Evolution of the high energy nucleus–nucleus collision is schematically depicted in

Figure 16. Two Lorentz-contracted pancakes of nuclear matter collide, thermalize, and form a deconfined QGP medium which expands, cools down, and hadronises to final state hadrons. Experimentally we do not observe each stage separately, but only through the time-integrated final state quantities—the momentum spectra of hadrons, photons, or leptons, particle multiplicities, energy flow, etc. Nevertheless, some time ordering of different processes giving rise to the final state observables exists. At very early collision times when colliding matter thermalizes, the entropy is produced which later—after (almost) isotropic expansion— transforms into particle multiplicities [

28,

181,

205]. Early collision times also favour production of high

${p}_{T}$ partons [

29,

305] or heavy quarks (

c, b) [

306]. The formation of QGP reveals itself in many ways, including radiation of low momentum direct or virtual photons serving as a thermometers, enhanced production of hadrons containing strange (

s) quarks [

21,

307,

308], and melting of

$c\overline{c}$ or

$b\overline{b}$ mesons [

115,

116,

306] called quarkonia. The subsequent rapid expansion of deconfined matter having more than ten times the degrees of freedom than the hadronic matter (see Equation (

22))—and therefore also much higher internal pressure—produces a strong radial flow which leaves its imprint on the spectra of final state particles and their yields [

28,

183,

205,

309,

310].

In the following, we present several examples of observables related to different stages of dynamics of nucleus–nucleus collisions at high-energies.

#### 5.1. Bulk Observables

Traditionally, the very first measurements of heavy-ion collisions at a new energy regime comprise the charged-particle density at midrapidity

$d{N}_{\mathrm{ch}}/d\eta $ ${|}_{\eta =0}$, also including its centrality dependence. Its collision-energy dependence for the 5% (6%) most central heavy-ion collisions—normalized per participant pair (i.e.,

$\u2329{N}_{\mathrm{part}}\u232a$/2)—is presented in

Figure 17 (left panel). The right panel of

Figure 17 shows that the normalized charged-particle density is rising with centrality, which means that the particle multiplicity at mid-rapidity increases faster than

${N}_{\mathrm{part}}$, presumably due to the contribution of hard processes to the particle production [

29]. However, this increase is very similar to that observed at the top RHIC energy.

One of the most celebrated predictions of the collective behaviour of matter created in non-central collisions of ultra-relativistic nuclei concerns its evolution in the transverse plane, which results from the pressure gradients due to spatial anisotropy of the initial density profile [

183,

314] (see

Figure 18). The azimuthal anisotropy is usually quantified by the Fourier coefficients [

315]:

where

ϕ is the azimuthal angle of the particle,

${\mathrm{\Psi}}_{n}$ is the angle of the initial state spatial plane of symmetry, and

n is the order of the harmonic. In a non-central heavy ion collision, the beam axis and the impact parameter define the reaction plane azimuth

${\mathrm{\Psi}}_{\mathrm{RP}}$. For a smooth matter distribution in the colliding nuclei, the plane of symmetry is the reaction plane

${\mathrm{\Psi}}_{n}={\mathrm{\Psi}}_{\mathrm{RP}}$, and the odd Fourier coefficients are zero by symmetry. However, due to fluctuations in the matter distribution (including contributions from fluctuations in the positions of the participating nucleons in the nuclei—see

Figure 11), the plane of symmetry fluctuates event-by-event around the reaction plane. This plane of symmetry is determined by the participating nucleons, and is therefore called the participant plane

${\mathrm{\Psi}}_{\mathrm{PP}}$ [

316]. Since the planes of symmetry

${\mathrm{\Psi}}_{n}$ are not known experimentally, the anisotropic flow coefficients are estimated from measured correlations between the observed particles [

299,

317].

In the following, we shall restrict ourselves to the properties of the Fourier coefficients

${v}_{n}$ with

$n=\phantom{\rule{3.33333pt}{0ex}}2$ and

$n=3$, which provide the dominant contributions to the observed azimuthal

elliptic and

triangular asymmetry, respectively. The sensitivity of

${v}_{2}$ to initial condition is illustrated on

Figure 15 (right panel), where the centrality dependence of the elliptic flow in Pb + Pb collisions at

$\sqrt{{s}_{NN}}$ =2.76 TeV is shown. For more details on the corresponding initial state models, see

Section 4.4.3.

