is shown with best-fit parameter values $a = 0.8 \pm 0.2 \text{ GeV}/(\text{fm}^2c)$ and $p = 0.44 \pm 0.07$, and $\chi^2/dof = 1.2/16$</figcaption></figure>

energy density $\varepsilon_{\text{LB}}$ times the formation time $\tau$ in the collisions [203]

$$ \varepsilon_{\text{LB}} \tau = \frac{1}{S_T} \frac{1}{f_{\text{total}}} \sqrt{1 + a^2} \langle m \rangle \frac{dN_{\text{ch}}}{dy}, \qquad (8) $$

where $\sqrt{1 + a^2 \langle m \rangle}$ is the effective transverse mass, and $f_{\text{total}} = 0.55 \pm 0.01$ is the fraction of the charged out of all particles [204]. Here, $a$ is the effective $p_T/m$ ratio extracted as part of the estimate. The transverse area $S_T$ is calculated using a Glauber model [205] in which the full area of participating nucleons is taken into account. The resulting $\varepsilon_{\text{LB}} \tau$ is shown in Fig. 14 for pp, p–Pb, and Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. We see an increase of roughly an order of magnitude from pp and most peripheral p–Pb collisions to the most central Pb–Pb collisions. A power-law ($aN_{\text{part}}^p$) fit to the data indicates a non-trivial increase in the energy density with an increasing transverse area of the initial overlap between the colliding nuclei.

In a system that follows hydrodynamic expansion, the expanding fluid performs work and the resulting time-dependence of the energy density in a system that undergoes boost-invariant longitudinal expansion (and no transverse expansion) follows a power law with $\tau^{-4/3}$ instead of $\tau^{-1}$,

$$ \varepsilon_{\text{Bj,hydro}}(\tau) = \varepsilon_0 \left(\frac{\tau_0}{\tau}\right)^{4/3}, \qquad (9) $$

where $\varepsilon_0$ and $\tau_0$ are the initial energy density and the formation time, respectively. These simple data-driven estimates are compared with the energy density from a hydrodynamical model calculation (IPGlasma+MUSIC+URQMD with the s95p-PCE equation of state [206]) for hydrodynamical evolution and the UrQMD model for final state scattering) in Fig. 15. The blue line and light gray band indicate the energy density estimated using the free-streaming estimate from Eq. 7 while the dark gray band indicates the behaviour of a longitudinally expanding hydrodynamical system. Since then the system freezes out gradually, there is no single time at which the expansion switches from hydrodynamical expansion to free streaming. The band was constructed by switching at $\tau = 6$ and 12 fm/c, which are the times at which approximately 10% and 90% of the system is frozen out according to the hydrodynamical calculation. The uncertainty from the measured $E_T$ has been added linearly. It can be seen that this simple data-driven estimate is in reasonable agreement with the full model calculation up to 2–3 fm/c.

### 2.1.4 Temperature of the system

Temperature is a key thermodynamic property that characterises the fireball at any given time during its evolution. To experimentally access the temperature of the early partonic phase one can rely on sensitive probes that are produced at the early stage of the collision, such as heavy flavour $q\bar{q}$
<figure><img src="image_20.png" /><figcaption>Fig. 15 Evolution of the average energy density as a function of the proper time of the system in hydrodynamic calculations. The green curve represents the average energy density profile from hydrodynamic model calculation from [207,208]. The blue line the grey bands represent the energy density estimated from the total transverse energy, $E_T$, measured in 0–5% central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [204] using Eq. 7 for the free-streaming expansion and Eq. 9 for hydrodynamical expansion. The hydrodynamical expansion is matched to the free-streaming behaviour at $\tau = 6$ and 12 fm/c</figcaption></figure>

states (quarkonia) and electromagnetic radiation. The study of quarkonium as a thermometer for the QGP was developed starting from the early idea [86] that the strong potential binding $q$ and $\bar{q}$ into pairs is screened by colour charges when immersed in the dense and hot coloured medium, ultimately leading to a “melting” of the $q\bar{q}$ states and offering the possibility to connect the production or suppression of quarkonia to the QGP temperature. The most recent developments and the ALICE results on quarkonia are discussed more in details in the Sect. 2.5. Electromagnetic radiation is emitted by a variety of sources during the evolution of the system in the form of real “direct” photons or virtual photons, measurable via their internal conversion into lepton–antilepton pairs. The invariant mass spectrum of dileptons is studied looking for any enhancement at intermediate (≈ 1–2 GeV for dielectrons) masses relative to expectations from hadronic decays in vacuum. Such an excess could be related to thermal emission from the QGP and thus provide information on its temperature. However, as dilepton measurements at the LHC are not yet sensitive to possible thermal signals [209], a precise measurement of the low-mass dielectron continuum will be pursued by ALICE during the LHC Runs 3 and 4. In the rest of this section, we focus on results obtained by measuring real direct photons.

Temperature can be accessed experimentally by measuring the yield of thermal photons that are emitted by the hot plasma during its entire evolution. As the mean free path of a photon in hot matter is much larger than the typical sizes of the created fireball, photons escape the collision zone unaffected, delivering information on the QGP conditions at early times, as well as on the development of collective flow during the evolution of the hot matter. Direct photons are the photons that do not originate from parton fragmentation nor hadronic decays, and are produced in electromagnetic interactions during different stages of the collision. The low-$p_T$ region of the spectrum ($p_T \lesssim 3$ GeV/c) is dominated by “thermal” photons and follows an approximately exponential behaviour, $d^2N_{\gamma_{dir}}/(p_Tdp_Tdy) \propto e^{-p_T/T_{eff}}$, characterised by the inverse logarithmic slope $T_{eff}$. The latter represents the effective temperature of the fireball and can be related to the “true” temperature by accounting for the radial expansion of the system, which causes a blue-shift of the emitted photons [210]. At higher momentum ($p_T \gtrsim 5$ GeV/c), “prompt” photons dominate and follow a power law spectrum. These photons are produced in the initial hard scatterings between the colliding nuclei. Other direct photon production mechanisms, like the interaction of hard scattered partons with the medium (“jet-photon conversion”) [211], may be important for $p_T \lesssim 10$ GeV/c [212,213]. These contributions to the spectrum need to be disentangled, such that the measured excess of thermal photons can be related to the effective temperature of the system.

