<figure><img src="image_35.png" /><figcaption>**Fig. 29** Non-linear flow mode coefficients $\chi_{4,22}$ (left) and $\chi_{5,23}$ (right) in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV compared to various hydrodynamic calculations [288]</figcaption></figure>

ments are primarily affected by intra-jet, resonance decays, quantum-statistics correlations, and radial flow. In addition, correlations induced earlier in the evolution of the system will show longer range $\Delta\eta$ correlations compared to those induced later. The fact that particles are created in pairs due to local charge conservation introduces additional charge-dependent effects into these correlations. The balance function observable, a two-particle charge dependent correlator, which studies the distribution of balancing charges in momentum space, has been suggested to be sensitive to all these effects [328–330]. Measurements of the balance function at RHIC [331] and SPS [332,333] energies showed a longitudinal narrowing when going from peripheral to central collisions, while other results from RHIC [334,335] also revealed a similar trend on the azimuthal dimension.

The top panel of Fig. 30 presents the balance function widths for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV from ALICE [335]. They are compared to predictions from the TRENTo + VISHNU model (initially shown in Fig. 24). In particular, the $\langle\Delta\phi\rangle$ measurements show some differences between the data and the predictions. Since other observables sensitive to radial flow have been well described by this model earlier in this section, these differences in $\langle\Delta\phi\rangle$ might arise from charge-diffusion effects in the QGP, which are another QGP transport property. These effects are not implemented in this model, but were independently studied elsewhere regarding the electric diffusivity of light

quarks [336,337]. These differences are also observed for the $\langle\Delta\eta\rangle$ measurements, where charge diffusion effects are expected to be present. The differences in both cases between the data and hydrodynamic model calculations therefore might place constraints on this specific transport property of the QGP.

The charge independent (CI) differential two-particle transverse momentum correlator $G_2^{\text{CI}}$ was constructed to be sensitive to the diffusion of transverse momentum currents [338,339]. Under the assumption that more central collisions generate the QGP that lasts for a longer time, the longitudinal width of $G_2^{\text{CI}}$ can provide a measure of viscous diffusion as the effects of diffusion become more pronounced. The proposed way of extracting $\eta/s$ under this assumption contrasts with the methods shown previously, involving comparisons to results from hydrodynamic models which strongly rely on the correct modelling of the initial conditions. Measurements of $G_2^{\text{CI}}$ by the STAR collaboration [340] have shown a longitudinal broadening for more central collisions, which are in agreement with the expected viscous effects inferred from a value of $\eta/s$ close to the AdS/CFT limit.

The measurements of $G_2$ were performed in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV for charge independent (CI) and charge dependent (CD) hadron combinations [341]. The CI results are shown in the bottom panel of Fig. 30. The azimuthal behaviour of $G_2^{\text{CI}}$ shows a narrowing trend, while the longitudinal evolution conversely demonstrates a
<figure><img src="image_36.png" /></figure>

<figure><img src="image_37.png" /><figcaption>**Fig. 30** (Top) Evolution of longitudinal and azimuthal widths as a function of collision centrality for the balance function $B^{+−}$. (Bottom) The two-particle transverse momentum correlator $G_2^{CI}$ as a function of centrality. ALICE data [335] are compared to calculations from the $T_{RENTO} + VISHNU$ model chain [49]</figcaption></figure>

clear broadening towards central collisions. Extracting the expected widths of the correlator for the most central collisions for different values of $\eta/s$ and interpreting the longitudinal broadening of $G_2^{CI}$ as originating exclusively from viscous effects, suggests that the measured longitudinal width is compatible with a value of $\eta/s$ close to the AdS/CFT limit. These features are not reproduced by the HIJING and AMPT models (not shown). The bottom panel of Fig. 30 also shows that the longitudinal width is compatible with the $T_{RENTO} + VISHNU$ model, which uses an $\eta/s$ close to the AdS/CFT limit, and therefore should incorporate the aforementioned viscous effects. On the other hand, the same model cannot reproduce the azimuthal width, which requires further investigation.

### 2.2.8 Polarisation of hyperons and vector mesons

The spin-orbital angular momentum interaction is one of the most well-known effects in nuclear, atomic, and condensed

matter physics. In non-central heavy-ion collisions, a large initial angular momentum ($O(10^7) \hbar$) is expected to be created perpendicular to the reaction plane. Due to any possible spin-orbit coupling, particles produced in such a collision can become globally polarised [342–345]. Global polarisation measurements of produced particles provide important information about the initial conditions and dynamics of the QGP, as well as the hadronisation processes [346–348]. Considering the importance of initial conditions, it is imperative that these are determined using other measurements. The three-dimensional spatial profile of the initial conditions and its effect on the space-time evolution of a heavy-ion collision can be further probed by the directed flow, $v_1$, of hadrons containing light (u, d, and s) and heavy (c) quarks. The charge dependence of $v_1$, as well as the different magnitude of global polarisation of particles and anti-particles, is sensitive to the effects of the strong magnetic fields caused by colliding nuclei and will be discussed in Sect. 2.6.

The collision symmetry requires that the directed flow is an anti-symmetric function of pseudorapidity, $v_1^{\text{odd}}(\eta) = -v_1^{\text{odd}}(-\eta)$. Due to event-by-event fluctuations in the initial energy density of the collision, the participant plane angle defined by the dipole asymmetry of the initial energy density [269,349] and that of projectile and target spectators, are different from the geometrical reaction plane angle $\Psi_{\text{RP}}$. As a consequence, the directed flow can develop a rapidity-symmetric component, $v_1^{\text{even}}(\eta) = v_1^{\text{even}}(-\eta)$, which does not vanish at midrapidity [269,349–351]. Results for rapidity-even $v_1$ of charged particles at midrapidity in Pb–Pb collisions at $\sqrt{s_{\text{NN}}} = 2.76$ TeV [352] show strong evidence of dipole-like initial density fluctuations in the overlap zone of the nuclei. Similar trends in the rapidity-even $v_1$ and the estimate from two-particle correlation measurements at midrapidity [268,321,353,354] indicate a weak correlation between fluctuating participant and spectator symmetry planes. The observed negative slope of the rapidity-odd $v_1$, with approximately a 3 times smaller magnitude than found at the highest RHIC energy [355], suggests a smaller longitudinal tilt of the initial system and disfavors the strong fireball rotation predicted for the LHC energies [356–358].

Most of the recent calculations of the global polarisation assume complete thermal equilibrium and the validity of the hydrodynamic description of the system [347,362–365]. They relate polarisation of a charged particle to the thermal vorticity of the system at the hadronisation time. The global polarisation is determined by the average vorticity component along the orbital momentum of the system and is perpendicular to the collision reaction plane. Both the magnitude and the direction of the vorticity can strongly vary within the system [254]. In particular, a significant component along the beam direction can be acquired due to the transverse anisotropic flow [254,255]. The vorticity of the system can be linked with $v_1$ through the asymmetries in the initial veloc-
<figure><img src="image_38.png" /><figcaption>**Fig. 31 (Top)** The global hyperon polarisation ($P_H$) as a function of collision energy compared with the STAR data at lower energies [359,360]. The insert shows zoomed-in comparison with the data at the top RHIC energy. The systematic uncertainties are shown as shaded boxes. Points are slightly shifted along the horizontal axis for better visibility. (Bottom) Centrality dependence of the hyperon polarisation component along the beam direction $P_z$ measured in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV compared with STAR data and two hydrodynamic calculations [361]</figcaption></figure>

ity fields. Hydrodynamic simulations show that the orbital angular momentum stored in the system and the directed flow of charged particles are almost directly proportional to each other [346]. This allows for an empirical estimate of the collision energy dependence of the global polarisation [254]. The STAR results for the directed flow [366,367] and the hyperon global polarisation [359,360] show that the slopes of $v_1$ at midrapidity ($d\nu_1/d\eta$) for charged hadrons (pions) and the hyperon polarisation are indeed strongly correlated.

The global polarisation of the $\Lambda$ and $\bar{\Lambda}$ hyperons was measured in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV [368]. The results are reported differentially as a function of collision centrality and transverse momentum of the hyperon for the 5–50% centrality interval, and the dependence on collision energy is shown in the top panel of Fig. 31. The average hyperon global polarisation for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 and 5.02 TeV is found to be consistent with zero, $\langle P_H \rangle (\%) \approx -0.01 \pm 0.05$ (stat.) $\pm 0.03$ (syst.) in the

15–50% centrality interval. The results are compatible with expectations based on an extrapolation from measurements at lower collision energies at RHIC [352], hydrodynamic model calculations, and empirical estimates based on collision energy dependence of directed flow [254]. All these calculations predict the global polarisation values at the LHC energies of the order of 0.01%.

In addition to the global hyperon polarisation, a polarisation of $\Lambda$ and $\bar{\Lambda}$ along the beam direction was predicted in non-central heavy-ion collisions due to the strong elliptic flow [254,255]. The bottom panel of Fig. 31 presents the polarisation of hyperons in the z-direction of Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [361]. The polarisation is non zero, thus exhibits a clear second harmonic sine modulation as expected due to elliptic flow, and is of similar magnitude as the one measured at RHIC [369]. Comparisons with two different hydrodynamic calculations [370] are also shown. The strange quark scheme assumes that the hyperon polarisation
is inherited from the strange quark prior to hadronisation, while in the other approach the polarisation is calculated for the $\Lambda$ and $\bar{\Lambda}$ at the freeze-out using the hyperon mass for the mass of the spin carrier. The data favour the strange quark scheme, both quantitatively and qualitatively, which perhaps demonstrates that these measurements are sensitive to the z-polarisation of strange quarks in the QGP phase.

Another consequence of quarks being polarised due to the spin-orbital angular momentum interactions leads to a preferential alignment of the intrinsic angular momentum (spin) of vector mesons. These are formed by hadronisation of quarks, along the direction of global angular momentum [342–344,371]. The spin alignment of a vector meson is quantified by measuring the spin density matrix element $\rho_{00}$ which is the probability of finding a vector meson in spin state 0 out of 3 possible spin states (–1, 0, 1). In the absence of spin alignment all 3 spin states are equally probable which makes $\rho_{00} = 1/3$. The present measurements of the $\rho_{00}$ values for $K^{*0}$ mesons are shown in Fig. 32. They are found to deviate from 1/3 at low-$p_T$ in midcentral Pb–Pb collisions, whereas at high-$p_T$ the $\rho_{00}$ values are consistent with 1/3. Various systematic tests such as $\rho_{00}$ measurements for $K_S^0$ (meson with spin zero) in Pb–Pb and $\rho_{00}$ measurements for $K^{*0}$ mesons in pp collisions (see the left panel of Fig. 32) yield $\rho_{00} = 1/3$, indicating no spin alignment as expected. The observed $p_T$ dependence of $\rho_{00}$ is qualitatively consistent with the expectation from the hadronisation of polarised quarks via a recombination mechanism in the presence of initial angular momentum. At low-$p_T$, the maximum deviation (at a $3\sigma$ level) of $\rho_{00}$ from 1/3 occurs in midcentral collisions, whereas in central and peripheral collisions, the measurements are consistent with 1/3 (see the right panel of Fig. 32). The centrality dependence of $\rho_{00}$ is consistent with the impact-parameter dependence of initial angular momentum [373]. The expected order of magnitude of $\rho_{00}$ can be estimated from the previously shown polarisation measurements of $\Lambda$ hyperons. In the quark recombination model [342,344,371], the $\Lambda$ polarisation ($P_\Lambda$) is equivalent to strange quark polarisation. Considering the same quark polarisation ($P_q$) for a light and strange quark the $\rho_{00}$ of vector mesons are related to quark polarisation as $\rho_{00} = \frac{1-P_q^2}{3+P_q^2}$. In a non-relativistic thermal approach [363], the vorticity ($\omega$) and temperature ($T$) are related to $P_\Lambda$ and $\rho_{00}$ as $P_\Lambda = \frac{\omega}{4T}$ and $\rho_{00} \approx \frac{1}{3} \left(1 - \frac{(\omega/T)^2}{3}\right)$. Substituting the inputs from the $\Lambda$ hyperon measurements shown previously, the estimated $\rho_{00}$ value from these models are found to be close to 1/3. Therefore, the large effect observed for the $\rho_{00}$ of vector mesons requires further theoretical input.

Recently, spin alignment studies were extended to the charmonium sector, by performing measurements on the angular distribution of decay muons of the $J/\psi$, in the rapid-

ity interval $2.5 < y < 4$ [374]. Results were obtained for Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV, for three transverse momentum intervals in $2 < p_T < 10$ GeV/c and as a function of centrality, in $2 < p_T < 6$ GeV/c. The $\lambda_\theta \propto 1 + \cos^2 \theta$ quantity was studied, with $\theta$ being the polar angle emission of the positive decay muon, in the $J/\psi$ rest frame. The $\theta$ angle was measured with respect to an axis perpendicular to the event plane. It can be shown that the finite spin-alignment condition $\rho_{00} \neq 1/3$ is equivalent to having $\lambda_\theta \neq 0$. As a function of centrality $\lambda_\theta$ deviates from zero with a $3.5\sigma$ significance in the 40–60% centrality interval. For the measurement as a function of $p_T$, the significance reaches $3.9\sigma$ for $2 < p_T < 4$ GeV/c (30–50% centrality), where $\lambda_\theta = 0.23 \pm 0.05$ (stat.) $\pm 0.03$ (syst.). Having $\lambda_\theta > 0$ implies a partially transverse polarisation for the $J/\psi$. For a decay involving spin-1 particles, such a positive value also implies $\rho_{00} < 1/3$, i.e. an effect similar to that observed for the $K^{*0}$ and ascribed to vorticity effects. However, the $J/\psi$ production process in nuclear collisions at the LHC is different from that of $K^{*0}$ (and $\phi$), with the charm–anticharm pair being created very early in the collision process and the bound $c\bar{c}$ state being produced both in the QGP phase and at hadronisation. Due to the earlier production, charmonium may, in addition to vorticity, be more sensitive than strange hadrons to the large magnetic field present in the first instants of the Pb–Pb collision. Quantitative theory estimates, still not available, will be necessary to elucidate this possibility.