The left panel of

Figure 19 shows the measured energy dependence of the integrated elliptic flow coefficient

${v}_{2}$ in one centrality bin. Starting from

$\sqrt{{s}_{NN}}$ ≈ 5 GeV, there is a continuous increase of

${v}_{2}$. Below this energy, two phenomena occur. At very low energies, due to the rotation of the compound system generated in the collision, the emission is in-plane (

${v}_{2}>0$). At the laboratory kinetic energy around 100 MeV/nucleon, the preferred emission turns into out-of-plane, and

${v}_{2}$ becomes negative. The slowly moving spectator matter prevents the in-plane emission of participating nucleons or produced pions, which appear to be sqeezed-out of the reaction zone [

318]. As the spectators move faster, their shadowing disappears, changing the pattern back to the in-plane emission.

Let us note that at RHIC, for the first time, the magnitude of the elliptic flow (

Figure 19) was found to be consistent with the EoS expected from the QGP [

16,

183]. The integrated value of

${v}_{2}$ for the produced particles increases by 70% from the top SPS energy to the top RHIC energy (see left panel of

Figure 19), and it appears to do so smoothly. In comparison to the elliptic flow measurements in Au + Au collisions at

$\sqrt{{s}_{NN}}=200$ GeV, at the LHC,

${v}_{2}$ increases by about 30% at

$\sqrt{{s}_{NN}}=2.76$ TeV. However, this increase is not seen in the differential elliptic flow of charged particles shown on the right panel of

Figure 19. Thus, the bulk medium produced at RHIC and LHC has similar properties, and the 30% increase of

${v}_{2}$ between the two energies is due to an enlarged available phase space, resulting in the same increase of the average transverse momentum of particles <

${p}_{T}$> between the RHIC and LHC energies.

As was first noted in Ref. [

314], at high energies, only the interactions among the constituents of matter formed in the initially spatially deformed overlap can produce

${v}_{2}>0$. A transfer of this spatial deformation into momentum space provides a unique signature for re-interactions in the fireball, and proves that the matter has undergone significant nontrivial dynamics between its creation and its freeze-out [

183]. The rapid degradation of the initial spatial deformation due to re-scattering causes the “self-quenching” of elliptic flow: if the elliptic flow does not develop early (when the collision fireball was still spatially deformed), it does not develop at all [

183]. In particular, the transformation of the rapidly expanding ideal gas of non-interacting quarks and gluons into strongly interacting hadrons is unable to produce a sufficient elliptic flow. The elliptic flow thus reflects the pressure due to re-scattering—the induced expansion and stiffness of the EoS during the earliest collision stages. Its continuous rise with the energy up to its highest value at the LHC indicates that the early pressure also increases.

The energy dependence of the integrated triangular flow coefficient

${v}_{3}^{2}\left\{2\right\}$ of charged hadrons is shown on the left panel of

Figure 20 in four bins of centrality, 0%–5%, 10%–20%, 30%–40%, and 50%–60%. As

${v}_{3}^{2}\left\{2\right\}$ is sensitive to the fluctuations in the initial matter distribution, it is interesting to observe that at

$\sqrt{{s}_{NN}}$ = 7.7 and 11.5 GeV, values of

${v}_{3}^{2}\left\{2\right\}$ for 50%–60% centrality become consistent with zero. For more central collisions, however,

${v}_{3}^{2}\left\{2\right\}$ is finite—even at the lowest energies—and changes very little from 7.7 GeV to 19.6 GeV. Above that, it begins to increase more quickly, and roughly linearly with

$log(\sqrt{{s}_{NN}})$. Generally, one would expect that higher energy collisions producing more particles should be more effective at converting the initial state geometry fluctuations into

${v}_{3}^{2}\left\{2\right\}$. Deviations from that expectation could indicate interesting physics, such as a softening of the EoS [

113,

114] discussed already in

Section 3.3. This can be investigated by scaling

${v}_{3}^{2}\left\{2\right\}$ by the charged particle rapidity density per participating NN pair,

${\mathrm{n}}_{\mathrm{ch},\mathrm{PP}}$ =

$d{N}_{\mathrm{ch}}/d\eta $/(0.5

${N}_{\mathrm{part}}$), see the left panel of

Figure 20. A local minimum of

${v}_{3}^{2}\left\{2\right\}$/

${\mathrm{n}}_{\mathrm{ch},\mathrm{PP}}$ in the region near 15–20 GeV observed in the centrality range 0%–50% and absent in the more peripheral events could indicate an interesting trend in the pressure developed inside the system.