The invariant yield of direct photons is determined by first measuring the direct photon excess over decay photons, $R_\gamma$, which is defined by

$$R_{\gamma} = \frac{\gamma_{\text{inc}}}{\pi_{\text{param}}^{0}} / \frac{\gamma_{\text{dec}}}{\pi_{\text{param}}^{0}} = \frac{\gamma_{\text{inc}}}{\gamma_{\text{dec}}}, \qquad (10)$$

where $\gamma_{\text{inc}}$ is the inclusive photon invariant yield, $\gamma_{\text{dec}}$ the decay photon invariant yield, and $\pi_{\text{param}}^0$ is the parametrised neutral pion invariant yield. Here the neutral pion invariant yield has been measured in the $\pi^0 \to \gamma\gamma$ channel. The result is reported in Fig. 16 (bottom) for central and semi-central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. At high-momenta, the $R_\gamma$ is consistent with prompt photon production (pQCD) and JETPHOX calculations [214,215]. For 0–20% collision centrality and considering all data points in $0.9 < p_T < 2.1$ GeV/c, the significance of the direct photon excess is about $2.6\sigma$, which indicates that there is an excess of thermal photons coming from the QGP. From $\gamma_{\text{dir}} = (1 - 1/R_\gamma)\gamma_{\text{inc}}$, the invariant yield of direct photons is calculated for central and semi-central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, and is reported in Fig. 16 (top) and compared to measurements in central and semi-central Au–Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. Harder photon spectra are measured at the LHC than those at RHIC. Hydrodynamical model calculations for Pb–Pb collisions [216,217] are con-
sistent with the data (Fig. 16). The region that is dominated by thermal direct photons is fitted by an exponential function (see solid lines in Fig. 16) to obtain the effective temperature ($T_{eff}$). For central and semi-central Pb–Pb collisions, the fits lead to $T_{eff} = (304 \pm 41)$ MeV and $T_{eff} = (407 \pm 114)$ MeV, respectively, and the extracted temperatures are consistent within uncertainties. The large relative uncertainty on the temperature in semi-central collisions is due to the large uncertainties on the fitted photon spectrum at low-$p_T$. Comparing these values to the slopes of the spectra measured by PHENIX and reported in Fig. 16, one can see an increase in the effective temperature from RHIC to the LHC. However, obtaining the initial temperature of the fireball is only possible indirectly by invoking model calculations that incorporate the evolution of the QGP medium as well as radial flow effects that blue-shift the direct photon spectra, and has not yet been attempted.

As direct photons escape the medium unaffected and are predominantly produced when the system is hot, it is expected that they don't have a large flow component. Measuring otherwise poses large questions to our understanding of the space-time dynamics of the QGP. The elliptic flow of direct photons ($v_2^{\gamma,dir}$) was calculated by combining the elliptic flow of inclusive photons together with that of decay photons ($v_2^{\gamma,dec}$) and the measured $R_\gamma$ using the scalar product method (see [221] for details). The measured $v_2^{\gamma,dir}$, reported in Fig. 17, appears to be close to the decay photon flow for both centrality classes and similar to the measurements by the PHENIX collaboration at RHIC [223]. Hydrodynamic and transport models [216,217,222] that include thermal photon production related to the local temperature of the fluid show a smaller $v_2^{\gamma,dir}$ than observed. For models to increase $v_2^{\gamma,dir}$ one can lower the initial temperature of the fluid, such that more photons are emitted when the system is more developed, but in turn would lead to a larger discrepancy in the prediction of the direct photon production. Especially at RHIC energies, theoretical predictions are not yet able to describe the direct photon production and the flow simultaneously. This is what is referred to as the “direct photon puzzle” [218,221]. Due to the size of the experimental uncertainties in Fig. 17, it is not yet confirmed if this puzzle is present at LHC energies, although the results are consistent with the measurements at RHIC. In Run 3, statistical uncertainties are expected to be reduced by a factor of ten, and the systematic uncertainties by about a factor of two. If the $R_\gamma$ measured with the new datasets will more strongly reject the null hypothesis, i.e. $R_\gamma = 1$, it will be very probable to establish if there is a direct photon puzzle at the LHC.

In summary, the results of the fits to the direct photon spectra indicate that the effective temperature of the fireball is larger than the critical temperature and increases from RHIC to LHC energies.

### 2.1.5 Size and lifetime of the system

The size and lifetime of the system created in a collision can be inferred from femtoscopy, which measures particle momentum correlations at kinetic freeze-out. The study of momentum correlations of particles emitted from a common source was historically referred to as “HBT interferometry” in the heavy-ion community, named after the original technique proposed by Hanbury-Brown and Twiss in the 1950s and 1960s to determine the size of laboratory and stellar sources by studying the interference of emitted photons [224,225].

In the case of two-particle femtoscopy [226–228], if two particles are emitted from a pp or heavy-ion collision and detected, the two-particle count rate is used to form the momentum correlation function, $C(k^*)$, given by $C(k^*) = \mathcal{N} \frac{A(k^*)}{B(k^*)}$, where $A(k^*)$ is the measured distribution of particle pairs from the same event, i.e. the two-particle count rate, $B(k^*)$ is the reference distribution of pairs created from particles coming from different events (referred to as “mixed events”), $\mathcal{N}$ is the normalisation factor, and $k^*$ is the magnitude of the momentum of each of the particles in their pair rest frame. Note that for identical particle pairs the invariant momentum difference is denoted as $q_{inv} = 2k^*$.

In order to extract the explicit spatial information and implicit time information about the emitting particle source at kinetic freeze-out, the measured two-particle correlation function is, in general, fitted with a formula that includes a quantum statistics term for identical particles, and a parameterisation which incorporates strong final-state interactions between the particles (FSI), for cases where they are important [229,230]. For example, for uncharged particles:

$$C(k^*) = 1 + \lambda e^{-4k^* R^2} + \lambda \alpha \left[ \frac{1}{2} \left| \frac{f(k^*)}{R} \right|^2 + \frac{2 \mathcal{R} f(k^*)}{\sqrt{\pi} R} F_1(2k^* R) - \frac{\mathcal{I} f(k^*)}{R} F_2(2k^* R) + \Delta C \right], \quad (11)$$

where $f(k^*)$ is the s-wave scattering amplitude, $\alpha = 0.5$ in the case of identical bosons, $R$ is the source radius parameter assuming a spherical Gaussian source distribution, and $\lambda$ is the correlation strength. The term $\Delta C$ is a calculated correction factor that takes into account the deviation of the spherical wave assumption used in the inner region of the short-range potential in the derivation of Eq. 11 [226]. The second term in Eq. 11 describes the quantum statistics if identical bosons are considered, and the third one is the FSI term. In Eq. 11, the $R$ parameter is related to the effective size of the source. In order to improve the description of the particle emitting source, two-particle femtoscopy can be reformulated in three-dimensions. The experimental correlation function is then obtained in terms of the components of the
<figure><img src="image_21.png" /><figcaption>**Fig. 16** Direct photon spectra (top) and direct photon excess $R_{\gamma}$ (bottom) measured in Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV [218] and $\sqrt{s_{NN}}=0.2$ TeV [219,220] in the 0–20% (left) and 20–40% (right) centrality classes. The slope of the exponential function fitted to the ALICE</figcaption></figure>

data was determined without subtracting any pQCD contribution. The slope of the spectrum measured by PHENIX was determined after subtracting a pQCD contribution determined by parameterising a direct photon measurement in pp collisions