### 2.2.9 Conclusions

**Global collectivity and kinetic freeze-out temperatures.** The highest ever values of anisotropic and radial flow in heavy-ion collisions are achieved at the LHC. The radial-flow velocities derived from $p_T$ spectra are up to about 70% of the speed of light (in the lab frame), while light hadron $v_2$ measurements, which determine the magnitude of elliptic flow, are 30% higher than at the top RHIC energy. The kinetic freeze-out temperature extracted from Blast-Wave parameterisations depends strongly on whether resonances are included for feed-down. The differences are on the order of 60 MeV for central collisions, yielding the range $90 < T_{kin} < 150$ MeV. The extracted radial flow velocities are much less sensitive to such an inclusion.

**Emergent features of the dynamical evolution of the QGP.** New observables measured (and in some cases developed) by ALICE over the last decade such as finite higher-order anisotropies $v_{n \ge 3}$, correlations between different-order anisotropies, and a translation of the angular momentum of the QGP to the polarisation of its outgoing hadrons, reveal an extremely rich pattern of the dynamical evolution of the QGP. Higher-order anisotropies were measured up to the ninth order, and show an exponential decrease, which can
<figure><img src="image_39.png" /><figcaption>**Fig. 32** (Left) $\rho_{00}$ as a function of $p_T$ for K\*0 in Pb–Pb and pp collisions, and for K\*0 in Pb–Pb collisions. (Right) Centrality dependence of $\rho_{00}$ for K\*0 mesons in Pb–Pb collisions. From Ref. [372]</figcaption></figure>

be understood in the context of a finite shear viscosity over entropy ratio $\eta/s$. It was also demonstrated how higher-order anisotropies can be understood from lower contributions, with such a decomposition playing an important role in accessing the properties of the QGP just before hadronisation. The observations of correlations between different-order anisotropies and hadron polarisation offer unique constraints on the initial state, on the QGP evolution, and on the hadronisation mechanisms.

**Hydrodynamic description.** Hydrodynamic calculations, involving the QGP equation of state, describe a wide variety of these results. Such results include measurements sensitive to anisotropic and radial flow, the energy dependence of $\Lambda$ global polarisation values, and $\Lambda$ longitudinal polarisation measurements at the LHC. The assumed dependence of $\eta/s$ and $\zeta/s$ on temperature used in these calculations for such a description will be further discussed in Sect. 2.7. In a number of cases, the original assertion assumes the modeling of the initial state is sufficiently accurate and will be addressed in Sect. 4. On the other hand, the same hydrodynamic calculations cannot describe balance-function widths, which might be explained by charge diffusion effects not implemented in these calculations. The differences with respect to the data can therefore provide a crude measure of charge diffusion, a key QGP transport parameter, notwithstanding other correlation mechanisms not present in hydrodynamic models (e.g. mini-jets).

**Global polarisation and spin alignment.** Measurements of the hyperon global polarisation at the LHC are consistent with zero, and compatible with expectations based on an extrapolation from measurements at lower collision energies at RHIC, hydrodynamic model calculations, and empirical estimates based on collision energy dependence of directed flow. On the other hand, the K\*0 spin alignment measurements show a potential sensitivity to the global angular momentum of the system, not seen in the polarisation measurements. These differences remain a challenge to interpret,

and are in need of further theoretical attention. The significant transverse polarisation of the $J/\psi$ with respect to the event plane may provide insight on the relative contributions of the global angular momentum and of the initial magnetic field.

## 2.3 Hadronisation of the QGP

While the QGP formed from a heavy-ion collision, with a finite energy, pursues its explosive expansion, the energy density of the medium reaches the pseudo-critical level of the transition (≈ 0.5–1 GeV/fm³ according to lattice-QCD calculations [169,375]). The hadronisation of the plasma, i.e. the process of hadron formation from the previous partonic phase, starts and makes the transition from the deconfined medium to hadronic matter; such a process leads to a system which is still collective but has different degrees of freedom: a hadron gas. The hadronisation, in which the confinement phenomenon of QCD sets in, involves quark and gluon processes characterised by small momentum transfers and hence large values of the strong coupling constant, such that a perturbative approach is not applicable. Thus, one must resort to phenomenological models, such as the statistical hadronisation models and the recombination models presented in this section. These models describe successfully several properties of the final-state hadrons produced in heavy-ion collisions, suggesting that they can capture some key features of the process of hadron formation [376]. After the hadronisation, which at LHC energies occurs 7–10 fm/c (see Sect. 2.1.5) after the initial collision, the created hadrons can still interact via inelastic processes, implying that the overall chemical composition can evolve further. Such interactions live on, until the temperature of the chemical freeze-out is reached, i.e. until the moment at which the hadronic species and their respective populations become settled. If the most vigorous inelastic interactions have markedly ceased by then, the mildest inelastic ones can continue, affecting at most the resonance population. In parallel, though, significant elastic collisions between hadrons still occur, alter-
ing the momentum distributions of any given species. Such momentum transfers persist down to the kinetic freeze-out, occurring at a typical time of ≳ 10 fm/c [239]. After this ultimate freeze-out, the system vanishes into free-streaming hadrons that propagate towards the detection apparatus.

The late stages of the collision that are presented here come along with several questions. To illustrate the current spurs, a few typical questions can be underlined: (i) what is the level of chemical equilibration and/or kinetic thermalisation achieved by the hadron populations, be it for the light-flavour or, conversely, for the heavy-flavour species?, (ii) to what extent is the temperature for the chemical freeze-out, $T_{\text{chem}}$, a unique temperature, shared by any hadron species?, (iii) what is the level of interplay among the quark flavours at the hadronisation stage?, (iv) what is the level of interplay between the momentum domains connecting the soft and hard sectors?, (v) similarly to $T_{\text{chem}}$, can we assume a univocal value of $T_{\text{kin}}$, the temperature of a sudden kinetic freeze-out? and (vi) how long does the hadronic phase last?

In concrete terms, the properties of both aspects, the hadronisation itself as well as the resulting medium, can be studied in high-energy nuclear collisions utilising a set of observables essentially related to the production of the different hadron species. To that end, the total multiplicity per rapidity unit $dN_{\text{ch}}/dy$ together with the differential distributions (in momentum, $d^2N/dp_T dy$ and/or in azimuth, $d^3N/dp_T d\phi dy$) can be analysed. This will be discussed further in the following subsections.

### 2.3.1 Hadron gas composition and QCD thermodynamics

The measurement of the multiplicity per rapidity unit $dN_{\text{ch}}/dy$ of different hadron species provides access to the chemical composition of the hadron gas at the stage where the aforementioned inelastic collisions cease and the abundances of the different hadron species are frozen (except for resonance decays). QCD thermodynamics in the confined phase, below the pseudo-critical temperature, can be well approximated as an ideal hadron-resonance gas (HRG) of all known hadrons and resonances [377, 378]. The relative abundances of particle species measured in heavy-ion collisions over a broad centre-of-mass energy range were observed to closely follow the equilibrium populations of a HRG [122]. This is very suggestive of a thermal origin of particle production, which implies that the properties of the fireball are governed by statistical QCD, and allows for the calculation of the chemical composition of the system within the framework of the statistical hadronisation approach. This approach is based on the partition function of the HRG and on the assumption that the hadro-chemical composition of the fireball is frozen out on a hypersurface defined by a uniform temperature [122, 379–382]. This allows for the determination of thermodynamic parameters characterising the chem-

ical freeze-out, which are relevant for the QCD phase diagram. In particular, such freeze-out can be characterised by only three macroscopic parameters: the temperature $T_{\text{chem}}$ and the volume $V$ of the fireball, together with the baryochemical potential $\mu_B$, which guarantees the conservation on average of the baryon number in the grand-canonical formulation of the partition function (other chemical potentials are fixed by the conservation laws). Notably, the statistical hadronisation model with a canonical formulation, incorporating local quantum number conservation, can also describe the measured yields of hadron species in pp and $e^+e^-$ collisions [383–385], albeit with a worse $\chi^2/ndf$ compared to heavy-ion reactions. In small collision systems, a fireball in thermal and chemical equilibrium is not formed; for instance, in $e^+e^-$ annihilation, quark-antiquark pairs are produced and subsequently fragment into jets of hadrons. The observation of statistical features in these cases suggests that the underlying hadronisation process (i.e. the way an excited system populates the hadronic states) follows for a good part statistical laws, which are governed by entropy maximization and determined by phase space dominance. Moreover, the temperatures resulting from statistical model fits to pp and $e^+e^-$ data (of about 175–180 MeV [383, 384]) exceed the pseudocritical temperature from lattice QCD calculations, indicating that this case is different from the hadronisation of a QGP at the crossover boundary.

**Light-flavour hadron yields and chemical freeze-out properties.** The multiplicities of hadron species containing only light (u, d and s) quarks measured at midrapidity in central (0–10%) Pb–Pb collisions at $\sqrt{s_{\text{NN}}}$ = 2.76 TeV are shown in Fig. 33 for non-strange and strange mesons ($\pi^\pm$ [273], $K^\pm$ [273], $K_S^0$ [274], $\phi(1020)$ [275]) and baryons (p [273], $\Lambda$ [274]) including multi-strange hyperons ($\Xi^\pm$ [386], $\Omega^\pm$ [386]), as well as light nuclei (d, $^3$He [387] and $^4$He [388])<fnref n="6" /> and $^3\Lambda$ hypernuclei [389] and their antiparticles. The particle yields at midrapidity, integrated over the full $p_T$ range, were obtained from their measured $p_T$-differential distributions, which were fitted individually with a blast-wave function [276], used for the extrapolation to zero $p_T$ (except for $^4$He and $^3\Lambda$ H for which the blast-wave parameters from the combined fit to d and $^3$He were used). The yields were then calculated by adding to the integral of the data in the measured $p_T$ region, the integral of the fit function outside that region. The fraction of extrapolated yield is small for most hadron species (ranging from 4% for protons to 20% for $\Xi^\pm$) and it is larger for $^3\Lambda$ H (about 28%) and $\Omega^\pm$ (about 50%). Particle and antiparticle measurements are reported separately wherever possible to underline that matter and antimatter are

<footnote n="6">⁶ The yields of antinuclei $\bar{d}$ and $^3\bar{He}$ have been derived based on published data [387].</footnote>
observed to be produced in equal amounts within experimen-
tal uncertainties at LHC energies [390].

The measured multiplicities are well described by sta-
tistical hadronisation models (SHM), as shown in Fig. 33,
where the data are compared to the results of four different
implementations of such models, namely THERMUS [391,
391,392], SHARE [393–396], Thermal-FIST [397,398], and
GSI-Heidelberg [72,120,399]. The SHM describes the yields
of all the measured species over nine orders of magnitude
in abundance values. Among the parameters, the value for
the baryochemical potential has been fixed to μ<sub>B</sub> ≈ 0 in
most models, given the almost equal abundances of parti-
cles and antiparticles at LHC energies; as a consequence,
the corresponding fits are performed using the average of
particle and antiparticle yields. An exception is the GSI-
Heidelberg model, where μ<sub>B</sub> is a free parameter, determined
to be zero with an uncertainty of about 4 MeV. For a com-
parison on equal basis, the χ<sup>2</sup>/NDF value was recalculated
for the other models considering both particles and antiparti-
cles. The chemical freeze-out temperature obtained from the
fit is T<sub>chem</sub> ≈ 156 MeV, with uncertainties of 2–3 MeV and
very small differences (±1 MeV) among the different imple-
mentations of the SHM. It should be noted that T<sub>chem</sub> is very
close to the pseudo-critical temperature T<sub>pc</sub> = 156–158 MeV
for the transition from the QGP to the hadron gas obtained
from lattice-QCD calculations for μ<sub>B</sub> = 0 [119,169]. As
pointed out in [400], this can be related to the rapid drop of
the particle density and, consequently, of the multi-particle
scattering rates with decreasing temperature. In particular,
multi-hadron scatterings are expected to be substantial only
in the vicinity of T<sub>pc</sub>, while for temperatures lower than T<sub>chem</sub>
only two-particle interactions and decays play a relevant role,
but they are too slow to maintain the system in equilibrium
catching-up with the decreasing temperature. Therefore, the
chemical freeze-out hallmarks a moment near the hadronisa-
tion of the QGP itself. The last parameter estimated from the
SHM fits is the volume of the fireball for one unit of rapid-
ity at the chemical freeze-out, which is of about 4500 fm<sup>3</sup>
in SHARE, Thermal-FIST, and GSI-Heidelberg results. A
significantly larger volume is obtained from the fits with
THERMUS, which comprises an excluded-volume (Van-der-
Waals like) correction [401] to account for the short-range
repulsive interactions between hadrons, resulting in a lower
particle density in the fireball. As discussed in more detail
in Sect. 2.7, the volume of one unit of rapidity at chemi-
cal freeze-out extracted from the SHM fit can not be directly
compared to the homogeneity region estimated from the fem-
toscopy measurements described in Sect. 2.1.5, since the lat-
ter does not represent the source volume at a precise instant
during the fireball evolution. A SHM fit to the yields of
pions, kaons, and protons measured in Pb–Pb collisions at
√s<sub>NN</sub> = 5.02 TeV [402] gives a larger volume and a value of
T<sub>chem</sub> compatible with the one observed at √s<sub>NN</sub> = 2.76 TeV,

as expected from the trend established from the statistical
hadronisation analysis of the measured hadron yields at lower
collision energies, which shows that T<sub>chem</sub> increases with
increasing energy for √s<sub>NN</sub> < 20 GeV and saturates at a
value of 155–160 MeV for higher √s<sub>NN</sub> [122].