As was already briefly mentioned in

Section 4.4.4, recent years have witnessed a growth of interest in studying collective phenomena on event-by-event basis. In addition to the key publications [

165,

166,

205,

206,

207,

208,

300,

301,

302,

303], it is worth mentioning the study [

320] where the anisotropic flow coefficients

${v}_{1}$–

${v}_{5}$ were computed by combining the IP-Glasma flow with the subsequent relativistic viscous hydrodynamic evolution of matter through the quark–gluon plasma and hadron gas phases. The event-by-event geometric fluctuations in nucleon positions and intrinsic color charge fluctuations at the sub-nucleon scale are expected to result in experimentally measurable event-by-event anisotropic flow coefficients [

321]. Let us note that fluctuating initial profiles observed in over-many-events integrated triangular and higher odd flow coefficients are also revealed in difference between various participant plane angles

${\mathrm{\Psi}}_{n}$ introduced in Equation (

66). Correlations between the event plane angles

${\mathrm{\Psi}}_{n}$ of different harmonic order can not only yield valuable additional insights into the initial conditions [

322], but are also experimentally measurable [

323]. The same is also true for the correlations between different flow harmonics [

324,

325,

326,

327].

Enhanced production of hadrons with the quantum numbers not present in colliding matter is one of the oldest signals of the deconfined QGP medium [

307,

308]. Measurements of the yields of multistrange baryons were carried out at CERN SPS by WA85, and later on by WA97/NA57 collaborations since the mid-eighties. After 2000, more data came from RHIC, and starting from 2010 also from the LHC. The current status is summarized in the five panels of

Figure 21. In the top left and middle panels (

Figure 21a,b), a compilation of the results from SPS, RHIC, and the LHC in terms of

strangeness enhancement defined as normalized (to

p +

p or

p + Be) yield per participants is presented. On the top right (

Figure 21c), the hyperon-to-pion ratios as functions of

$\u2329{N}_{\mathrm{part}}\u232a$ for Pb + Pb, Au + Au, and

p +

p collisions at the LHC and RHIC energies are displayed. The normalized yields are larger than unity for all the particles, and increase with their strangeness content. This behaviour is consistent with the picture of enhanced

$s\overline{s}$ pair production in a hot and dense QGP medium [

307,

308].

The two bottom plots represent a comparison between the hyperon-to-pion ratios from

p +

p,

p + Pb, and Pb + Pb collisions. Interestingly, the ratios in

p + Pb collisions increase with multiplicity from the values measured in

p +

p to those observed in Pb + Pb. The rate of increase is more pronounced for particles with higher strangeness content. Let us note that the Grand canonical statistical description of Pb + Pb data shown as full and dashed lines in

Figure 21 may not be appropriate in small multiplicity environments such as those produced in the

p + Pb case. It appears that for the latter case, the evolution of hyperon-to-pion ratios with the event multiplicity is qualitatively well described by the Strangeness Canonical model implemented in THERMUS 2.3 [

328]. In this case, a local conservation law is applied to the strangeness quantum number within a correlation volume

${V}_{c}$, while treating the baryon and charge quantum numbers grand-canonically within the whole fireball volume

V [

330].

#### 5.2. Hard Probes

Heavy quarks, quarkonia, and jets—commonly referred to as

hard probes—are created in the first moments after the collision, and are therefore considered as key probes of the deconfined QCD medium. Production of these high transverse momentum (

${p}_{T}\gg {\mathrm{\Lambda}}_{QCD}$) objects occurs over very short time scales (

τ ≈

$1/{p}_{\mathrm{T}}$ ≈ 0.1

$fm/c$), and can thus probe the evolution of the medium. Since the production cross-sections of these energetic particles are calculable using pQCD, they have been long recognised as particularly useful “tomographic” probes of the QGP [

331,

332,

333].