<figure><img src="image_22.png" /><figcaption>**Fig. 17** Elliptic flow of direct photons, $v_2^{\gamma,dir}$, in the 0–20% (left) and 20–40% (right) centrality classes for Pb–Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV [221]. The vertical bars and the boxes indicate the</figcaption></figure>

statistical and the total uncertainties on the data, respectively. ALICE data are compared to model calculations [216,217,222] and PHENIX data [223] in the same respective centrality classes
<figure><img src="image_23.png" /><figcaption>Fig. 18 Homogeneity volume (top) and decoupling time $\tau_f$ (bottom) measured at $\sqrt{s_{NN}}=2.76$ TeV [239,240] compared to those obtained for central Au–Au and Pb–Pb collisions at lower energies at the AGS [232], SPS [95,233,234], and RHIC [185,235–238]. The homogeneity region is determined as the product of the three pion femtoscopic radii at $\langle k_T \rangle = 0.3$ GeV/c for 0–5% central events, whereas the decoupling time $\tau_f$ is extracted from $R_{long}$ ($k_T$) according to Eq. 12</figcaption></figure>

pair momentum difference vector: $q_{out}$ (along the direction of the sum of the transverse momenta of the particles) and $q_{side}$ (perpendicular to the direction of the sum of the transverse momenta of the particles), and parallel to the beam direction, $q_{long}$.

Femtoscopy measures the volume of the emitting source, which is in general not equivalent to the total volume occupied by the system at freeze-out. For an expanding source, with strong flow gradients, particles with similar momenta are emitted from a region referred to as the homogeneity volume, which is smaller than the total volume [231].

The charged-particle pseudorapidity density dependence of the femtoscopic pion radii and the decoupling time $\tau_f$ of pions for central Pb–Pb and Au–Au collisions were measured at AGS [232], SPS [95,233,234], RHIC [185,235–238] and LHC [239,240] energies. The decoupling time of the system is typically approximated with the decoupling time of pions $\tau_f$, since pions are the most abundant species (≈ 80%) constituting the bulk of the system. The size of the homogeneity

region, obtained as the product of the three pion radii $R_{out}$, $R_{side}$, and $R_{long}$ measured at $\langle k_T \rangle = 0.3$ GeV/c, has a linear dependence on the charged-particle pseudorapidity density shown in the top panel of Fig. 18. The magnitude increases three times from AGS energies to the LHC.

According to the implementation of the system evolution in hydrodynamic models, the $R_{long}$ at midrapidity is proportional to the total duration of the longitudinal expansion [241]. The decoupling time $\tau_f$ can be obtained from the relation between $R_{long}$ and the transverse mass ($m_T$) that is

$$R_{long}^2(m_T) = 2 \frac{\tau_f^2 T_{kin}}{m_T} \frac{K_2(m_T/T)}{K_1(m_T/T_{kin})}, \quad (12)$$

where $T_{kin}$ is the kinetic freeze-out temperature, taken to be 0.12 GeV, and $K_1$ and $K_2$ are the integer order modified Bessel functions [241]. The decoupling time obtained from the fit to the $m_T$ distribution of the pion $R_{long}$ [239] is shown in the bottom panel of Fig. 18, together with world data. The $\tau_f$ increases linearly with the cube root of the charged-particle pseudorapidity density. The time measured at AGS is about 4–5 fm/c increasing gradually up to 7–8 fm/c at top RHIC energies, and finally it reaches 10–11 fm/c in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV. It should be noted that corrections to Eq. 12, due to the transverse expansion and the finite pion chemical potential can increase the decoupling time by up to 25% [242]. An additional source of uncertainty originates from the chosen value of the kinetic freeze-out temperature. Varying it by 0.02 GeV in both directions leads to an increase of $\tau_f$ by 13% or a decrease by 10%, respectively, without changing the overall observed scaling behaviour. Taking into account these uncertainties leads to the $\tau_f$ range 10–13 fm/c.

The LHC centrality-dependent Pb–Pb data are compared to the world heavy-ion data, as well as to p–Pb and pp data at various energies, in Fig. 19. As argued in [227], each of the three-dimensional radii is observed to scale roughly with the cube root of the measured charged-particle multiplicity density ($\langle dN_{ch}/d\eta \rangle^{1/3}$) across a wide range of collision energies and colliding systems with different initial geometries. For similar $\langle dN_{ch}/d\eta \rangle^{1/3}$ in different Å collision systems, differences in the magnitude of the radii, more significant in the transverse direction ($R_{out}$), were predicted by hydrodynamic calculations [247] and are attributed to distinct freeze-out shapes. The pp and Å data sets exhibit significantly different scaling behaviour, although both are linear in $\langle dN_{ch}/d\eta \rangle^{1/3}$. The radii in p–Pb and pp collisions agree at low multiplicities and diverge by up to 10–20% at the highest multiplicities reached in these systems. The application of the femtoscopic technique to the study of the source in small systems will be further discussed in Sect. 5 while the rest of this section is dedicated to discussing the radii dependence on the transverse momentum (or transverse mass).
<figure><img src="image_24.png" /><figcaption>**Fig. 19** Multiplicity dependence of pion radii in pp, p-Pb and Å collisions for a number of collision systems and energies [234, 235, 239, 240, 243–246]. Various experiments use different $k_{T}$ ranges. On the plot are the values for the range for which the average $k_{T}$ is closest to the selected value of 0.25 GeV/c</figcaption></figure>

In a hydrodynamical picture, the collective expansion of the fireball reduces the size of the homogeneity region due to the interplay between the collective and thermal velocities of particles. Due to this, each of the femtoscopic radii is expected to decrease with $m_{T}$ following a power-law behaviour [227]. This is shown in Fig. 20, where the dependence of $R_{\text{out}}$, $R_{\text{side}}$ and $R_{\text{long}}$ on the pair transverse mass $m_{T}$ is reported on the left, right top and right bottom panels, respectively, for different centralities.

To discuss the general applicability of hydrodynamics for describing the transverse-mass dependence of the femtoscopic radii, we show that one finds similar results for different implementations of the model. In Fig. 20, data are compared to the calculations [249] from the (3+1)D hydrodynamic model coupled to the THERMINATOR 2 statistical hadronisation code [251]. The slope of the $m_{T}$ dependence of the transverse radii $R_{\text{out}}$ and $R_{\text{side}}$, which depends on the amount of radial flow in the system, is well reproduced by the model. The calculation also reproduces well the magnitude

of $R_{\text{out}}$ for all centralities, whereas it tends to underestimate the magnitude of $R_{\text{side}}$, which lies on the lower edge of the systematic uncertainty associated with the data. The intercept of $R_{\text{side}}$ at low-$m_{T}$ is usually associated with the overall geometrical size of the system. The model reproduces the values of $R_{\text{long}}$ for all centralities, but it overpredicts the magnitude and the slope of the $m_{T}$ dependence, especially at low-momentum. This is an indication that also the longitudinal dynamics is reasonably well described in the model, both in momentum and space-time sectors. Altogether, the good agreement of the model with the femtoscopy data supports the validity of the hydrodynamic approach.