Experimental information on the QCD phase bound-
ary, complementary to the one extracted from the analysis
of hadron yields within the SHM, can be obtained from
measurements of event-by-event fluctuations of conserved
quantum charges and their probability distributions (see
e.g. [404]). The second-order fluctuations can be directly
connected to susceptibilities, defined as the second-order
derivatives of thermodynamic pressure, which can be cal-
culated on the lattice, thus providing a direct link between
measurements and predictions from lattice-QCD calcula-
tions. Second-order fluctuations and correlations of con-
served charges, which are established at the pseudo-critical
temperature T<sub>pc</sub>, can be extracted from data notably if the
number of particles N<sub>q</sub> and antiparticles N<sub>q̄</sub> are uncorrelated
and both distributed according to Poisson statistics. In this
case, the probability P(N<sub>q</sub> − N<sub>q̄</sub>) becomes a Skellam distri-
bution, which is completely determined by the average values
⟨N<sub>q</sub>⟩ and ⟨N<sub>q̄</sub>⟩. So, under the assumption of Skellam distri-
bution (which is supported by event-by-event measurements
of net-proton fluctuations from the STAR Collaboration at
RHIC [405,406] and by ALICE at the LHC [407]), inclu-
sive measurements of particle yields can provide access to
susceptibilities, which can be directly compared with lattice-
QCD predictions at T<sub>pc</sub> [378,408]. The fluctuations of con-
served charges (e.g. in net baryon and net strangeness num-
bers) extracted from the measured yields of different hadron
species (K<sup>±</sup>, K<sub>S</sub><sup>0</sup>, K<sup>*</sup>(892)<sup>0</sup>, φ(1020), p+p̄, Λ + Λ̄, Ξ<sup>±</sup>, Ω<sup>±</sup>)
in Pb–Pb collisions at √s<sub>NN</sub> = 2.76 TeV were calculated
in [404]. They were found to be consistent with the sus-
ceptibilities from lattice-QCD calculations in the interval of
pseudo-critical temperature 150 < T<sub>pc</sub> ≤ 163 MeV,
comparable to the chemical freeze-out temperature obtained
from the SHM fit to the hadron yields. This indicates that
the hadrons produced in central Pb–Pb collisions at the LHC
originate from the hadronisation of the QGP.

**Production of protons, nuclei, and strange particles.** A sig-
nificant deviation between the measured yields and the SHM
calculations can be seen in Fig. 33 for protons in the THER-
MUS and SHARE models, which are based on the ideal HRG
implementation consisting of non-interacting hadrons and
resonances. Such calculations predict a 25% higher proton
yield as compared to the measured one, resulting in a data-
to-model difference of about 2–3.5 standard deviations. This
tension between the data and the equilibrium value of the p/π
ratio has been dubbed the “proton-yield anomaly” and dif-
ferent explanations were proposed for it in the past years. It
was argued that it was possibly connected to the annihilation
<figure><img src="image_40.png" /><figcaption>**Fig. 33** Measured multiplicity per unit of rapidity of different hadron species and light nuclei [273–275,386–389] compared to SHM fits from THERMUS [391,391,392], SHARE [393–396], Thermal-FIST [397, 398], and GSI-Heidelberg [72,120,399]. Differences between the</figcaption></figure>

of baryons and antibaryons in the hadronic phase after the chemical freeze-out [409,410]. Using simulations with the microscopic transport model for hadrons, UrQMD [71,124], correction factors were determined for the effects of interactions before the kinetic freeze-out, resulting in an improved description of the data, but with a value of the hadrochemical equilibrium temperature of about 165 MeV [411]. The effect of baryon-antibaryon annihilation was also studied in the framework of a hydrodynamic calculation forking to UrQMD for the hadronic phase, showing that the annihilations help in achieving a better description of the data [412]. Another proposed explanation was based on different particle eigen-volumes for different hadron species, giving rise to a species-dependent excluded-volume correction, which provides an improved description of the data [413]. In the Thermal-FIST results shown in Fig. 33, an energy-dependent Breit-Wigner (eBW) scheme has been implemented to model the influence of finite resonance widths in the HRG, which leads to a suppression of the proton yields, mainly due to a reduced feed-down from $\Delta$ resonances [397]. With this approach, a significantly improved agreement between the SHM and the data for the proton yield is obtained at $T_{chem} = 155$ MeV. The eBW scheme, however, induces a larger discrepancy with the data for the yield of $\Lambda$ hyperons. According to Ref. [397], this discrepancy may be mitigated by including in the SHM calculations additional, yet undiscovered, strange baryonic

model calculations and the measured yields are shown in the bottom panels. The hypertriton yield is obtained using a theoretical estimation of $BR = 25\%$ for the branching ratio of the $^{3}_{\Lambda}H \rightarrow ^{3}\text{He}\pi^{-}$ decay channel [403]

resonances not reported in the PDG lists. Along the line of improving the description of the interactions in the HRG, the S-matrix formulation of statistical mechanics offers a natural implementation of interactions, with proper treatment of resonance widths and of non-resonant contributions. In particular, pion-nucleon interactions can be incorporated into the model via the empirical scattering phase shifts based on $\pi^{\pm}$-N scattering data. As pointed out in [414], the S-matrix approach allows one to obtain an improved matching with the lattice-QCD results for the correlation of the net baryon number with the electric charge, which is not well reproduced by an ideal HRG model. At a temperature of 156 MeV, the treatment of $\pi$-N interactions via the S-matrix approach leads to a reduction of the proton yield by about 17% relative to the ideal HRG result [399]. The corresponding reduction of the pion yield is only about 1%. The implementation of the S-matrix correction in the GSI-Heidelberg SHM fits, shown in Fig. 33, provides here the best match to the measured proton yields, together with a good description of all other measured particles.

An alternative approach was proposed considering that, in addition to the overestimation of the measured proton yield, the various SHM calculations tend to underestimate the data for strange baryon production, as it can be seen in Fig. 33 for the $\Lambda$, $\Xi^{\pm}$ and $\Omega^{\pm}$ yields, which differ by about 1.5–2 standard deviations from the predictions. This approach
advocates different hadronisation temperatures for up and down versus strange quarks, based on studies of flavour-specific fluctuations and correlations of conserved charges in lattice-QCD simulations with physical quark masses. These simulations indicate that strange quarks experience deconfinement at slightly larger temperatures compared to light quarks, implying that strange hadrons may be formed in the QGP slightly above $T_{pc}$ [415]. As a consequence, a univocal temperature cannot be assigned to the QCD crossover transition and a flavour hierarchy in the hadronisation is anticipated, with the characteristic temperature being higher by 10–15 MeV for the strange quarks compared to up and down quarks [416]. These studies motivated the usage of two different temperatures for strange and non-strange hadrons in the SHM fit [123,417]. This provided an improved description of the data at RHIC and LHC energies compared to the single-temperature description, with the difference between the two temperatures decreasing with decreasing $\sqrt{s_{NN}}$, i.e. increasing $\mu_B$. However, it should be pointed out that the data-based S-matrix correction to treat the interactions in the HRG was not included in the fits with two freeze-out temperatures. In this context, a recent study of the effect of S-matrix corrections employing a coupled-channel model for K, Λ and Σ interactions predicts an enhancement in the $\bar{\Lambda}+\Sigma^0$ yields relative to the HRG baseline, thus allowing for a more accurate description of the measured particle yields within the THERMUS SHM framework [418].

The production of light nuclei is quantitatively well described by the statistical hadronisation model. However, it is debatable whether the light nuclei should be included in the thermal model fit. The production mechanism for such loosely bound states ($\mathcal{O}(\text{MeV})$) is presently a subject of intense investigation. Models have been proposed which predict that nuclei are formed at the late stages of the fireball expansion via coalescence of nucleons that are close to each other in phase space [419] (see Sect. 5 for a detailed discussion). It should be noted that if the yields of light nuclei are excluded from the fit, the temperature and volume extracted from the SHM do not change significantly. Therefore, the inclusion of nuclei in the fit does not alter the considerations that were made above on the characteristics of the hadron gas at the chemical freeze-out. Conversely, the yields of nuclei are very sensitive to the temperature: a small variation in temperature leads to large variations of the yields of nuclei. In particular, a SHM fit with the GSI-Heidelberg framework including only the yields of light nuclei gives $T_{\text{chem}}^{\text{nuclei}} = 159 \pm 5$ MeV, which is consistent with the value of $T_{\text{chem}}$ extracted from all particles [122]. These observations indicate that the same thermal parameters governing light-hadron yields also determine the production of light composite objects which are loosely bound states. However, this poses a question about how the relatively loosely-bound light nuclei formed at the QGP hadronisation can survive in

the hostile environment of the hadronic phase that follows the chemical freeze-out. This was dubbed the “snowball in hell” problem, as the binding energies of light nuclei (e.g. 2.2 MeV for the deuteron) are much lower than the temperature of the hadron gas at the chemical freeze-out. A possible explanation assumes that the reactions involving break-up and formation of light nuclei proceed in relative chemical equilibrium after the chemical freeze-out, so that nuclei are disintegrated and regenerated at similar rates during the hadronic phase, thus preserving the relative abundances of nuclei and nucleons at $T_{\text{chem}}$ [420–422]. A different proposed possibility is that, at the QGP hadronisation, compact and colourless bound states of quarks are produced and evolve into the final state nuclei only after a formation time of some fm/c [122]; such compact objects can survive a short-lived hadronic phase after hadronisation.

It is important to note that the measured yields of strange and multi-strange particles at midrapidity in Pb–Pb collisions are described (within 2 standard deviations) by the full-equilibrium SHM model, corresponding to the *grand-canonical* ensemble. The relative abundance of strange hadrons in heavy-ion collisions is larger compared to the suppressed strangeness yield observed in collisions between elementary particles, as discussed in detail in Sect. 3.2. Strangeness production in pp collisions is consistent with a *canonical* formulation of the statistical model, in which a correlation volume parameter is introduced to account for the locality of strangeness conservation (a strangeness undersaturation parameter is also often employed for lower energies), see [418] for a recent analysis. In the framework of the SHM, the lifting of strangeness suppression from pp to A–A collisions can therefore be understood in terms of the transition from the canonical to the grand-canonical limit [423]. An enhanced strangeness production in heavy-ion collisions, as compared to pp collisions, was one of the earliest proposed signals for the QGP formation [87,424–426]. In particular, in a deconfined state, the abundances of light partons, including the strange quark, are expected to quickly reach their equilibrium values due to the low energy threshold to produce s¯s pairs. It was also shown that the strangeness content of the QGP in equilibrium is similar to that of a chemically equilibrated hadron gas at the same entropy-to-baryon ratio, although in both cases the production level of strange hadrons is higher than that observed in pp collisions [427–429]. However, it should be considered that the time needed to achieve the equilibrium for multi-strange baryons via two-body hadronic collisions is estimated to be much longer than the lifetime of the fireball produced in heavy-ion collisions [424]. The equilibration times for Λ, Ξ, and Ω baryons via hadronic processes are instead expected to be substantially smaller ($\lesssim 1$ fm/c) if multi-hadron scatterings play a relevant role [400], which is the case when the particle densities are high. Such conditions are realised only for tempera-
tures near to $T_{\text{pc}}$. This confirms the conclusions in [424,425] that strange hadron abundances are established very close in time to the transition from the QGP to hadronic matter and are not appreciably changed by hadronic interactions at temperatures below $T_{\text{pc}}$. Hence, the observed validity of a grand-canonical description of strange-hadron production in Å collisions can be seen as a natural consequence of the formation of the QGP in the early phases of the collision evolution [72].

**Heavy-flavour hadron yields.** An additional question of the statistical nature of hadron production is whether the yields of hadrons containing heavy quarks can be described with the statistical hadronisation approach discussed above for light-flavour particles. The masses of charm and beauty quarks are substantially larger than the temperature scale of the QCD medium attained at LHC energies, and therefore thermal production of these quarks in the QGP is strongly suppressed. Thus, the charm and beauty content of the fireball is determined by the initial production of c and b quarks in hard-scattering processes, i.e. very far from a chemical equilibrium population for a temperature of the order of $T_{\text{pc}}$. The heavy c and b quarks produced in initial hard scatterings traverse the QGP and interact with its constituents, exchanging energy and momentum with the medium. These interactions can lead the charm quarks to (partially) thermalise in the QGP [69,430–433] such that, at the phase boundary, they can be distributed into hadrons with thermal weights as discussed above for the light quarks. In the statistical hadronisation approach, the absence of chemical equilibrium for heavy quarks is accounted for by a fugacity factor $g_c$, which ensures that all initially produced charm quarks are distributed into hadrons at the phase boundary [434]. The value of $g_c$ is determined from a balance equation containing the charm quark production cross section as input. With this approach, the yield of hadrons containing charm quarks can thus be calculated using as input values: (i) the thermal parameters obtained from the analysis of light-flavour hadrons, and (ii) the charm-quark production cross section, which is an external input to the model.

The yields of charm hadrons (non-strange [435] and strange D mesons [436], $\Lambda_c^+$ baryons [437] and $J/\psi$ mesons [438]) measured in central (0–10%) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV are shown in Fig. 34. The yields per unit of rapidity of $D^0$ and $J/\psi$ mesons were obtained by integrating the $p_T$-differential measurements performed down to $p_T = 0$. Instead, the $p_T$-integrated yields of $D^+$, $D^*(2010)^+$, $D_s^+$ mesons, and $\Lambda_c^+$ baryons were extrapolated exploiting the measured $p_T$-differential production ratios relative to the $D^0$ mesons, using different assumptions on their shapes in the $p_T$ range where the measurements were not performed (see [435–437] for details). The fraction of extrapolated yield is different for the different hadron species, and it is about

19% for the $\Lambda_c^+$ (which is measured down to $p_T = 1$ GeV/c), about 70% for $D^+$ and $D_s^+$ (measured down to $p_T = 2$ GeV/c), and about 85% for $D^*(2010)^+$ mesons, which are measured for $p_T > 3$ GeV/c.