Let us start our discussion with the results on the inclusive production of high-

${p}_{T}$ hadrons. The latter are interesting on their own because it was there where for the first time the suppression pattern was observed [

14,

15,

16,

17]. In an inclusive regime, the comparison between

${d}^{2}N$/

$d{p}_{T}d\eta $ (the differential yield of high-

${p}_{T}$ hadrons or jets per event in A+B collisions) to that in

p +

p collisions is usually quantified by introducing the nuclear modification factor

For collisions of two nuclei behaving as a simple superposition of

${N}_{\mathrm{coll}}$ nucleon–nucleon collisions, the nuclear modification factor would be

${R}_{AB}$ = 1. The data of

Figure 22 reveal a very different behaviour. The left panel shows a compilation of

${R}_{AA}$ from Au + Au and Pb + Pb collisions, the right panel the result of

${R}_{pPb}$ from three LHC experiments at the same energy

$\sqrt{{s}_{NN}}$ = 5.02 TeV. In the

${R}_{AA}$ case, the suppression pattern of high-

${p}_{T}$ (>2–3 GeV/c) hadrons in the deconfined medium—predicted many years ago [

331,

332,

333] as a

jet quenching effect—is clearly visible at RHIC and the LHC. However, for proton–nucleus collisions (

Figure 22, right panel), no suppression is seen, even at the highest LHC energy. Moreover,

${R}_{AA}$ in the 5% most central Pb + Pb collisions at the LHC shows a maximal suppression by a factor of 7–8 in the

${p}_{T}$ region of 6–9 GeV. This dip is followed by an increase, which continues up to the highest

${p}_{T}$ measured at

$\sqrt{{s}_{NN}}$ = 5.02 TeV, and approaches unity in the vicinity of

${p}_{T}$ = 200 GeV [

334].

#### 5.2.1. High-${p}_{T}$ Hadrons and Jets

The suppression of high-

${p}_{T}$ hadrons in the deconfined medium was thoroughly studied at RHIC using azimuthal correlations between the trigger particle and associated particle; see

Figure 23. Near-side peaks in central (0%–5%) Au + Au collisions present in all panels of

Figure 23 (left) indicate that the correlation is dominated by jet fragmentation. An away-side peak emerges as

${p}_{T}^{\mathrm{trig}}$ is increased. The narrow back-to-back peaks are indicative of the azimuthally back-to-back nature of dijets observed in an elementary parton–parton collision. Contrary to the latter, the transverse-momentum imbalance of particles from the jet fragmentation due to different path lengths of two hard partons in the medium is apparent. The azimuthal angle difference

$\Delta \varphi $ for the highest

${p}_{T}^{\mathrm{trig}}$ range (8 <

${p}_{T}^{\mathrm{trig}}$ < 15 GeV/c) for mid-central (20%–40%) and central Au+Au collisions—as well as for d+Au collisions—is presented in

Figure 23 (right panel). We observe narrow correlation peaks in all three

${p}_{T}^{\mathrm{assoc}}$ ranges. For each

${p}_{T}^{\mathrm{assoc}}$, the nearside peak shows a similar correlation strength above background for the three systems, while the away-side correlation strength decreases from

d + Au to central Au + Au. For the

d + Au case, the yield of particles on the opposite side

$\Delta \varphi $ =

π prevails over the same side. Moreover, for Au + Au collisions, the nearside yields obtained after subtraction of the background contribution due to the elliptic flow show a little centrality dependence, while the away-side yields decrease with increasing centrality [

335].

Unfortunately, the advantage of the large yield of dijets is offset by a loss of information about the initial properties of the probes (i.e., prior to their interactions with the medium). Correlating two probes that both undergo an energy loss also induces a selection bias towards scatterings occurring at—and oriented tangential to—the surface of the medium. It is thus interesting to study correlations when one of the particles does not interact strongly with the medium. Triggering on the high-

${p}_{T}$ isolated photon (i.e., not from

${\pi}^{0}\to 2\gamma $ decays) would do the job. While in

p +

p collisions an emerging quark jet should balance its transverse momentum with the photon, in the heavy-ion collisions, much of its momentum is thermalized while the quark traverses the plasma. This is illustrated in

Figure 24 (left panel), where a single hard photon with

${p}_{T}$ = 402 GeV emerges unhindered from the de-confined medium produced in Pb+Pb collisions at the LHC. The accompanying quark jet produced via the QCD Compton scattering

$qg\to q\gamma $ loses 1/3 of its energy (≈140 GeV!) inside the hot and dense matter.

The measurement presented on the middle and right panels of

Figure 24 shows that for more central Pb + Pb collisions, a significant decrease in the ratio of jet transverse momentum to photon transverse momentum—

$\u2329{x}_{\mathrm{J}\gamma}\u232a$—relative to the PYTHIA reference [

200] is observed. Furthermore, significantly more photons with

${p}_{T}$ > 60 GeV/c in Pb + Pb are observed to not have an associated jet with

${p}_{T}$ > 30 GeV/c jet, compared to the reference. However, no significant broadening of the photon + jet azimuthal correlation has been observed.