The calculation from the HKM model [252], also based on a hydrodynamic formalism, is shown in Fig. 20 for the 5% most central collisions. It differs from the previous model in the implementation of the freeze-out process. Two variants are compared: one in which hadronic interactions in the late hadron-gas phase are neglected (dashed lines, labelled as “w/o rescattering” in Fig. 20), and one in which hadronic
<figure><img src="image_25.png" /><figcaption>**Fig. 20** Pair transverse mass dependence of the pion [240] and kaon [248] femtoscopic radii for different event centralities in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. The measured $R_{out}$, $R_{side}$, $R_{long}$ are reported in the left, right top and bottom panel, respectively. The experimental data are reported with solid symbols together with statistical and systematic uncertainties. Bands represent theoretical predictions of</figcaption></figure>

pion radii by a (3+1)D hydrodynamic model coupled to the Thermiator code [249] for the same centralities as in data, selected based on the impact parameter $b$ in the calculation. Lines represent calculations for central collisions by the HKM model with and without rescattering [250]

interactions are implemented using the UrQMD transport model (continuous lines, labelled as “W/ rescattering” in Fig. 20). Both calculations are compatible with the pion femtoscopy data for $R_{out}$ within uncertainties. However, predictions of HKM with rescattering are also compatible with the measured $R_{side}$ and $R_{long}$, whereas calculations without rescattering slightly underpredict them. Therefore, the approximate agreement of the hydrodynamics-based models with data is a universal feature of such calculations, not of a particular implementation. It is also to be noted that the particular choice of initial conditions and the equation of state for these models was motivated by the analysis of the RHIC femtoscopic measurements and is essential for the correct description of the data. The results also indicate that the details of the freeze-out process have limited influence on pion femtoscopy. Some studies suggest that femtoscopy of heavier particles might be a more sensitive probe in this case [252].

In addition to the pion radii for the 0–5% most central events, Fig. 20 also shows the $m_T$ dependence of the three radii for charged and neutral kaons in comparison with HKM predictions [250]. The HKM calculation without rescattering

exhibits an approximate $m_T$-scaling but does not describe the data. The measurements are well reproduced by the full hydro-kinetic model calculations that includes the effect of the hadronic rescattering in the late stages of the system evolution, thereby showing the importance of the hadronic phase at the LHC.

### 2.1.6 Conclusions

**Particle production.** The multiplicity of final-state particles is closely related to the entropy produced in the collision. Central heavy-ion collisions are more effective in converting the longitudinal beam energy into particle production at midrapidity than pp collisions. The final state multiplicity spans from few tens of charged particles per unit of rapidity in peripheral heavy-ion collisions (a “small system”) to few thousands in central heavy-ion collisions (a “large system”). Centrality gives an experimental handle to control the amount of energy deposited in the collision region and thus the properties of the resulting system.
**Energy density and temperature.** The energy density estimate based on the free-streaming Bjorken equation, which provides a lower bound for this quantity, and with the assumption of a formation time of 1 fm/c [112], results in $\epsilon = (12.3 \pm 1.0)$ GeV/fm$^3$ in 0–5% Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV. This value exceeds by more than a factor of 30 the critical energy density from lattice QCD in most central Pb–Pb collisions at the LHC. The slope of the thermal direct photon spectra represents an effective temperature, which can be related to the true temperature of the system once radial flow is taken into account via full hydrodynamic model calculations. The effective temperature in central Pb–Pb collisions at the LHC is $T_{\text{eff}}=(304 \pm 41)$ MeV. The latter is larger than at RHIC and than the critical temperature from lattice QCD. The observations above provide an experimental demonstration that two necessary conditions for the formation of the quark–gluon plasma are met at the LHC.

**Freeze-out.** The increase in beam energy by a factor of 25 from top RHIC energy to the LHC produces a homogeneity region approximately twice larger in most central collisions. The decoupling time for midrapidity pions lies in the range 10–13 fm/c at the LHC, which is about 40% larger than at RHIC. These results indicate that the fireball formed in central Pb–Pb collisions at the LHC lives longer and expands to a larger size at freeze-out compared to lower energies. The quantitative agreement between models based on relativistic hydrodynamics with statistical hadronisation and femtoscopic measurements supports this picture of the evolution of the system created in heavy-ion collisions.

## 2.2 QGP evolution and its dynamical properties

The dynamical properties of the QGP offer an essential insight into the strongly-coupled matter. They are principally characterised by measurements sensitive to anisotropic and radial flow, which contribute to the collective motion observed in heavy-ion collisions. In addition, these properties control how the initial state angular momentum manifests in the global polarisation of the produced particles. Measurements of anisotropic flow at RHIC drew immense attention [109–112] since the observed values were large enough to indicate the formation of a “perfect liquid” in the laboratory, a claim which was further validated by the estimated values of $\eta/s$ being close to zero for the dense phase of the QGP. Around the same time, AdS/CFT calculations performed using string theoretical techniques showed that $\eta/s$ for any fluid has a lower limit of $1/4\pi$ [107]. This originates from the uncertainty principle: the decrease in shear viscosity with the mean free path of the fluid constituents is limited because of the finite momentum of the constituents, thus preventing it to be zero.

Subsequent calculations have predicted the temperature dependence of $\eta/s$ and $\zeta/s$ in the frameworks of AdS/CFT (infinite coupling), QCD (strong coupling), and pQCD (weak coupling) [50–54]. They generally show that the weaker the coupling, the higher the values of $\eta/s$ are and the stronger the temperature dependence is. They also show that the strong/infinite coupling approaches lead to a rise in $\zeta/s$ when the QGP is near the transition temperature to the hadronic phase. Strong and weak-coupling theories can also predict different dependencies of the $\zeta/\eta$ ratio in relation to the speed of sound [253]. These predictions can be tested in the QGP since the viscosities affect the macroscopic properties of many-body systems. Therefore, the collective motion of the QGP inferred from final state hadron measurements, coupled with comparisons to hydrodynamic calculations, offer a unique opportunity to constrain the fundamental properties of strongly interacting matter.

Beyond what was described in Sect. 1.2.2, more differential techniques characterising anisotropic and radial flow have emerged over the last 10 years, which promise to place even greater constraints on QGP transport properties, and will be detailed in this section. Two key examples regarding anisotropic flow are Symmetric Cumulants and Non-linear flow mode coefficients. The former evaluates event-by-event correlations between different orders of anisotropic flow coefficients, while the latter quantifies the non-linear contributions to higher order anisotropic flow from lower orders. Measurements of two-particle correlations can provide additional constraints on the quantitative deduction of effects due to radial flow and shear viscosity. These can also reveal information about the charge diffusion as the created positive-negative charge pairs move through the QGP. An investigation into the polarisation of hyperons in the beam direction [254,255] also provides a new avenue to further explore the hydrodynamic response.