The data are compared to SHM calculations [434] where: (i) thermal parameters come from the fit to light-flavour hadrons at $\sqrt{s_{NN}} = 2.76$ TeV extrapolated to $\sqrt{s_{NN}} = 5.02$ TeV (same $T_{\text{chem}}$ and larger volume V following the increase of $dN_{\text{ch}}/d\eta$ with increasing $\sqrt{s_{NN}}$), and (ii) the charm production cross section is determined from the measured $D^0$ yield in the 10% most central Pb–Pb collisions [435], leading to $d\sigma_{cc}/dy = 0.579$ mb. The solid band in the SHM predictions in Fig. 34 represents the uncertainty related to the determination of the charm production cross section at midrapidity. For the measured data points, the empty boxes represent the systematic uncertainty associated to the measurement in the visible $p_T$ range, while the shaded boxes stand for those associated to the $p_T$-extrapolation procedure. The measured yields of open-charm mesons are compatible with the SHM calculations within the uncertainties, which is to some extent expected since the charm-quark production cross section used in the model is tuned to reproduce the measured $D^0$-meson yield. A larger difference is observed for the production yield of $\Lambda_c^+$ baryons, which is underestimated by the SHM predictions. A possible solution for this difference within the framework of the statistical hadronisation model was proposed in [439], where an augmented set of excited charm baryon states beyond those listed in the PDG [440] is considered, as discussed in Sect. 2.3.2. In fact, as presented in [434], if a charm-quark production cross section of $d\sigma_{cc}/dy = 0.68$ mb is considered instead of $d\sigma_{cc}/dy = 0.579$ mb and the augmented set of charm baryon states is used, the SHM provides a satisfactory description of the measured $\Lambda_c^+$ baryon yield. Particularly interesting for the statistical hadronisation approach is the $J/\psi$ meson. In fact, an enhanced production of hidden-charm states has been observed in Pb–Pb collisions at LHC energies compared to the predicted suppression with respect to pp collisions due to the colour-charge screening in the QGP [86] (see Sect. 2.5.1). The SHM provides a good description of the $J/\psi$ production yield within uncertainties, supporting the statistical (re)combination of charm and anticharm quark pairs at the phase boundary [441].

Recalling that charm quarks need to be (to some extent) thermalised in order to be distributed into hadrons according to thermal weights, one can conclude that the fair description of the measured open charm and charmonium yields within the SHM suggests that the interactions with the medium constituents bring charm quarks in (or close to) kinetic equilibrium with the QGP. This is further supported by the measurements of charm-hadron $R_{AA}$ and $v_2$ at low-$p_T$ reported in Sect. 2.4.1.
**Fig. 34** $p_T$-integrated yields per unit of rapidity measured at midrapidity for different charm-hadron species in the 10% most central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [435–438] compared to SHM predictions [434]. Systematic uncertainties from data (open boxes) and from the $p_T$ extrapolation (shaded boxes) are shown separately. The coloured band in the SHM predictions represents the uncertainty on the charm-quark production cross section

<figure><img src="image_41.png" /></figure>

### 2.3.2 Particle momentum and angular distributions: connection to microscopic hadronisation mechanisms

Measurements of the production of different hadron species as a function of $p_T$ can provide insight into the transition, at the pseudo-critical temperature $T_{pc}$, from the partonic degrees of freedom of the QGP to the hadron gas phase. The process of hadron formation from the QGP medium is expected to be different from other cases of hadronisation, such as the (in-vacuum) fragmentation of hard-scattered partons in elementary collisions, in which no bulk of thermalised partons is present [442]. For processes with a momentum transfer much larger than $\Lambda_{QCD}$, the QCD factorisation theorems allow one to separate short and long distance dynamics and to express the cross section for inclusive hadron production as a convolution of two terms: the hard-scattering cross section at the partonic level, which can be calculated as a perturbative series in powers of the strong coupling constant $\alpha_s$, and the fragmentation functions, which account for the non-perturbative evolution of a hard-scattered parton into the given hadron in the final state. Fragmentation functions are not calculable in QCD, but they can be parameterised after experimental measurements, mostly from $e^+e^-$ collisions. In microscopic models (i.e. models involving explicitly quarks and gluons, which are the underlying fundamental degrees of freedom in QCD), the (in-vacuum) fragmentation of the hard-scattered parton occurs via the creation of $q\bar{q}$ pairs through string breaking or gluon radiation and splitting, thus forming colour singlet states that evolve into hadrons. In the case of the hadronisation of the QGP created in heavy-ion collisions, a bulk of deconfined partons is present when the transition temperature $T_{pc}$ is reached. So, there is no need for the creation of

additional partons through string breaking, and the quarks of the bulk that are close to each other in phase space can simply combine into hadrons [443–446]. Models of hadronisation via recombination (also known as coalescence) were originally developed [447] to describe the relative abundances of hadrons measured at very forward rapidity in $\pi$-nucleus collisions at the SPS [448] and at FNAL [449]. These models are based on the concept that the presence of a reservoir of partons (the valence quarks of the projectile in the considered case, concerning a kinematic region at very forward rapidity) can induce significant modifications in the hadronisation process. In the case of the hadronisation of the QGP, the bulk of deconfined partons constitutes this reservoir.

Recombination models are built on the assumption that quarks, at the hadronisation of the QGP, can be treated as effective degrees of freedom having a dynamical mass approaching the constituent mass [450,451], while gluons disappear as dynamical degrees of freedom and are converted into $q\bar{q}$ pairs [445,452]. With the additional assumption of dominance of the lowest Fock states in the hadron wave function, effective constituent quarks (and antiquarks) are combined into mesons and baryons taking into account only the valence structure of the hadron. Hence, the probability of emitting a hadron from the QGP is proportional to the probability of finding its valence quarks in the bulk medium. As discussed in [453], the hadron spectra produced via recombination of quarks from a thermal medium are not significantly affected by the inclusion of higher Fock states to take into account the complexity of the internal structure of hadrons. The recombination models have provided a natural explanation for some unexpected measurements of baryon and meson production yields and elliptic flow in the intermediate-$p_T$
region ($1.5 < p_T < 5$ GeV/c) at RHIC [109–112]. Notably, also global polarisation measurements are expected to be sensitive to the hadronisation mechanism and, as discussed in Sect. 2.2.8, the results on $K^*(892)^0$ polarisation as a function of $p_T$ are qualitatively consistent with the expectation of hadronisation of polarised quarks via recombination [372].

In the hadronisation of the fireball, the recombination process competes with the hadronisation of energetic (high-$p_T$) partons that escape from the QGP and hadronise in the vacuum via fragmentation. Final hadron spectra are a mixture of hadrons from recombination (with a momentum essentially given by the sum of the momenta of the valence quarks) and from string fragmentation (having a momentum lower than that of the parent parton). As discussed in [443], the recombination process is expected to dominate over fragmentation at low- and intermediate-$p_T$ (up to few GeV/c), while at higher momenta ($p_T > 8–10$ GeV/c), where the particle spectra exhibit a power-law trend, a transition occurs to a regime dominated by fragmentation of jets. This transition is predicted to take place at higher values of $p_T$ for baryons as compared to mesons. This naturally explains the baryon versus meson grouping of $v_2$ and the baryon-to-meson enhancement at intermediate-$p_T$ that will be discussed later in this paragraph. At low-$p_T$, the hadronisation is expected to occur completely via recombination. In this region, the hadrons formed by combination of thermalised quarks are found to have momentum spectra reproducing the thermal equilibrium limit, and this naturally leads to a good agreement with spectra from hydrodynamic calculations [454]. Also note that the dominance of recombination at low- and intermediate-$p_T$ is completely consistent with what was discussed in Sect. 2.3.1 about the thermal origin of particle production, based on the agreement between SHM calculations and the measured integrated yields (see Figs. 33 and 34). As pointed out in [452], the recombination mechanism connects a thermal partonic phase with the observed thermal hadronic phase, and therefore it can be seen as a microscopic manifestation of statistical hadron production. In this thermal-limit regime, the hadron $p_T$ spectra and $v_2$ are the result of a system which evolved through equilibrium stages, thus not preserving the memory of the previous stages and of the underlying microscopic dynamics that drove the system evolution. Instead, the intermediate-$p_T$ region, which is located above the thermal-limit regime but below the high-momentum region where fragmentation dominates, offers a window where non-equilibrium effects in the hadronic observables can provide sensitivity to the hadronisation mechanism at a microscopic level.

**Light-flavour particle spectra and ratios.** The transverse-momentum spectra of $\pi^\pm$, $K^\pm$, $p+\bar{p}$, and $\phi(1020)$ in 0–10% and 70–80% central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [402,455] are shown in the left panel of Fig. 35. As

commented in Sect. 2.2.2, the spectra harden with increasing centrality and multiplicity (radial flow signature, more evident for heavier particles): the maximum of the proton spectrum shifts from ~0.6 GeV/c to ~1.3 GeV/c going from peripheral to central collisions. The right panel of Fig. 35 shows the $p_T$-differential $(p+\bar{p})/(\pi^+ + \pi^-)$ and $(p+\bar{p})/(2\phi)$ ratios (referred to as $p/\pi$ and $p/(2\phi)$ in the following$^7$) in inelastic pp and centrality-dependent Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. The $p/\pi$ ratios in Pb–Pb collisions exhibit a bump structure in $1.5 \lesssim p_T \lesssim 8.0$ GeV/c, gradually increasing with increasing centrality, reaching a maximum of about 0.82 at $p_T \approx 3.5$ GeV/c in the most central collisions. This can be explained as an interplay between the collective motion of the system (heavier particles are boosted to higher momenta) and the recombination mechanism that is expected to be the dominant one for the hadronisation in this momentum region. It is interesting to note that the fraction of yield in the peak is about 1.5–2% for pions and 7–21% for protons, depending on centrality, and therefore the $p_T$-integrated $p/\pi$ ratio shows only a mild dependence on centrality. In particular, the $p_T$-integrated ratio decreases by about 20% from the most peripheral to the most central collisions [402], consistent with SHM calculations including either antibaryon-baryon annihilations in the hadronic phase [409,410] or the S-matrix approach to account for non-resonant interactions in the HRG [399]. This suggests that baryon enhancement at intermediate-$p_T$ is mainly due to the redistribution of baryons and mesons over the momentum range. Further insight can be obtained from the ratios of the yields of p and $\phi$, which have similar masses but different quark content. At low-$p_T$ ($p_T \lesssim 3.5$ GeV/c), the $p/(2\phi)$ ratio is rather independent of $p_T$ in central Pb–Pb collisions. Such behaviour can be expected from hydrodynamic-based models [456,457] in which the particle mass is the main variable in the determination of the spectral shapes (even though interactions in the hadronic phase could modify this picture because they are expected to affect differently protons and $\phi$ mesons). In contrast to this and as a confirmation of the radial flow signature, the 70–80% Pb–Pb ratio is similar to that in pp collisions showing a decreasing trend with $p_T$. At higher momenta, $p_T \gtrsim 8$ GeV/c, all the Pb–Pb and pp baryon-to-meson ratios shown in Fig. 35 are consistent within the uncertainties suggesting the dominance of vacuum-like fragmentation processes in this region [458].

The $p_T$-differential $p/\pi$ and $p/(2\phi)$ yield ratios were also measured in Xe–Xe collisions at $\sqrt{s_{NN}} = 5.44$ TeV [281] and compared to the results in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [402]. The baryon-to-meson ratios in the

<footnote>$^7$ At LHC, the particle yield ratios are quoted for the sum of particles and antiparticles since both populations of a same species are produced in almost equal amount. See for example discussion in Sect. 2.3.1 or in [273].</footnote>
<caption>**Fig. 35 (Left)**</caption>

Transverse-momentum spectra of $\pi^{\pm}$, $K^{\pm}$, $p+\bar{p}$ [402] and $\phi(1020)$ [455] in 0–10% (filled markers) and 70–80% (empty markers) central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. (Right) Proton-to-pion and proton-to-phi $p_T$-differential ratios in inelastic pp collisions at $\sqrt{s} = 5.02$ TeV and in several centrality intervals in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. Systematic uncertainties are shown as boxes while statistical uncertainties as vertical bars. In the $(p+\bar{p})/(\pi^+ + \pi^-)$ ratio, only the systematic uncertainties uncorrelated across centrality intervals are shown in the error boxes, in order to better highlight the evolution of the ratio with the collision centrality

<figure><img src="image_42.png" /></figure>

two collision systems are compatible between each other within the uncertainties, if events with similar charged-particle multiplicity densities are selected. As concluded in [281], this indicates that the radial flow and the hadronisation dynamics, which determine the shape of the $p_T$-spectra of different hadron species, are mainly driven by the density of produced particles, in contrast to the elliptic flow which depends also on the eccentricity of the collision region.

The baryon-to-meson ratios in the strange-hadron sector show similar features as those discussed above for the $p/\pi$ ratios. The measured $(\Lambda + \bar{\Lambda})/(2K_S^0)$ ratios (referred to as $\Lambda/K_S^0$ in the following) as a function of $p_T$ in central (0–5%), semi-central (40–60%), and peripheral (80–90%) Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV are reported in the left panel of Fig. 36 and compared to the ratio measured in minimum-bias pp collisions at $\sqrt{s} = 7$ TeV [274]. The ratio measured in the most peripheral Pb–Pb collisions is compatible with the pp result. For more central collisions, the peak at low/intermediate-$p_T$ becomes more pronounced and its position shifts towards higher momenta, in qualitative agreement with the effect of increased radial flow. The ratios of the $p_T$-integrated $\Lambda$ and $K_S^0$ yields (after extrapolation to $p_T = 0$) are independent of centrality within the experimental uncertainties [274], suggesting, as mentioned before, that the baryon enhancement at intermediate-$p_T$ is predominantly due to a redistribution of baryons and mesons over $p_T$ rather than to an enhanced production of baryons. In addition, the $\Lambda/K_S^0$ ratio measured in pp collisions at $\sqrt{s} = 7$ TeV within jets with $p_{T,jet}^{ch} > 10$ GeV/c reconstructed with a resolution parameter $R = 0.4$ [459] is shown for comparison. The ratio in jets is almost independent of the transverse momentum of the particles produced in the jet and it does not show a maximum at intermediate-$p_T$. For $p_T \lesssim 5$ GeV/c, it

is lower than the ratio measured in inclusive (minimum bias) pp collisions as discussed in Sect. 3.2. For $p_T > 7–8$ GeV/c, the $\Lambda/K_S^0$ ratio in jets is consistent with the inclusive one in pp collisions, as well as with the values measured in Pb–Pb collisions independently of the centrality. This confirms that the high-$p_T$ hadrons stem from the fragmentation of hard-scattered partons, which escape from the QGP, and indicates that the relative fragmentation into baryons and mesons at high-$p_T$ is essentially vacuum-like and is unmodified – or at least, not significantly – by the medium formed in the heavy-ion collisions.