An important progress in the theoretical understanding of the suppression of energetic partons traversing a deconfined matter was the introduction of the diffusion coefficient

$\widehat{q}$ relevant for the transverse momentum broadening and collisional energy loss of partons (jets) [

338]. This quantity, which is commonly referred to as the jet quenching parameter, can be determined either via weak coupling techniques [

339,

340,

341], a combination of lattice simulations and dimensionally-reduced effective theory [

342], or from the gauge/gravity duality [

343]. Typical estimates for this quantity at RHIC and LHC energies range between 5 and 10 GeV

${}^{2}$/fm, demonstrating the currently still sizable uncertainties in these calculations.

#### 5.2.2. Quarkonia

Melting of the quarkonia—bound states of heavy quark and anti-quark

$q\overline{q}$ where

q =

$c,b$—due to a colour screening in the deconfined hot and dense medium was proposed thirty years ago as a clear and unambiguous signature of the deconfinement [

115]. However, shortly after that, it was noticed that not only diffusion of the heavy quarks from melted quarkonium, but also the drag which charm quarks experience when propagating through the plasma is important [

344].

The latter might lead to an enhancement instead of a suppression. This is in variance with the original proposal that the heavy quarks, once screened, simply fly apart. With the advent of the strongly-interacting QGP, the Langevin equation model of quarkonium production was formulated, where the charm quark–antiquark pairs evolve on top of a hydrodynamically expanding fireball [

345]. A heavy quark and anti-quark interact with each other according to the screened Cornell potential and interact, independently, with the surrounding medium, experiencing both drag and rapidly decorrelating random forces. An extension of this approach to bottomonium production [

346] shows that a large fraction of

$b\overline{b}$ pairs that were located sufficiently close together during the initial hard production will remain correlated in the hot medium for a typical lifetime of the system created in heavy-ion collisions. The distribution of the correlated

$b\overline{b}$ pair in relative distance is such that it will dominantly form 1S bottomonium. A study of quarkonia production in heavy-ion collisions thus provides an interesting window not only into static, but also into dynamical properties of the hot, dense, and rapidly expanding medium [

116,

306].

On the left panels of

Figure 25, the invariant-mass distributions of

${\mu}^{+}{\mu}^{-}$ pairs (di-muons) produced in the

p +

p (a) and Pb + Pb (b) collisions at the LHC are presented. A prominent peak due to the production of the heavy quarkonium state, the bottomonium

$\mathrm{{\rm Y}}(1\mathrm{S})$, can be clearly seen in both

p +

p and Pb + Pb data. Peaks from the higher excited states of Υ,

$\mathrm{{\rm Y}}(2\mathrm{S})$, and

$\mathrm{{\rm Y}}(3\mathrm{S})$, although discernible in the

p +

p case, are barely visible in the Pb + Pb data. More quantitative information on this effect can be found in the right panels of

Figure 25, where the centrality dependence of the double ratio

${\left[\mathrm{{\rm Y}}(2\mathrm{S})/\mathrm{{\rm Y}}(1\mathrm{S})\right]}_{\mathrm{PbPb}}/{\left[\mathrm{{\rm Y}}(2\mathrm{S})/\mathrm{{\rm Y}}(1\mathrm{S})\right]}_{\mathrm{pp}}$ (top) and of the nuclear modification factors

${R}_{AA}$ of

$\mathrm{{\rm Y}}(1\mathrm{S})$ and

$\mathrm{{\rm Y}}(2\mathrm{S})$ (bottom) are displayed. Let us note that the observed suppression of the relative yield is in agreement with the expectations that different quarkonium states will dissociate at different temperatures with a suppression pattern ordered sequentially with the binding energy; i.e., the difference between the mass of a given quarkonium and twice the mass of the lightest meson containing the corresponding heavy quark [

347]. Moreover, the observed pattern is now also confirmed in Pb + Pb collisions at

$\sqrt{{s}_{NN}}$ = 5.02 TeV [

348]. The double ratio is significantly below unity at all centralities, and no variation with kinematics is observed, confirming a strong Υ suppression in heavy-ion collisions at the LHC.