### 2.2.1 First ALICE results of anisotropic flow

Anisotropic flow is quantified by the $v_n$ coefficients and symmetry plane angles $\Psi_n$. Of particular interest are non-central heavy-ion collisions, which are characterised by a dominant ellipsoidal geometry in the initial state. Theoretical expectations for the LHC energies predicted an increase of 10–50% for the $p_T$ integrated $v_2$ when compared to the previous RHIC measurements at lower energies. As can be seen from the top panel of Fig. 21, the first $v_2$ measurements at the LHC reported by ALICE in midcentral collisions show an increase of about 30% compared to the top energy at RHIC [256], and this result was in agreement only with predictions from hydrodynamic models which included values of $\eta/s$ close to the AdS/CFT limit of $1/4\pi$. The further increase of $v_2$ by about 3% in all centralities for Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [257], indicated that the elliptic flow did
<figure><img src="image_26.png" /><figcaption>**Fig. 21** (Top) Energy dependence of $v_2$ for midcentral heavy-ion collisions integrated over $p_T$ [256–263]. (Bottom) Decomposition of the two-particle correlation function in terms of even and odd anisotropic flow harmonics ($v_n$) from very central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [264]</figcaption></figure>

not saturate in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, and was in qualitative agreement with most of the hydrodynamic models with similarly low values of specific shear viscosity.

Another interesting aspect of anisotropic flow is the nature of its higher order coefficients $v_3–v_5$. A double-peaked structure in two-particle azimuthal correlations, which was first observed at RHIC [265], was initially interpreted as the signature of a Mach-cone response from fast partons. However, the ALICE results (bottom panel of Fig. 21), in conjunction with a reanalysis of RHIC data [266], and other LHC experiments [264,267,268], demonstrated that a more natural explanation stems from the non-vanishing values of these higher flow coefficients. One of the first extractions of the two-particle correlation function at the LHC is shown in the bottom panel of Fig. 21. The amplitude of the Fourier

components provides a measure of $v_n^2$ for the corresponding momentum ranges. The higher harmonic contributions originate from event-by-event fluctuations in the number and distribution of nucleons in the overlap region of the colliding nuclei, and hydrodynamic models predicted that their magnitude and transverse momentum dependence are sensitive to $\eta/s$ [269,270]. In addition, event-by-event fluctuations violate symmetries linked to an idealistic ellipsoidal geometry, and as a consequence each symmetry plane $\Psi_n$ is distinct.

Measurements of spatial anisotropies of the final state using femtoscopic techniques were also performed in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [271,272]. While the momentum anisotropies shown previously reflect the initial spatial anisotropies, the azimuthally-differential femtoscopic measurements are sensitive to spatio-temporal characteristics of the source as well as to the collective velocity fields at freeze out. These femtoscopic measurements revealed that the second order spatial anisotropy remains finite in the final state [271], however, it is reduced compared to values expected from the initial state. The third order final state spatial anisotropy appears washed out [272], being significantly smaller than the typical initial state third order anisotropy. All of these observations are consistent with hydrodynamic expectations. These indicate that the finer features of the initial state anisotropy dissipate more quickly for higher eccentricity orders compared to the lower orders. The associated predictions will be compared to other data later in this section.

### 2.2.2 Identified hadron spectra, radial flow, and kinetic freeze-out temperatures

The hydrodynamic expansion of the QGP and the late, dissipative hadronic rescattering phase are reflected in the motion of final-state hadrons. Radial flow leads to flatter transverse momentum ($p_T$) distributions with increasing mass, particularly at low values of $p_T$. In this region, the $p_T$ distribution has contributions from the random thermal motion, and the collective expansion. While the former depends on the decoupling temperature, the latter is dependent on the hadron mass because all hadrons acquire an additional transverse momentum given by their mass multiplied by the common radial flow velocity. Figure 22 shows the $p_T$ spectra of various particles<fnref n="5" /> in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV for the 0–5% and 80–90% centrality intervals [273–275]. The spectral shapes depend on centrality with the maxima located at higher transverse momenta in central compared to peripheral collisions. Furthermore, the flattening of the spectra in the low-$p_T$ region is mass dependent, and is more pronounced for heavier par-

<footnote n="5">⁵ Only $\pi^+$, $K^+$, and p are shown as the distributions of positively and negatively charged particles are compatible within uncertainties at all $p_T$ at the LHC.</footnote>
<figure><img src="image_27.png" /><figcaption>**Fig. 22** Transverse momentum distributions of $\pi^+$, $K^+$, p [273], $K_S^0$ and $\Lambda$ [274], and the $\phi$ meson [275] for the 0–5% and 80–90% centrality intervals in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV. The data points are scaled by various factors for better visibility. Statistical and systematic uncertainties are shown as bars and boxes, respectively</figcaption></figure>

ticles. This observation is in line with the expected effect of increasing radial flow with collision centrality.

The hadron spectra can be studied in the context of the Boltzmann–Gibbs Blast-Wave (BG-BW) parameterisation [276]. In this parameterisation, particles are produced by a thermalised medium, which expands radially and undergoes an instantaneous kinetic freeze-out at a temperature $T_{\text{kin}}$. In the simplest case, the expansion is radial and is prescribed by a common velocity field profile, $\beta(r) = \beta_s (r/R)^n$, where $\beta_s$, $R$, and $n$ are the expansion velocity on the surface of the fireball, its radius, and the exponent regulating the shape of the velocity profile, respectively. It is well known that particle spectra, especially at low-transverse momenta, are populated by the decay products of resonance decays. These decays modify the spectral shapes of hadrons, and thus the simple Boltzmann–Gibbs prescription may fall short. So far, several attempts have been made to include the products of resonance feed-down into the Blast-Wave parameterisation, but they usually treat the production of primary hadrons and short-lived hadronic resonances separately. The calculation of the latter usually resorts to either Monte Carlo generators or a semi-analytical treatment of decay integrals which are both computationally intensive. An alternative approach was proposed in Ref. [277], enabling a computationally-

<figure><img src="image_28.png" /><figcaption>**Fig. 23** Blast-Wave model parameters of the kinetic freeze-out temperature $T_{\text{kin}}$ and radial flow velocity $\beta_T$. These extracted values involve simultaneous fits to $\pi$, K, and p spectra and $v_2$ for the two models in Pb–Pb collisions</figcaption></figure>

efficient fitting procedure to measured particle spectra (BW FastReso) [278]. In addition, the procedure was extended to include anisotropic flow, particularly the $v_2$ of different particle species (see Sect. 2.2.4). The extracted parameters from the simultaneous fits to the $p_T$ spectra and to $v_2$ of $\pi$, K, and p in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV are shown in Fig. 23 from both BG-BW and BW FastReso approaches. A key aspect to note is that the radial flow velocities appear rather independent of the BW parameterisation in more central collisions, which correspond to larger $\beta_T$ values. The highest ever radial flow velocities in heavy-ion collisions are observed at the LHC (a comparison to RHIC results can be found elsewhere [273]). However, the kinetic freeze-out temperatures depend strongly on the BW parameterisation. The BG-BW approach determines the kinetic freeze-out temperature directly, whereas BW FastReso in this implementation assumes the same chemical and kinetic freeze-out temperature for hadrons. BW FastReso allows for these temperatures to be two distinct free parameters, however, they happen to be consistent for fits utilising this option (not shown). The parameters shown in Fig. 23 indicate that the inclusion of resonance feed-down denoted by the filled and open circles results in an increase of the kinetic freeze-out temperature by ~ 60 MeV. This demonstrates that this temperature is highly dependent on the inclusion of resonance feed-down.