A further illustration of the previously outlined physics message concerning particle yield ratios can be observed in the right panel of Fig. 36 which allows for a direct comparison of the $p_T$-differential baryon-to-meson ratios for non-strange and strange hadrons ($p/\pi$ and $\Lambda/K_S^0$) thanks to a double ratio Pb–Pb/pp calculated at $\sqrt{s_{NN}} = 5.02$ TeV and 2.76 TeV, respectively [274,402]. The two double ratios are compatible between each other within uncertainties in all centrality intervals and they are consistent with unity for $p_T > 8–10$ GeV/c, where fragmentation dominates. For $p_T \lesssim 1.5$ GeV/c, a hierarchy of the baryon-to-meson ratios is observed as a function of centrality: the ratios are strongly reduced in central collisions compared to peripheral collisions, thus compensating the enhancement at intermediate-$p_T$.

**Comparison to models.** The results related to $\pi^{\pm}$, $p+\bar{p}$, and $\phi(1020)$ spectra and yield ratios in different centrality classes (0–10%, 10–20%, and 40–50%) are compared with several models in Figs. 37 and 38. These models aim at bridging the gap between the partonic phase with its hydrodynamic motion and the ultimate hadronic stages. The models considered in this section are: VISHNU [326,460], EPOS v3.4 (simply called EPOS3 in the following) [199], Catania coa-
<figure><img src="image_43.png" /><figcaption>**Fig. 36** (Left) $p_T$-differential $\Lambda/K_S^0$ [274] ratio in Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV compared to pp collisions at $\sqrt{s} = 7$ TeV for the inclusive production [274] and for the production in jets [459]. (Right) $p_T$-differential double ratios (baryon/meson ratios in Pb–Pb divided by the pp ones at a same colliding energy) for $p/\pi$ at $\sqrt{s_{NN}} = 5.02$ TeV [402] and $\Lambda/K_S^0$ at $\sqrt{s_{NN}} = 2.76$ TeV [274]. Results are shown for several centrality</figcaption></figure>

<figure><img src="image_44.png" /></figure>

intervals. Systematic uncertainties are shown as boxes while statistical uncertainties as vertical bars. The relative systematic uncertainty on the pp results in the double ratios, which is correlated among the different centrality intervals of the Pb–Pb measurements is displayed separately in the lower panel

lescence plus independent-fragmentation model [457], and CoLBT [461]. In the following, the description of the main features of each model is provided:

- The VISHNU model, introduced in Sect. 2.2.3, is a viscous hydrodynamic calculation of the expansion of the fireball, handing over to a microscopic hadron cascade model (UrQMD) [71,124] for the late phases of the expansion of the hadron resonance gas. The initial energy density profile is modelled with $T_R$ENTo [462]. To convert the hydrodynamic output into particles (propagated then with UrQMD) first an isothermal freeze-out surface is found and then the hadron spectra are calculated with the Cooper-Frye formula [463]. Note that this model addresses only the low-$p_T$ part of the spectra (< 3 GeV/c).

- The EPOS3 model is a general-purpose event generator based on the Gribov–Regge theory of multiple scattering [197,464], perturbative QCD, and string fragmentation [465]. The dense region in the system, the so-called core, is treated as the QGP [466] and modelled with a hydrodynamic evolution (since EPOS2 [198]) incorporating viscosity (EPOS3 [199]), followed by statistical hadronisation which conserves energy, momentum, and flavours thanks to a microcanonical formulation [466]. Particle production from the low-density regions of the system, called corona, is treated as in proton–proton collisions. EPOS3 implements further a saturation in the

initial state as predicted in the Colour Glass Condensate model [467] and a hadronic cascade for the late stages. The model also accounts for jets and their interactions with the hydrodynamically expanding bulk matter. This is important for particle production at intermediate-$p_T$ [468] and it is reminiscent of the recombination mechanism [443,445].

- The Catania model [457] describes the QGP hadronisation via the competing processes of parton fragmentation and quark recombination. The latter is implemented by employing the instantaneous coalescence approach described in [445,469], which is based on the Wigner-function formalism [470] to calculate the spectrum of hadrons from that of quarks. The approach is based on a Monte Carlo implementation that allows one to include a 3D geometry as well as the radial flow correlation in the partonic spectra and the effect of the main resonance decays. At low-$p_T$, the partonic spectra are obtained with a blast-wave approach assigning the mass of constituent quarks (330 MeV/$c^2$ for u and d quarks and 450 MeV/$c^2$ for s quarks). The contribution of mini-jets is also considered, which is important to model the intermediate momentum region where soft–hard recombination processes (involving a quark from a mini-jet and one from the bulk) play a role in the determination of the spectral shapes until independent fragmentation takes over at high-$p_T$. This is achieved by using a parton $p_T$ spectrum calculated at next-to-leading order
in a pQCD scheme. In A–A collisions, the modification due to the jet quenching mechanism is included (see Eq. 3 in [457]). Note that this approach does not guarantee the hadronisation of the full bulk of particles of the QGP medium. Moreover, the underlying kinematics of the instantaneous projection of massive quarks into hadrons, which does not include interactions with the surrounding medium, makes it impossible to conserve the four-momentum in the instantaneous coalescence process. Therefore, the model has to be considered fully applicable only at $p_T > 1.5$ GeV/c even if the description of the spectra at low-$p_T$ is reasonable.

– The CoLBT model [461] is based on a state-of-the-art coupled linear Boltzmann transport model (CoLBT-hydro) [471] to simulate the concurrent evolution of the bulk medium and parton showers. The hadronisation is modelled via the Hydro-Coal-Frag hybrid approach [472] that includes: (i) particle formation from the hydrodynamical fields at the freeze-out hypersurface at low-$p_T$, (ii) instantaneous coalescence of effective constituent quarks at intermediate-$p_T$, and (iii) fragmentation at high-$p_T$. The interplay between hadron formation at the hydrodynamical freeze-out and parton dynamics is regulated with a $p_T$ threshold for the effective constituent quarks. Above this threshold, viscous corrections to the equilibrium distributions are large and recombination and fragmentation become the relevant hadronisation mechanisms. The recombination includes thermal–thermal, thermal–shower, and shower–shower coalescence. Shower partons that do not coalesce are hadronised via string fragmentation. Finally, UrQMD is used to model the hadronic phase until the kinetic freeze-out.

Figure 37 shows the transverse-momentum spectra of $\pi^{\pm}$, $p+\bar{p}$, and $\phi(1020)$ in Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV for the 10–20% centrality class compared with the VISHNU, EPOS3, and CoLBT models described above. For the CoLBT model, the curves obtained with the separate contributions of hydrodynamics, coalescence and fragmentation are also shown. The VISHNU calculations reproduce the pion and proton spectra fairly well in the low-$p_T$ region ($p_T < 3$ GeV/c), where a hydrodynamic description of the fireball evolution is expected to be valid. It should be noted that the measured spectral shapes of protons could not be reproduced by viscous hydrodynamic calculations lacking an explicit description of the hadronic phase, and therefore the inclusion of UrQMD to model the hadron transport has been crucial to obtain a good description of the measured spectra [473]. EPOS3 includes both soft and hard physics processes and, as a consequence, it is anticipated to give a good description of data from low- to high-$p_T$; this is observed here up to 4 GeV/c or 8 GeV/c depending on the considered particle species. The CoLBT model provides a good description of the measured

<figure><img src="image_45.png" /><figcaption>Fig. 37 Comparison of $\pi^{\pm}$, $p+\bar{p}$ [402], and $\phi(1020)$ [455] transverse-momentum spectra in 10–20% central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV to VISHNU [326,460], EPOS v3.4 [199], and CoLBT [461] models. The systematic uncertainties are represented as boxes while the statistical uncertainties are shown as vertical bars</figcaption></figure>

pion and proton spectra. The hydrodynamics contribution is the dominant one for $p_T < 2–3$ GeV/c, while fragmentation dominates for $p_T > 6–8$ GeV/c. At intermediate-$p_T$ the coalescence contribution is needed to reproduce the data. As noted in [461], in the CoLBT model the transition of the hadron production mechanism from the hydrodynamics domain to the region dominated by coalescence and fragmentation occurs at higher $p_T$ for more central collisions. This is due to the stronger radial flow in central collisions that pushes the hydrodynamics regime up to higher momenta.

Despite some difficulties in the description of the individual transverse-momentum spectra [402], the majority of the models considered here are able to describe adequately (in their $p_T$ range of applicability) the baryon-to-meson ratios $p/\pi$ shown in the left panels of Fig. 38. For the 0–10% centrality class, the measured $p/\pi$ ratios are shown for two different centre-of-mass energies, namely $\sqrt{s_{NN}} = 5.02$ and 2.76 TeV. The trend and the magnitude of the yield ratios are similar at the two collision energies, notably with a slight shift of the peak towards higher $p_T$ at the higher $\sqrt{s_{NN}}$, consistent
<figure><img src="image_46.png" /><figcaption>**Fig. 38** Comparison of $p/\pi$ (left) and $(p+\bar{p})/(2\phi)$ (right) in 0–10%, 10–20%, and 40–50% Pb–Pb collisions to the VISHNU [326,460], EPOS v3.4 [199], Catania [457], and CoLBT [461] models as illustrated in the</figcaption></figure>

legend. More details on the different Catania model components can be found in the text. The systematic uncertainties are represented as boxes while the statistical uncertainties are shown as vertical bars

with the larger radial flow. The VISHNU model is able to reproduce the low-$p_T$ rise of the $p/\pi$ ratio, but it can quantitatively reproduce the data only for $p_T < 1.0–1.5$ GeV/c since, above this range, the contributions from jet showers and mini-jets to the pion spectra start to be important [474]. For $\sqrt{s_{NN}} = 2.76$ TeV, the Catania model with the contributions of radial flow and soft–hard coalescence qualitatively describes the peak in the $p/\pi$ ratio in the $2 < p_T < 4$ GeV/c region, where recombination processes are dominant. On the other hand, as expected, the Catania calculations without flow cannot reproduce the peak, while the removal of the soft–hard coalescence worsens the agreement with data in $6 < p_T < 8$ GeV/c. The Catania model with only the fragmentation component can reproduce the data only for $p_T > 8$ GeV/c. The EPOS3 event generator is able to describe the shape of the $p/\pi$ ratio, despite the peak position being shifted to lower $p_T$ indicating lower radial-flow effects in

the model than that measured in data. On the other hand, in the 40–50% centrality class, where radial flow is smaller than in central collisions, EPOS3 accurately describes the peak position but overestimates the magnitude of the $p/\pi$ ratio at intermediate-$p_T$. The CoLBT model describes well the measured yield ratios both in the 10–20% and 40–50% centrality classes, with a steep increase at low-$p_T$ due to the mass ordering induced by the radial flow, and a decrease for $p_T > 3$ GeV/c, which results from the interplay among hydrodynamic expansion, quark recombination, and parton fragmentation.

Considering the $p/(2\phi)$ ratio shown in the right panels of Fig. 38, both EPOS3 and Catania can reproduce the measured ratio at very low-$p_T$ and at high-$p_T$, but fail in describing the intermediate-$p_T$ region, mostly because the spectral shapes are underestimated for protons. The same considerations hold true in 40–50% Pb–Pb collisions concerning the EPOS3
model. It should be noted that, as pointed out in [457], similar slopes of proton and $\phi$ meson spectra (i.e. a flat $p/(2\phi)$ ratio as a function of $p_T$) are expected for $p_T < 2$ GeV/c in case of coalescence (as in the case of hydrodynamics) because the difference between a combination of three quarks with $330 \text{ MeV/c}^2$ mass and that of two quarks with $450 \text{ MeV/c}^2$ mass flowing with the same collective velocity is marginal.

**Anisotropic flow coefficients.** A further benchmark for the recombination/fragmentation scenario is provided by the comparison of the measured elliptic flow coefficient $v_2$ of identified particles with theoretical models. Figure 39 shows the $v_2$ as a function of $p_T$ for $\pi^\pm$, $K^\pm$, $K_S^0$, p+$\bar{p}$, $\Lambda+$+$\bar{\Lambda}$, and $\phi$ in Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV for different centrality classes [290]. The main features of the $p_T$ dependence of $v_2$ are the same in all the centrality intervals displayed in the figure. For $p_T < 2-3$ GeV/c, the strong radial flow results in a mass-ordered $v_2$ as reproduced by the VISHNU hydrodynamic model with the initial energy density profile from TRENTo (see Sect. 2.2.4). At intermediate momenta ($3 \lesssim p_T < 8-10$ GeV/c), a grouping of the baryons versus mesons is observed, as expected in the case of hadron formation via recombination. The $\phi$ meson plays a special role because its mass is similar to that of the proton. This is reflected in the trend of its $v_2$ which follows the one of the proton at low-$p_T$; on the other hand, at intermediate-$p_T$, the $v_2$ of the $\phi$ becomes compatible with meson $v_2$ indicating that the elliptic flow is driven by quark content rather than by mass in this momentum range. At higher $p_T$ ($p_T > 8-10$ GeV/c), where $v_2$ originates mainly from the path-length dependence of the in-medium energy loss of partons produced in hard-scattering processes and fragmentation takes over from recombination as the dominant hadronisation mechanism, the $v_2$ values of the different hadron species become compatible within uncertainties.

As shown in Sect. 2.2.6, the same features discussed above for the $p_T$-differential elliptic flow $v_2$, namely the mass ordering at low-$p_T$ followed by baryon versus meson grouping at intermediate-$p_T$, are observed also for the higher-order flow harmonics $v_3$ and $v_4$, as well as for the non-linear flow modes $v_{4,22}$ and $v_{5,32}$ [291]. In particular, the observation of the same particle type grouping for $v_4$ and $v_{4,22}$ is in line with the expectation that quark recombination is the dominant particle production mechanism in the intermediate-$p_T$ region and affects both flow modes similarly.

The top right and the bottom panels of Fig. 39 show the comparison of $v_2$ to the Catania, EPOS3, and CoLBT models for the 20–30%, 30–40% and 40–50% centrality classes, respectively. For EPOS3, the predictions from a simulation without the hadronic cascade stage (modelled with UrQMD) are also shown and they indicate that the interactions in the hadronic phase are crucial to obtain a good description of the measured $v_2$. EPOS3 is able to reproduce the mass-

ordering observed up to $p_T = 2-3$ GeV/c. It cannot quantitatively reproduce the $v_2$ of pions and protons at intermediate-$p_T$ but a good agreement with the data is observed for the other hadrons up to $p_T = 5-6$ GeV/c. At higher momenta, EPOS3 always underestimates the data, indicating that a larger amount of energy loss is needed in the model. On the other hand, the Catania and CoLBT models, which both include recombination and fragmentation, describe well the measured $v_2$ of the different hadron species in a wide $p_T$ interval, extending from the hydrodynamic realm at low momentum to the fragmentation region at high-$p_T$. Conversely, the results of CoLBT simulations without the coalescence component, shown in the bottom-right panel of Fig. 39, underestimate the measurements for $p_T > 4$ GeV/c, indicating that in the hybrid hadronisation approach implemented in this model quark recombination is crucial to describe the data.

The recombination of flowing quarks provides a natural explanation for the observed $p_T$ dependence of $v_2$ and of the baryon versus meson grouping at intermediate-$p_T$ [443,475]. In the recombination picture, mesons (M) and baryons (B) at a transverse momentum $p_T$ reflect the properties of partons with an average transverse momentum of $p_T/2$ (for mesons) or $p_T/3$ (for baryons). At low-$p_T$, their elliptic flow, to a first approximation, should be determined by that of their constituent quarks (q): $v_{2,M}(p_T) = 2v_{2,q}(p_T/2)$ for mesons and $v_{2,B}(p_T) = 3v_{2,q}(p_T/3)$ for baryons. As a consequence, $v_2$ attains its climax value at higher momenta for baryons than for mesons. These simple formulae for $v_2$ in the recombination picture hold true under the assumptions of coalescence of quarks with the same velocity (i.e. neglecting the width of the wave functions) and of completely direct production (i.e. neglecting the contributions from resonance decays). This motivated the picture of constituent-quark-number scaling of $v_2$ as a function of the transverse kinetic energy, which was observed to hold with good approximation (within ± 10% at intermediate-$p_T$) in the data at RHIC energies [110–112]. At the LHC, up to ±20% deviations from the exact scaling were observed [290]. After the precise measurements obtained at the LHC for heavy-flavour hadrons, the studies of the scaling of the flow coefficients with the number of constituent quarks could be extended to charm mesons and charmonia. Such studies (see e.g. those reported in Sect. 2.5.5) can shed light on the role of the hadronisation mechanism in the hierarchy observed in the $v_2$ of pions, D, and $J/\psi$ mesons at low- and intermediate-$p_T$.

It should be noted that the constituent-quark number scaling is a prediction based on naïve coalescence calculations with several simplifying assumptions. Deviations from this simple scaling and a consequent break-up of the quark-number scaling are expected to appear when some of these assumptions are relaxed by performing a full calculation with the Wigner function formalism and by accounting for multiple extra-considerations, such as: the effects of the
Figure 40 shows the $p_T$-differential yield ratios $D_s^+/D^0$ (left panel) and $\Lambda_c^+/D^0$ (right panel) measured in minimum-bias pp collisions [481–484] and in the 10% most central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [436,437]. The ratio between the $p_T$-differential yields of strange and non-strange D mesons in pp collisions does not show any marked $p_T$ dependence within the experimental uncertainties and is compatible with previous measurements at $e^+e^-$ colliders [485]. A hint of a larger $D_s^+/D^0$ ratio (at $2.3\sigma$ level of the combined statistical and systematic uncertainties) is observed for $p_T < 8$ GeV/c in central Pb–Pb collisions compared to pp collisions, which can be explained in the case of charm-quark hadronisation via recombination within a strangeness-rich medium. At higher $p_T$, the $D_s^+/D^0$ ratio converges to the values measured in pp collisions.

Unlike the ratio between charm meson yields, the $\Lambda_c^+/D^0$ ratio measured in minimum-bias pp collisions exhibits a $p_T$ dependence, which is around 0.4–0.6 at low-$p_T$ and decreases to 0.2–0.3 at high-$p_T$, similarly to what was observed in the light-flavour sector for the baryon-over-meson ratios $p/\pi$ and $\Lambda/K_S^0$ [482]. In addition, the measured $\Lambda_c^+/D^0$ ratio is significantly higher than the measurements done at $e^+e^-$ and ep colliders [485,486]. This enhancement, observed for different charm-baryon species in pp collisions [487–489], as discussed in Sect. 6.2.2, suggests that the fragmentation of charm quarks into hadrons is not universal across collision systems, and further mechanisms that enhance the baryon productions have to be taken into account with respect to those considered for the $e^+e^-$ and $e^\pm p$ predictions. A hint of enhancement in the baryon-over-meson ratio is observed for $4 < p_T < 8$ GeV/c in the 0–10% most central Pb–Pb collisions with respect to pp collisions. This possible enhancement at intermediate-$p_T$ is consistent both with the radial expansion of the system that imparts the same velocity to all particle species, boosting heavier particles to higher momenta, and the hadronisation of the charm quarks via the recombination mechanism that is expected to be dominant in this momentum region.

In Fig. 40, the $p_T$-differential $D_s^+/D^0$ and $\Lambda_c^+/D^0$ yield ratios are compared to theoretical predictions obtained with four different models. The Catania model [490,491] assumes that a colour-deconfined state of matter is formed in both pp and Pb–Pb collisions and implements the heavy-quark transport via the Boltzmann equation (see Sect. 2.4 for more details). As for the light-flavour particles, the hadronisation in the Catania model can occur via instantaneous coalescence, implemented through the Wigner formalism, in addition to the fragmentation. The coalescence mechanism is predicted to be the dominant one at low- and intermediate-$p_T$, while at high-$p_T$ charm quarks are expected to mainly hadronise via fragmentation. The Catania model reproduces within the uncertainties the measured $D_s^+/D^0$ ratios both in pp and Pb–Pb collisions. They also describe the $\Lambda_c^+/D^0$ ratios in

pp collisions while, in central Pb–Pb collisions, they slightly overestimate the measurements at low-$p_T$ and underestimate them at intermediate-$p_T$. A similar description of the heavy-quark transport and hadronisation in Pb–Pb collisions is provided by the TAMU model [492]. In this case, the charm-quark transport in a hydrodynamically expanding medium is described by the Langevin equation and the hadronisation via recombination is implemented with a Resonance Recombination Model (RRM) [454]. Within this formalism, the hadronisation proceeds via formation of resonant states when approaching the pseudo-critical temperature, whose rates are governed by the Boltzmann equation. In this case, the recombination process is not instantaneous and it is governed by a time scale which is the inverse of the width of the resonant states formed and is therefore different for different resonant states. The resonant states include heavy-flavour baryon states that have not been measured so far, but they are predicted by the Relativistic Quark Model (RQM) [494] and consistent with lattice-QCD predictions. It is also important to notice that, differently from the Wigner formalism, the RRM guarantees 4-momentum conservation for all $p_T$ values. In pp collisions, in contrast to the Pb–Pb case, the hadronisation in the TAMU model is not described with the RRM: the abundances of the different charm-hadron species are instead determined with a statistical hadronisation approach [439], which takes into account an augmented set of charm-baryon excited states based on guidance from the RQM and is able to reproduce the measured $\Lambda_c^+/D^0$ ratio. The TAMU predictions reproduce the magnitude and shape of the $p_T$-differential $\Lambda_c^+/D^0$, while they overestimate the $D_s^+/D^0$ ratios both in pp and Pb–Pb collisions. The POWLANG model [493] implements a Langevin-based transport of heavy quarks in the QGP followed by in-medium hadronisation. At the hadronisation stage, occurring around the QCD pseudo-critical temperature, charm quarks are recombined with light thermal quark or di-quark states from the medium into colour-singlet clusters. Low-mass clusters are then decayed into a charm hadron and a light particle, while heavy clusters are fragmented according to the Lund model [495,496]. In order to provide a benchmark relevant for pp collisions, the POWLANG model is compared to the results obtained with the POWHEG-BOX event generator [497] matched to PYTHIA 6 [76] for the parton shower and for the hadronisation, the latter being implemented only via fragmentation. In central Pb–Pb collisions the model overestimates the $D_s^+/D^0$ ratios, while it better describes the $\Lambda_c^+/D^0$ measurements. Finally, the predictions for the $D_s^+/D^0$ and $\Lambda_c^+/D^0$ ratios in Pb–Pb collisions obtained from an approach based on the statistical hadronisation model are reported in Fig. 40 as solid lines for the calculations using the list of charm-baryon states from the PDG [440], and as a dash-dotted line for the $\Lambda_c^+/D^0$ ratio obtained considering an augmented set of excited charm baryon states along with a larger charm production cross section. Such calculations
<figure><img src="image_47.png" /><figcaption>**Fig. 40** $D_s^+/D^0$ [436,483,484] (left) and $\Lambda_c^+/D^0$ [437,481,482] (right) yield ratios as a function of $p_T$ in pp collisions and in the 10% most central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV compared to dif-</figcaption></figure>

<figure><img src="image_48.png" /></figure>

ferent model calculations, namely Catania [490,491], TAMU [492], POWLANG [493], and the GSI-Heidelberg statistical hadronisation model [434]

use the GSI-Heidelberg SHM described in Sect. 2.3.1 for the hadron yields, while the $p_T$ spectra of charm hadrons are modelled with a core-corona approach [434]. The core contribution, especially important at low-$p_T$, is parameterised with a blast-wave function under the assumption of local thermal equilibrium of charm quarks in the fireball formed in the collision. Hence, the charm hadrons at hadronisation inherit the thermal motion of the charm quarks superimposed with the collective velocity of the hydrodynamically expanding QGP. The corona contribution is instead parameterised from measurements in pp collisions. Overall, this approach provides a fair description of the gross features of the data, even though it predicts a $p_T$ dependence of the $D_s^+/D^0$ ratio that is more pronounced than the measured one and it underestimates the measured $\Lambda_c^+/D^0$ ratio at intermediate-$p_T$. In particular, the calculations with an augmented set of charm baryons predict a larger $\Lambda_c^+/D^0$ ratio than the ones using the PDG list for $p_T < 6$ GeV/c, but both of them undershoot the data points in $4 < p_T < 8$ GeV/c.

The $p_T$-integrated $\Lambda_c/D^0$ ratios in Pb–Pb collisions in the two considered centrality classes were calculated by extrapolating the measured $\Lambda_c$ $p_T$-differential yields down to $p_T = 0$ as discussed in Ref. [437]. These ratios are compatible with the ones measured in pp and p–Pb collisions [481,482] within one standard deviation of the combined uncertainties. This behaviour is similar to the one discussed above for the $\Lambda/K_S^0$ baryon-to-meson ratio in the strangeness sector and suggests that, also in the charm sector, the modifications of the hadronisation in presence of the QGP mainly result in a redistribution of the $\Lambda_c$ and $D^0$ yields over $p_T$ rather than an over-

all increase in baryon production. However, considering the relatively large uncertainty of the $p_T$-integrated $\Lambda_c$ yield in Pb–Pb collisions, the current data do not allow for a discrimination among different charm-baryon formation scenarios.

Insight into the hadronisation of beauty quarks was obtained from recent measurements of the $p_T$-differential yield ratios of non-prompt $D_s^+$ and non-prompt $D^0$ mesons [484,498], i.e. D mesons originating from the decay of beauty hadrons. This study is complementary to the measurements of the $B_S^0/B^+$ ratios reported by the CMS Collaboration [499] for $p_T > 7$ GeV/c, which show a hint of enhancement of beauty-strange meson production in Pb–Pb collisions. The measurement of non-prompt D mesons provides sensitivity to the production yields of different B-meson species because in pp collisions about 50% of non-prompt $D_s^+$ mesons are produced in $B_S^0$ decays, while most of the non-prompt $D^0$ mesons originate from non-strange B-meson decays [484]. The measurements reported in Ref. [498] show a hint of a larger non-prompt $D_s^+/D^0$ yield ratio in central Pb–Pb collisions relative to pp interactions, with a significance of $1.7\sigma$. This is consistent with an enhanced production of beauty-strange mesons in heavy-ion collisions, as expected in a scenario in which beauty quarks hadronise via recombination with quarks from the strangeness-rich QGP. The TAMU model [500] can describe qualitatively the data points, capturing the enhancement of non-prompt $D_s^+$ mesons and its trend with $p_T$.

The models that implement the heavy-quark hadronisation via recombination and fragmentation in the QGP provide a good description of the main features of heavy-flavour hadron
production measurements down to low transverse momentum, supplying important additional insights into the hadronisation mechanisms.