### 2.2.3 Hydrodynamic descriptions of global observables

In Fig. 24, hydrodynamic model calculations of the average transverse momentum of identified particles and of var-
<figure><img src="image_29.png" /><figcaption>**Fig. 24** Comparisons of ALICE measurements of identified particle $\langle p_T \rangle$ and charged hadron $v_n$ coefficients in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV (left) and Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV (right) [257,279–281] to hydrodynamic calculations [48,49]</figcaption></figure>

ious anisotropic flow coefficients for charged hadrons are compared to ALICE data from heavy-ion collisions. Hydrodynamic models differ from Blast-Wave parameterisations. While the former use properties of the QGP through an equation of state, Blast-Wave models are limited and attempt to assess the final-state properties under certain assumptions. It is shown in Ref. [282] that $v_n$ can be described by multiple combinations of $\eta/s$ and $\zeta/s$ values. However, these are constrained further by attempts to describe $\langle p_T \rangle$ data for identified hadrons, which is why such a simultaneous comparison is important. The precision of the data is such that the uncertainty is about 2% for most data points, while both models are able to describe the data within 10% of the measured values. The $T_{\text{RENT}}+\text{VISHNU}$ calculation uses a Bayesian analysis to tune the input parameters of $\eta/s$ and $\zeta/s$ (among others) to best match the data in Pb–Pb $\sqrt{s_{NN}} = 2.76$ and 5.02 TeV collisions [49]. This tune does not include the Xe–Xe data [279–281], and the subsequent model calculations can be considered as predictions based on the Pb–Pb inference. The IP-Glasma+MUSIC uses a priori predictions of $\eta/s$ and $\zeta/s$ in an attempt to describe the same measurements at RHIC and the LHC [48]. Both models require an $\eta/s$ similar to the minimum AdS/CFT value of $1/4\pi$ with little or no dependence on the temperature, and a finite $\zeta/s$ that rises with decreasing temperature as it approaches the QGP transition temperature. Those parameterisations do not depend on the collision

centrality or the colliding system. Each of these approaches implicitly assumes a strongly-coupled QGP, and they appear to describe the data. It is also worth noting that the peak value of $\zeta/s$ close to the QGP transition temperature is larger in IP-Glasma+MUSIC compared to $T_{\text{RENT}}+\text{VISHNU}$. This may explain why $T_{\text{RENT}}+\text{VISHNU}$ appears to describe $\langle p_T \rangle$ slightly better, since lower values of $\zeta/s$ will increase $\langle p_T \rangle$, and this pushes $T_{\text{RENT}}+\text{VISHNU}$ closer to the data, particularly for kaons and protons. The values of $\eta/s$ and $\zeta/s$ explored for these hydrodynamic models will be discussed in more detail in Sect. 2.7, while the role of the initial-state models will be discussed in Sect. 4.

Even though the quality of these descriptions is impressive given the multitude of measurements they involve, such comparisons need to be further tested by more differential measurements. Figure 25 shows a comparison of measurements of anisotropic flow coefficients with $T_{\text{RENT}} + \text{VISHNU}$ predictions for very central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. Earlier hydrodynamic calculations could not describe the $v_3/v_2$ ratio for centralities less than 1%, a phenomenon sometimes referred to as the *ultra-central problem*, and the deviations compared to the data were around 30% [283]. For the ALICE measurements in the 0–1% range, more recent hydrodynamic predictions appear to do better, and describe the ratio on the level of 10%, which is similar to some of the other comparisons shown in Fig. 24. The value
<figure><img src="image_30.png" /><figcaption>**Fig. 25** Measurements of charged hadron $v_n$ coefficients in central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [279] compared with hydrodynamic calculations [49,283]. The arrow denotes predictions from Ref. [283] for 0–0.2% Pb–Pb collisions</figcaption></figure>

of $\eta/s$ in the TRENTO + VISHNU approach is roughly a factor of two smaller than the earlier calculation, which leads to a larger value of $v_3/v_2$. The lower value of $\eta/s$ is related to the implementation of bulk viscosity in recent hydrodynamic calculations, as concluded to be needed for very central Pb–Pb collisions [284]. These improvements might also have a contribution from advancements in the initial state modelling, which will be discussed in Sect. 4. The first measurements of the anisotropic flow power spectra up to the ninth order are shown in Fig. 26 for the 10–20% centrality interval in Pb–Pb collisions. As mentioned, higher order anisotropic flow coefficients are more sensitive to viscous damping, and the dampening rate depends on $\eta/s$. In addition to TRENTO + VISHNU, these data are also compared with the EKRT predictions. The EKRT model [285,286] describes anisotropic flow measurements equally well as TRENTO + VISHNU although requiring higher shear viscosity since it does not include bulk viscosity. However, even in its most recent implementation [287], it has difficulties describing average transverse momentum measurements of identified particles at the LHC, due to the assumption of $\zeta/s = 0$. More broadly, all of these investigations regarding the hydrodynamic response rely on a realistic description of

<figure><img src="image_31.png" /><figcaption>**Fig. 26** Higher order charged hadron $v_n$ coefficients in the 10–20% centrality interval [288]</figcaption></figure>

the initial state to extract QGP properties, and this will be addressed further in Sect. 4.

Finally, an extraction of the QGP speed of sound was performed with the ALICE data and hydrodynamic model comparisons using charged hadron $\langle p_T \rangle$ [289] measurements. The value squared of the speed of sound obtained from Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV is $(0.24 \pm 0.04) c^2$. This number is compatible with the Lattice QCD prediction of $0.25 c^2$ for a temperature of 222 MeV i.e. in the deconfinement regime [28]. Strong- and weak-coupling theories predict competing dependencies of the $\zeta/\eta$ ratio in relation to the speed of sound [253] even though the speed of sound is intrinsically not a dynamical physical quantity. Lattice QCD can predict its value in the QGP, which is consistent with the value obtained from ALICE data. It is also an important parameter for hydrodynamic models which require it as input for shear and bulk relaxation times. The value used in the IP-Glasma+MUSIC model [48] is consistent with what is extracted from ALICE data. Considering the uncertainty on the value obtained from ALICE data, it is not yet possible to discern between the weak- and the strong-coupling regimes.

### 2.2.4 Identified hadron anisotropic flow

A cornerstone for investigating the strongly-coupled QGP paradigm is a set of measurements of $v_2$ for various particle species. A characteristic mass ordering at low-$p_T$ is induced
<figure><img src="image_32.png" /><figcaption>**Fig. 27** The $p_T$-differential $v_2$ (a), $v_3$ (b), $v_4$ (c), and $v_{4,22}$ (d) measured by ALICE in semicentral (i.e., 20–30% centrality interval) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [290,291]. The data points are drawn with their statistical (error bars) and systematic (boxes) uncertainties.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>The curves represent estimations extracted from $T_{RENTO} + \text{VISHNU}$ that give the best description of other anisotropic flow measurements among all models studied</figcaption></figure>

at hadronization, after the development of radial flow in the QGP phase. Radial flow leads to a depletion in the particle $p_T$ spectrum at low values, which increases with increasing particle mass. When introduced in a system that exhibits azimuthal anisotropy, then radial flow develops an azimuthal profile, with larger velocity values in- than out-of-plane. This, in turn, leads to a depletion in the momentum spectrum that becomes larger in-plane than out-of-plane, resulting in a reduction in the differences of particle yields at a given $p_T$ in- versus out-of-plane and consequently in a reduction of $v_2$. Due to the dependence of momentum on the mass, this effect is more pronounced for heavier particles. The net result is that at a fixed value of $p_T$, heavier particles have smaller $v_2$ values compared to lighter ones. Therefore the focus in this section is on the low-$p_T$ region (i.e., $p_T < 3$ GeV/c), while the high-$p_T$ trends are discussed in Sect. 2.3.

Figure 27a, presents the transverse momentum dependence of $v_2$ as measured for various particle species in semicentral (i.e., 20–30% centrality interval) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [290]. Similar results are obtained in Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [292,293], Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV [294], and RHIC [295,296]. The characteristic mass ordering that develops due to an interplay between radial flow and the anisotropic expansion of the fireball is clearly seen for $p_T < 2.5$ GeV/c.

This mass ordering develops not only during the partonic evolution of the medium but also in the late hadronic re-scattering phase. The relative contribution, though, is expected to be related to the corresponding duration time of the two phases. Hence, at the LHC, the relative impact of the highly dissipative hadronic stage is expected to be weaker than at lower, RHIC energies [297]. An excellent testing ground for the contribution of the hadronic phase is provided by studying particles that are estimated to have small hadronic cross sections, such as the $\phi$ meson and the $\Xi$ baryon, and are thus expected not to be affected by this stage [298–301]. Figure 27a illustrates that the $\phi$ meson follows the observed mass ordering (similarly for the $\Xi$ baryon, see Ref. [292]). The current level of statistical and systematic uncertainties of these measurements, however, does not rule out the hypothesis that the mass ordering can be broken due to a significantly reduced contribution of the hadronic phase to the development of $v_2$ for the $\phi$ meson [297].

The interplay between radial flow and the anisotropic expansion of the system was further tested by investigating whether the mass ordering is reflected in measurements of higher order cumulants [302] or higher flow harmonics [290,293]. Panels (b) and (c) of Fig. 27 present the transverse momentum dependence of $v_3$ and $v_4$, respectively, for the same particle species (with the exception of the $\Lambda$ baryon for $v_4$ and the $\phi$ meson for both $v_3$ and $v_4$) and
the same centrality interval as before for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [290]. These results clearly illustrate that the effect can be seen in $v_2$, as well as $v_3$ and $v_4$ (but also for $v_5$ see Refs. [291,293]). This mass ordering develops also for the 1% most central Pb–Pb collisions [290,293], a category of events referred to as ultra-central collisions, where there is no dominant second order initial state geometry.

Hydrodynamic models have also shown that $v_2$ and to a large extent $v_3$ are linearly proportional to their corresponding initial spatial eccentricities, $\varepsilon_2$ and $\varepsilon_3$, for central and semicentral collisions [303]. However, the study of higher order flow coefficients (i.e., for $n > 3$) revealed that higher order eccentricities have a non-linear dependence on the lower order i.e., $\varepsilon_2$ and $\varepsilon_3$ [304,305]. This further supports the earlier ideas that the $v_n$ coefficients (i.e., for $n > 3$) receive contributions not only from the linear response of the system to $\varepsilon_n$, but also a non-linear response proportional to the product of lower order initial spatial anisotropies [304,305]. The non-linear contribution arises from lower order momentum anisotropies generating higher order spatial anisotropies as the system evolves, which in turn generate higher order momentum anisotropies. Figure 27d, presents the transverse momentum dependence of $v_{4,22}$, the non-linear flow mode of quadrangular flow, reported in Ref. [291] for the 20–30% centrality interval in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV. A similar mass ordering like the one reported for $v_n$ measurements develops also here for $p_T$ < 2.5 GeV/c. This arises from the interplay between radial flow and the initial spatial anisotropy, generated from both the geometry and the fluctuating initial energy density profile. Although the mass ordering should develop differently for $v_{4,22}$ and $v_4$ due to the dependence on $\varepsilon_2^2$ of the former, it is found to be quantitatively the same with maybe a hint of differences developing for $p_T$ < 0.8 GeV/c in the 0–30% centrality interval [291]. Non-linear flow modes are discussed in more detail in Sect. 2.2.6.

Figure 27 also present the comparison of the measurements with the same TRENTO + VISHNU calculations as shown in Fig. 24. The curves describe the trends of the data points for $v_2$, $v_3$, $v_4$, and $v_{4,22}$ within 20% for $p_T$ < 2 GeV/c. There are larger discrepancies at higher $p_T$ which have two potential sources. In hydrodynamic models, the momentum distribution of hadrons at freeze-out is principally modeled under the assumption of thermal equilibrium, and that distribution is labelled as $f_{\text{thermal}}$. A $\delta f$ term is introduced to account for non-equilibrium processes, thus expressing the final momentum distribution sampled as $f = f_{\text{thermal}} + \delta f$. The $\delta f$ corrections grow with $p_T$, and are highly model dependent [306]. Therefore, deviations at higher $p_T$ might be due to improper $\delta f$ corrections, or sub-optimal tunes of $\eta/s$ and $\zeta/s$.