### 2.3.3 The hadronic phase

The hadron-gas phase lasts approximately 5–10 fm/c from the chemical freeze-out, right after the hadronisation of the QGP, to the kinetic freeze-out, when all interactions cease and hadrons stream freely. The resonances with lifetimes of the same timescale are likely to decay before the kinetic freeze-out. They are therefore good probes of the dynamics of the hadronic phase [501]. In fact, the decay products of resonances are subject to elastic interactions in the hadron gas, which modify their momenta and prevent the reconstruction of the resonance signal by means of an invariant mass analysis. As a consequence, the measured resonance yield is suppressed with respect to the amount produced at the chemical freeze-out. In this context, it is also worth recalling that the line shapes of the invariant mass peaks of the reconstructed resonances in Pb–Pb collisions do not show any significant deviation, within the experimental uncertainties, with respect to the measurements in pp collisions and to the expectations based on the resonance mass and width from the PDG. The suppression effect due to rescattering can be compensated by regeneration processes, by which hadrons from the medium interact and form a resonance that will decay after the kinetic freeze-out. Regeneration could even be the dominant effect: in this case, the measured resonance yield would be enhanced. Resonance decays and regeneration reactions are not in balance, though, since chemical equilibrium is lost at the chemical freeze-out. A partial chemical equilibrium (PCE), however, has been proposed [502], for which the decays and the regeneration of the short-lived resonances obey the law of mass action, i.e. the abundances of the different resonances stay in equilibrium with the particles that are formed in the decays of these resonances. In this approach, the yields of resonances are related to the temperature at the kinetic freeze-out (see also [503]).

In a simple scenario of a sudden and instantaneous kinetic freeze-out common to all particle species, the amount of suppressed (or enhanced) yield that is measured depends on (i) the lifetime of the resonance, (ii) the cross sections for rescattering and regeneration processes, and (iii) the time span (duration) of the hadronic phase. The latter is expected to be longer for central collisions, where larger system volumes are created, than for peripheral collisions. The best figure of merit to quantify the net effect is the yield ratios of resonances to long-lived hadrons with the same quark composition; by measuring such ratios with resonances of increasing proper lifetime, like $\rho(770)^0$ ($\tau^{\text{rf}}$ = 1.3 fm/c), $K^*(892)^0$ ($\tau^{\text{rf}}$ = 4.16 fm/c), $\Lambda(1520)$ ($\tau^{\text{rf}}$ = 12.6 fm/c) and $\phi(1020)$ ($\tau^{\text{rf}}$ = 46.3 fm/c) [440], one can make use of different

sensitivities to the hadronic phase and its interactions. For simplicity, these resonances will be denoted as $\rho$, $K^*$, $\Lambda^*$, and $\phi$ in the following. The results of this investigation are shown in the left panel of Fig. 41, where the ratios of $p_T$-integrated$^8$ yields $\rho/\pi$ [504], $K^*/K$ [505], $\Lambda^*/\Lambda$ [506,507] and $\phi/K$ [505] are presented as function of the cubic root of the charged-particle multiplicity at $|y| \approx 0$, $(dN_{ch}/d\eta)^{1/3}$, which represents the radial extent of the system (see Fig. 18 in Sect. 2.1.5). Results are shown for Pb–Pb collisions at two different colliding energies, $\sqrt{s_{NN}}$ = 2.76 TeV and 5.02 TeV, and for pp collisions at $\sqrt{s}$ = 2.76 TeV and 7 TeV. Note that the $\phi$ meson is a hidden-strangeness hadron but, as far as hadro-chemistry is concerned, it behaves like the kaon that has strangeness $S$ = 1.

If no final-state effects such as rescattering and regeneration intervened during the hadron-gas phase to modify the resonance yields, the ratios in Fig. 41 would be constant as a function of $(dN_{ch}/d\eta)^{1/3}$ due to the similar production mechanisms for particles with the same strangeness and quark composition. What is instead observed in Fig. 41 is that the yields of $\rho$, $K^*$, and $\Lambda^*$ are progressively more suppressed with respect to the reference long-lived hadrons when going from peripheral to central collisions, whereas the $\phi/K$ ratio is almost flat. For the $\rho/\pi$, $K^*/K$, and $\Lambda^*/\Lambda$ ratios, the difference between the values measured in pp and in the most central collisions is larger than $3\sigma$ of the combined statistical and systematic uncertainties. The fact that the yield ratios involving $\rho$, $K^*$, and $\Lambda^*$ resonances decrease with increasing $(dN_{ch}/d\eta)^{1/3}$ suggests that, in the hadronic phase, the rescattering of the decay products is dominant over regeneration and that it is more effective in central collisions, where the hadronic phase lasts longer. The fact that the $\phi/K$ ratio remains approximately constant indicates instead that $\phi$ mesons are not affected by final state effects in the hadronic phase. This is likely due to the $\phi$ meson lifetime, one order of magnitude larger than that of the $K^*$ resonance, which causes $\phi$ mesons to decay after the kinetic freeze-out. Hence, the $\phi$ decay products do not rescatter and no regeneration of $\phi$ mesons occurs (since the regeneration of a resonance is likely to originate from rescatterings of resonance decay products). This is consistent with the PCE picture of Ref. [502], where the yield of the long-lived $\phi$ mesons relative to pions is expected to have a very mild dependence on the kinetic freeze-out temperature, i.e. to be minimally modified during the hadronic phase.

The yield ratios in Fig. 41 are also compared to EPOS3 simulations with and without a hadronic cascade phase mod-

<footnote>$^8$ Like for the ground states ($\pi$, K, $\Lambda$) used in the present ratio, the fraction of extrapolated yield to zero $p_T$, in pp as well as in Pb–Pb, remains small to moderate for the various resonances (typically 5% for $K^*(892)^0$, 6% to 16% for $\Lambda(1520)$, 14% for $\phi(1020)$ and 20–30% for $\rho(770)^0$).</footnote>
<figure><img src="image_49.png" /><figcaption>**Fig. 41** (Left) Ratios of midrapidity yields between resonances and ground-state hadrons of similar valence-quark content, as a function of $(dN_{ch}/d\eta)^{1/3}$. These are obtained for inelastic pp collisions $(\sqrt{s} = 2.76$ TeV and 7 TeV [508–511]) and for Pb–Pb collisions $(\sqrt{s_{NN}} = 2.76$ TeV [386,504,507] and 5.02 TeV [273,402,455]) ranging from peripheral to most central overlaps. Error bars and boxes represent the statistical and the total systematic uncertainty,</figcaption></figure>

<figure><img src="image_50.png" /></figure>

respectively. Model predictions are from EPOS3 with and without UrQMD [512]. (Right) Transverse-momentum distributions of $K^*(892)^0$ at $\sqrt{s_{NN}} = 5.02$ TeV [455] for central and peripheral collisions compared with predictions from EPOS3 (with and without UrQMD) and from a blast-wave model with a dedicated treatment of resonances (BW FastReso) [277]

elled by UrQMD [512]. The ratios obtained from EPOS3 without UrQMD are independent of multiplicity as expected for all resonances, including a flat behaviour for $\phi/K$, that further supports the similarity between $\phi$ and kaon as far as hadro-chemistry is concerned. When the hadronic cascade phase (UrQMD) is switched on, the model reproduces, at least qualitatively, the observed decreasing trend of the $\rho/\pi$, $K^*/K$ and $\Lambda^*/\Lambda$ ratios with increasing multiplicity. The $\phi/K$ ratio remains flat also with UrQMD, confirming the hypothesis of no final-state effects acting on the $\phi$ meson.

The hadronic rescattering effect is expected to be momentum dependent, with greater strength for low-$p_T$ resonances ($p_T < 2$ GeV/c) [275]. This can be studied by comparing the measured resonance $p_T$ spectra with the expectations from different models. The $p_T$ spectra of $K^*$ resonances are shown in the right panel of Fig. 41 for central (0–10%) and peripheral (70–80%) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. They are compared with predictions from the EPOS3 event generator (with and without UrQMD) and from an improved blast-wave model (BW FastReso) [277] hinging on the FastReso computation for resonance production [278] (described in Sect. 2.2.1). In the BW FastReso calculations, the common radial velocity and temperature at the kinetic freeze-out are evaluated from a simultaneous blast-wave fit to the measured $p_T$ spectra of pions, kaons, and protons and used to predict the shape of resonance spectra. The resulting distributions are then normalised so that their integrals are equal to the

measured yield of charged kaons in Pb–Pb collisions multiplied by the $K^*/K$ ratios given by a SHM fit to ALICE data [122]. For peripheral collisions, model predictions are in fairly good agreement with data over the full measured range. For central collisions, agreement is found only for $p_T > 2$ GeV/c, whereas at low-$p_T$, as expected, the measured yields are reduced due to the rescattering effects. EPOS3 without UrQMD fails to describe the data at low-$p_T$, for both central and peripheral collisions, whereas good agreement is obtained at high-$p_T$. When switching UrQMD on, a suppression is indeed observed at low-$p_T$, with increasing strength for decreasing $p_T$.

Rescattering in the hadronic phase is seen as the most probable cause of the measured suppression. This suggests to further use the $\rho/\pi$, $K^*/K$, and $\Lambda^*/\Lambda$ ratios to estimate the time span between chemical and kinetic freeze-out ($\tau_{kin} - \tau_{chem}$). This estimation is performed by means of an exponential decay law [455], $r_{kin} = r_{chem} \times e^{-(\tau_{kin} - \tau_{chem})/\tau_{res}}$, under the assumptions of negligible regeneration, and sudden chemical and kinetic freeze-outs occurring at the same instant for all particle species. The ratios $r_{kin}$ and $r_{chem}$ are any of the $\rho/\pi$, $K^*/K$ or $\Lambda^*/\Lambda$ measured yield ratios at the kinetic and chemical freeze-out, respectively. In the calculations, the ratio in pp collisions is taken as a substitute for the ratio $r_{chem}$ at the chemical freeze-out, while $r_{kin}$ is the measured ratio in Pb–Pb collisions. The lifetime of the resonance is dilated by the Lorentz factor, when going from the
resonance rest frame to the laboratory reference frame; such a factor is associated with $\langle p_T \rangle$ of the individual $p_T$-spectrum of each resonance, so that $\tau_{res} = \tau_{res}^{rf} \cdot \gamma_{res}^{\langle p_T \rangle}$. Results for the estimated duration of the hadronic phase are shown in the left panel of Fig. 42 as a function of $\langle dN_{ch}/d\eta \rangle^{1/3}$. For all particles an increasing trend is observed, which mirrors the decreasing behaviour of the yield ratios for increasing system size. Unlike the naïve expectation of a hadronic-phase duration common to all resonances for a given system size, different values are obtained from the $\rho$, $K^*$, and $\Lambda^*$ yields. The estimated times differ by up to an order of magnitude, being larger for longer-lived resonances. The predictions for the time span between chemical and kinetic freeze-out obtained from EPOS3 simulations with UrQMD [512] are also shown for reference in the left panel of Fig. 42. The hadronic-phase duration from EPOS3+UrQMD has a similar magnitude (of few fm/c) and trend with multiplicity as compared to the estimates obtained from the measured resonance yields, being closer to the estimate calculated from the $\Lambda(1520)$, which is the longest lived among the considered resonances.

The reason why the time durations obtained from $\rho/\pi$ and $K^*/K$ ratios are shorter than the time calculated from the $\Lambda^*/\Lambda$ ratio might be due to regeneration effects, neglected in the simple exponential decay model and affecting more the $\rho$ and $K^*$ resonances than the $\Lambda^*$. If regeneration plays a role, the hadronic-phase duration estimated from the yield ratios can be interpreted as the lapse of time between the on-average delayed resonance production due to regeneration and the kinetic freeze-out, therefore representing a lower limit for the duration of the hadronic phase. In order to estimate the actual duration of the hadronic phase, the delay due to the regeneration has to be added to the time span estimated from the ratios.

Sensitivity to the delay in the production of $K^*$ due to regeneration can be obtained from femtoscopy measurements (see Sect. 2.1.5 and also [248]). In particular, it is expected from hydrokinetic model (HKM) simulations [252,515] that the interferometry radii are little affected by the decay of $K^*$ resonances, but they are sensitive to regeneration, which shifts the emission time of pions and kaons. Since pions from $K^*$ decays have a negligible effect on the pion interferometry measurements due to the large amount of primary pions, a longer emission time for kaons than for pions is expected due to regeneration through the $K^*$ resonance. Regeneration can also explain [516] the $m_{\tau^-}$ scaling violation seen in Fig. 20 in Sect. 2.1.5, where the radii of kaons are systematically higher than those of pions in the same centrality class. The difference in the emission time of pions and kaons is quantified by the pion-kaon emission asymmetry, presented in the right panel of Fig. 42 for Pb-Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV as a function of $\langle dN_{ch}/d\eta \rangle^{1/3}$ [513]. The non-zero emission asymmetry

observed in the figure is the consequence of several effects, including the collective expansion of the system, the presence of short-lived resonances decaying into the considered particles, and the radial flow of these resonances. All these features are only qualitatively reproduced by a (3+1) viscous hydrodynamic model [517], coupled to the statistical hadronisation, resonance decay, and propagation code THERMINATOR 2 [514]. In order to reproduce quantitatively the measured emission asymmetry, an emission delay $\Delta\tau$ of 1.0–2.1 fm/c for the kaons has to be specifically included in the model. This $\Delta\tau$ can be interpreted as the delay due to the decay of $K^*$ resonances produced by regeneration in the late stages of the hadronic phase.