### 2.2.5 Symmetric cumulants

The studies of the individual flow amplitudes $v_n$ can be extended to correlations between event-by-event fluctuations of flow coefficients [303,310–313]. Model calculations show that while $v_2$ and $v_3$ exhibit an approximately linear dependence on the corresponding eccentricities $\varepsilon_2$ and $\varepsilon_3$, respectively, the higher order $v_n$ coefficients (i.e., for $n > 3$) have also non-linear contributions from $\varepsilon_2$ and $\varepsilon_3$ in addition to the linear ones from $\varepsilon_n$ [270,305,314,315]. These observations lead to non-trivial correlations between different flow coefficients which result in new and independent constraints on initial conditions and $\eta/s$. In addition, they have the potential to separate the effects of $\eta/s$ from fluctuations in the initial conditions. For such novel studies, ALICE introduced Symmetric Cumulants (SC) [310,316] and mixed harmonic cumulants [317]. These observables are independent of the symmetry plane angles $\Psi_n$, and are robust against systematic biases due to unwanted nonflow correlations (i.e., short-range correlations unrelated to the azimuthal asymmetry in the initial geometry, such as inter-jet correlations and resonance decays) in heavy-ion collisions.

The left panel of Fig. 28 presents the centrality dependence of correlations between $v_n$ coefficients (up to 5th order) using SC($k, l$) $\equiv \langle v_k^2 v_l^2 \rangle - \langle v_k^2 \rangle \langle v_l^2 \rangle$ in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [307,308]. The correlations among different flow coefficients depend on harmonic as well as collision centrality [308]. Positive values of SC(4,2), SC(5,2), and SC(5,3) and negative values of SC(3,2) and SC(4,3) are observed for all centralities. These indicate that event-by-event fluctuations of $v_2$ and $v_4$, $v_2$ and $v_5$, and $v_3$ and $v_5$ are correlated, while $v_2$ and $v_3$, and $v_3$ and $v_4$ are anti-correlated. Furthermore, the lower order harmonic correlations are much larger than the higher order ones. Precision measurements from Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [318] show similar trends.

The SC observables are compared with EKRT [309] and TRENTO + VISHNU [49] predictions in the right panel of Fig. 28. The EKRT calculations are shown for the two temperature dependent $\eta/s$ parameterisations that provide the best description of RHIC and LHC data: constant $\eta/s$ = 0.2 and “param1” [309]. The “param1” parameterisation is characterised by a moderate slope in the temperature dependence of $\eta/s$ which decreases (increases) in the hadronic (QGP) phase and the phase transition occurs around 150 MeV. The SC(3,2) and SC(4,2) are not described simultaneously in each centrality interval by the EKRT calculations, which points to a strong dependence on $\eta/s$ of both observables. The SC(3,2) is better described by the TRENTO + VISHNU predictions for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV although these measurements or the subsequent ones presented later in this section are not used in the Bayesian inference. In addition, normalised symmetric cumulants were also measured in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [318].
<figure><img src="image_34.png" /><figcaption>**Fig. 28** Centrality dependence of event-by-event flow harmonic correlations measured by ALICE in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [307,308] compared with various hydrodynamic calculations [49,309]. The SC(4,2) and SC(5,3) points are slightly shifted along the horizontal axis for better visibility in the left panel</figcaption></figure>

collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [307,308] and are reported in Sect. 4. Recently, ALICE measured multiparticle cumulants with three different flow harmonics in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV [318,319].

### 2.2.6 Non-linear flow modes and flow vector fluctuations

Higher order anisotropic flow coefficients (i.e., $n > 3$) have also contributions from the initial-state anisotropy of the lower orders as discussed in Sect. 2.2.4. These additional contributions can be characterised by measurements of the non-linear flow mode coefficients, $\chi_{n, mk}$ with the indices representing different harmonics, which in principle should primarily arise from the QGP evolution [304,320]. The lower order coefficients are more sensitive to the initial-state configuration, while the higher orders are more susceptible to the QGP transport properties at later times and therefore lower temperatures. Figure 29 shows the centrality dependence of the non-linear flow mode coefficients $\chi_{4,22}$ (left panel) and $\chi_{5,23}$ (right panel) in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [288]. A small decrease from central to peripheral collisions is found for both coefficients. In addition, the relationship of $\chi_{4,22} \approx \chi_{5,23}/2$ is approximately valid as predicted by hydrodynamic calculations [305]. Given both the TRENTO + VISHNU and IP-Glasma+MUSIC models appear equally competitive in describing the $\langle p_T \rangle$ and $v_n$ coefficients in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV (see left panel of Fig. 24), it is worth noting that the non-linear flow mode coefficients are also well described by both models. Furthermore, the EKRT model [309], which as mentioned does not implement bulk viscosities, does also equally well.

Fluctuating initial conditions can also generate fluctuations of the flow vector in different transverse momentum or pseudorapidity regions. This was first tested in Ref. [321], where two-particle correlation $V_{n\Delta}(p_T^a, p_T^t)$ (one associated particle from $p_T^a$ and the other trigger particle from $p_T^t$) measurements were fitted globally, and then compared to the product $v_n(p_T^a) \times v_n(p_T^t)$. If such a factorisation holds,

this implies all particles flow in the same direction irrespective of transverse momentum or pseudorapidity. This was indeed observed for the case of $n \ge 2$ at low values of $p_T$ ($\lesssim 2$ GeV/c) [321,322]. The degree to which factorisation is broken can also be investigated using two types of two-particle correlation measurements: (i) the factorisation ratio $r_n$ [323], which probes $V_{n\Delta}(p_T^a, p_T^t)$ with respect to the square root of the product of $V_{n\Delta}(p_T^a, p_T^a)$ and $V_{n\Delta}(p_T^t, p_T^t)$ that uses both particles from either $p_T^a$ or $p_T^t$; or (ii) $v_n\{2\}/v_n\{2\}$ [324], where $p_T^a \ne p_T^t$ for $v_n\{2\}$ and $p_T^a = p_T^t$ for $v_n\{2\}$. Both $v_n\{2\}/v_n\{2\}$ and $r_n$ were measured in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [325]. Clear deviations from unity are observed for the second order flow vector in the 0–5% centrality interval. This indicates the presence of possible $p_T$-dependent flow vector fluctuations, which can have an influence from the hydrodynamic response. Higher order measurements ($n > 2$) show no clear indication of $p_T$-dependent fluctuations within uncertainties. The results are described fairly well by hydrodynamic calculations, including the TRENTO + VISHNU approach [326]. Recently, ALICE reported the first measurements of $p_T$-dependent flow angle $\Psi_2$ and flow magnitude $v_2$ fluctuations, determined using new four-particle correlators, in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [327]. Deviations from unity are observed for both flow angle and flow-magnitude fluctuations in the presented centrality intervals for $p_T > 2$ GeV/c, being the largest in the most central collisions.

### 2.2.7 Charge dependent and independent two-particle correlations

Differential measurements of two-particle correlations can shed additional light on the system evolution. A number of physics processes contribute to the final shape of a differential two-particle correlation function. The two-particle correlation measurements involve the counting of the number of correlated particle pairs at relative azimuthal and longitudinal separation, $\Delta\varphi$ and $\Delta\eta$, respectively. These measure-