The interpretation of the femtoscopy results suggests that regeneration plays a relevant role until the late stages of the hadron gas evolution. This challenges the assumption of the simple formula used to estimate the duration of the hadronic phase from the resonance yields, in which regeneration was neglected, and confirms that regeneration provides a plausible explanation for the different values estimated from different resonances. A further consideration is that the findings from the resonance-yield and femtoscopy measurements may disfavour a scenario of a sudden kinetic freeze-out of all particle species at the same time. A more complex scenario could be figured out in which the kinetic freeze-out spans over a range of time, possibly different for different particle species. This is supported by hydrodynamic calculations coupled with UrQMD to model the hadron cascade [518] and by considering that the decoupling of particles from an expanding system is a continuous process, which takes place over a finite range of temperatures [502].

### 2.3.4 Conclusions

**Equilibration of light-flavour hadrons.** High-precision measurements of the yields of several species of hadrons composed of light (up, down, and strange) quarks were carried out, spanning nine orders of magnitude in production rate from pions to light nuclei and hypernuclei. The abundances are well described by statistical hadronisation models, confirming the thermal nature of particle production in heavy-ion collisions already observed at lower collision energies. The thermodynamic (macroscopic) parameters characterising the chemical freeze-out ($T_{chem}$ and $\mu_B$) follow the trend with $\sqrt{s_{NN}}$ established at SPS and RHIC. In particular, values of $T_{chem} \approx 156$ MeV and $\mu_B \approx 0$ are extracted from the SHM analysis of the hadron abundances measured in Pb-Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV. The high precision of the data revealed small tensions between the measured data points and the SHM predictions for protons, and to a lesser extent for strange baryons. This motivated several phenomenological studies, which pointed out the importance of including
<figure><img src="image_51.png" /><figcaption>**Fig. 42** (Left) Hadronic phase duration (time span between chemical and kinetic freeze-out) as a function of $\langle dN_{ch}/d\eta \rangle^{1/3}$ calculated from the $\rho/\pi$, K*/K and Λ*/Λ ratios in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [386,504,507]. The $\rho/\pi$ results are shifted horizontally by 0.1 units for visibility. (Right) Pion–kaon emission asymmetry</figcaption></figure>

<figure><img src="image_52.png" /></figure>

for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV as a function of $\langle dN_{ch}/d\eta \rangle^{1/3}$ [513]. The shaded areas show predictions from the THERMINATOR 2 model with default and selected values of additional delay $\Delta\tau$ for kaons [514]

the interactions among hadrons (relevant for temperatures close to $T_{pc}$) in the modelling of the hadron-resonance gas.

**Sensitivity of heavy quarks to thermalisation.** Heavy quarks (charm and beauty) are not produced in chemical equilibrium with the quark–gluon plasma. Nevertheless, the yields of charm hadrons can be described by the SHM using as input the production cross section of charm quarks in the initial hard-scattering processes, which determines the charm content of the fireball, and accounting for charm conservation in the fireball evolution. This supports a scenario in which charm quarks are to some extent thermalised in the QGP.

**Mechanisms of light-flavour hadron production in different $p_T$ domains.** In the light-flavour sector, the measurements in a wide momentum interval of the $p_T$-differential spectra of mesons and baryons, the baryon-to-meson ratios, as well as the anisotropic flow harmonics of different hadron species, allowed for a significant step forward in the understanding of the mechanisms of hadron formation at a microscopic level. At high-$p_T$ (> 8–10 GeV/c), the measurements in Pb–Pb collisions show that the $v_2$ coefficient is the same for all particle species and that particle yield ratios are consistent with those measured in pp collisions and in the fragmentation of jets. This confirms that, in this region, hadron formation is dominated by fragmentation. At low-$p_T$ (< 2 GeV/c), the measured spectra and flow coefficients are described by hydrodynamic calculations and are thus understood in a picture of a system which evolves in equilibrium. The intermediate-$p_T$ region, where equilibrium does not hold,

provides a window into the scenario in which the momentum and the angular distributions of hadrons can preserve a memory of their evolution, providing in particular sensitivity to the mechanism of hadron formation at the microscopic level. The features of the data in this $p_T$ region are qualitatively, and to a good extent also quantitatively, captured by models that include a hadronisation via recombination of effective constituent quarks at the phase boundary. The recombination involves soft partons emerging from the collective expansion of the QGP as well as partons which are produced in hard-scattering process and are quenched while traversing the hot and dense plasma.

**Charm-hadron production in different $p_T$ domains.** Measurements of the $p_T$-differential yields of several charm hadron species in Pb–Pb collisions show hints of an enhancement in the production charm-strange mesons ($D_s^+$) and charm baryons ($\Lambda_c^+$) at low and intermediate momentum relative to $D^0$ mesons with respect to pp collisions. This effect is naturally expected for the hadrons produced via quark recombination, and the measured particle yield ratios are described by models implementing heavy-quark hadronisation via both recombination and fragmentation. This is consistent with the conclusions drawn from measurements of $J/\psi$ production and angular distributions, which demonstrate the important role of the recombination mechanism to generate hidden-charm states during the QGP expansion or at the phase boundary.

**Resonance production altered by the hadronic phase.** Measurements of the yield of strongly-decaying resonances as
a function of the collision centrality show a suppression of short-lived resonances ($c\tau \sim 1$ to a few fm/c), increasing from peripheral to central collisions. This can be interpreted as a result of the rescattering of the resonance decay products in the hadronic phase, which dominates over regeneration. This allowed for a simple quantitative estimation of the duration of the hadronic phase, spanning from the chemical to the kinetic freeze-out, under the assumption that regeneration can be neglected. The hadronic phase duration estimated with this simple approach ranges from $\approx 1$ to $\approx 10$ fm/c, increasing from peripheral to central collisions. However, the times calculated from the yields of different resonances differ by up to an order of magnitude, and are larger for longer-lived resonances. These results, along with the ones obtained from kaon-pion femtoscopy, indicate that the regeneration contribution is likely to play a relevant role. Moreover, these results challenge the simple scenario of a sudden kinetic freeze-out of all particle species at the same time and may indicate that the decoupling of particles from the expanding hadron gas is a continuous process, which takes place over a range of times and temperatures that is different for different hadron species.

## 2.4 Partonic interactions in matter

This section discusses the application of processes generated in high momentum-transfer (high-$Q^2$) interactions as hard probes of the QGP. Hard probes are created far out of equilibrium with the QGP, and their in-medium interactions can be used to measure effects due to dynamical processes such as energy transport and equilibration. Because of their high-momentum they are short-wavelength phenomena, and their in-medium interactions are therefore sensitive to the microscopic structure of the QGP and its quasi-particle nature. It is important to note, however, that high-$Q^2$ refers only to the process of generation of such probes but not to their in-medium interactions, which in general span a broad range in $Q^2$.

The focus of this section is on the production and propagation in the QGP of high-momentum jets, where the high-$Q^2$ scale requires high-$p_T$, and heavy quarks, i.e. charm and beauty quarks, where the high-$Q^2$ scale is imposed by the quark masses. The essential characteristics of high-$Q^2$ processes as probes of the QGP are as follows:

1. Their production cross sections in pp collisions are calculable with controlled and improvable accuracy using the tools of pQCD;

2. Based on considerations of the uncertainty principle and QCD factorisation, high-$Q^2$ processes are expected to occur at the earliest stage of a heavy-ion collision, prior to equilibration of the QGP, and therefore probe its hottest and densest phase;

3. The interaction between a hard probe and the QGP can be calculated theoretically starting from the pQCD formulation for elementary collisions or from transport theory, giving a firm conceptual basis to such modeling approaches.

**Open heavy flavour in the QGP.** Because the masses of charm and beauty quarks are much larger than both $\Lambda_{QCD}$ and the medium temperature, their production even at low $p_T$ is governed predominantly by hard scattering processes early in the evolution of the system and additional thermal production is negligible. Heavy-flavour measurements probe the medium over a wide range of wavelengths, depending upon $p_T$. Theoretically, at low $p_T$ the interaction is governed largely by elastic (collisional) processes, whereas inelastic (radiative) processes predominate at high-$p_T$ (see [70,431] for recent topical reviews).

At long wavelengths, i.e. quark momenta less than a few GeV/c, heavy quarks exchange energy and momentum via multiple soft and incoherent collisions within the hydrodynamically expanding medium, picking up collective flow and approaching thermalisation. The relaxation time $\tau_Q$ of heavy quarks is expected to be close to the lifetime of the QGP, with $\tau_Q$ being significantly longer for beauty than for charm quarks due to their larger mass. Heavy-flavour observables thereby retain a memory of quark-medium interactions in an early phase of the QGP. The long-wavelength physics of the interaction of heavy quarks with the medium is treated theoretically in a diffusion approach based on Fokker-Planck or Langevin dynamics. The typical momentum exchange is small in individual scattering processes and, therefore, the dynamical behaviour can be described in terms of Brownian motion in the QGP medium [519,520]. In the intermediate-wavelength region, corresponding to heavy-flavour hadron transverse momenta up to about 10 GeV/c, measurements probe the mechanism of hadronisation from a deconfined medium (see Sect. 2.3). At short wavelengths, i.e. heavy-flavour hadron $p_T > 10$ GeV/c, heavy-quark interactions with the QGP probe the physics of jet quenching, notably the quark-mass dependence of medium induced energy loss, which is expected to be influenced by the dead-cone effect [40,521].

The theoretical description of heavy-flavour propagation through the medium is commonly realised in transport models, which start from the production of heavy quarks in hard scattering processes, followed by transport in a hydrodynamically expanding medium, hadronisation, and interactions during the hadronic phase.

The Brownian motion of heavy particles in a medium of light constituents is characterised by a diffusion process. The coupling between the medium and the heavy particle in this regime can be expressed by the dependence of the average
squared displacement of the particle, $\langle \vec{r}^2 \rangle$, on the time t

$$ \langle \vec{r}^2 \rangle = 6 D_s t \qquad (13) $$

where the spatial diffusion coefficient $D_s$ encodes the transport properties of the medium.

The description of heavy-quark transport in the QGP starts from the Boltzmann equation, which describes the space-time evolution of the heavy-quark phase space distribution in terms of external forces and a collision integral, encoding the interaction of the heavy quark with the medium partons. For moderate medium temperatures, the momentum transfer between the heat bath and a heavy quark is small in individual interactions, motivating the picture of soft and incoherent scatterings. In that case, the Boltzmann equation can be approximated by the Fokker-Planck equation [431]. This formulation incorporates momentum-dependent transport coefficients $A(\vec{p})$ and $B(\vec{p})$ characterising momentum friction (drag) and diffusion, which are denoted $\gamma$ and $D_p$ in the non-relativistic, momentum-independent limit. In this limit one also obtains the dissipation-fluctuation theorem, often called Einstein relation,

$$ D_p = m_Q \gamma T, \qquad (14) $$

which highlights the role of momentum frictions and diffusion by connecting the temperature $T$ of the heat bath with the momentum distribution of heavy quarks with mass $m_Q$.

Alternatively, heavy-quark transport in the QGP is implemented as a Langevin process with drag and diffusion terms. To ensure that the heavy-quark momentum distribution asymptotically approaches the equilibrium limit, the Einstein relation is usually enforced in such calculations, i.e. only the drag coefficient is treated as an independent parameter, with the momentum diffusion coefficient obtained via the equation above. In turn, the drag and spatial diffusion coefficients are related via

$$ D_s \propto \frac{T}{m_Q \gamma}. \qquad (15) $$

Since $\gamma$ is approximately proportional to $1/m_Q$, $D_s$ is almost independent of the quark mass and is therefore a parameter that characterises the transport properties of the medium. Furthermore, $D_s$ is proportional to the relaxation time $\tau_Q$ of heavy quarks in the medium, i.e. $\tau_Q = (m_Q/T)D_s$. If $D_s$ is sufficiently small, the relaxation time is smaller than the expansion rate of the medium, meaning that a heavy quark is likely to remain in the same fluid cell throughout the expansion of the medium. Consequently, heavy quarks pick up the large collective flow of the medium. If, on the contrary, $D_s$ and the relaxation time are large, both flow and yield suppression of heavy-flavour probes are small. Therefore, this QGP transport property can be determined by comparing measurements of the heavy-flavour yield suppression and flow

coefficients with corresponding results from model calculations. The product of $D_s$ and the thermal wavelength of the medium, $\lambda_{th} = 1/(2\pi T)$, is a dimensionless quantity that is proportional to the specific shear viscosity $\eta/s$; such investigations thereby also constrain $\eta/s$ [69,520].

ALICE measures open heavy-flavour production in a variety of channels. Heavy-flavour hadrons are fully reconstructed from their hadronic decays [484,522–530]. In addition, heavy-flavour production is measured using semileptonic decays into electrons [531–534] and muons [535–537], including partial reconstruction of semi-electronic weak decays of heavy-flavour baryons [488,489,526]. In addition, beauty production is measured via non-prompt $J/\psi$ mesons from the decay mode $B \to J/\psi + X$ [538–540] and non-prompt D mesons from the decay mode $B \to D+X$ [484,541]. For many of these measurements the excellent spatial resolution of the ITS [142] is instrumental since it allows to resolve the decay vertex of charm and beauty hadrons from the primary collision vertex.

**Jets in the QGP.** A scattered parton with $p_T$ greater than a few GeV propagates and evolves independently of other products of the same high-$Q^2$ interaction, starting at time earlier than 1 fm/c. The QGP has an extended size, with a lifetime in the order of 10 fm/c (see Sect. 2.1.5), so that the jet shower propagates through the expanding and cooling QGP. During this process the jet shower itself evolves and its colour-charged constituents interact with the colour-charged constituents of the QGP, resulting in modification of the shower. Such modifications, called “jet quenching,” are observable experimentally and calculable theoretically; comparisons of jet quenching data and calculations provide unique, penetrating probes of QGP structure and dynamics. See Refs. [542–544] for recent reviews of jet quenching.

Experimentally, jet quenching is manifest in several ways:

1. Medium-induced energy transport to large angles to the hard parton or jet direction, commonly called “energy loss”, observed through inclusive yield suppression;

2. Medium-induced modification of the distribution of jet constituents, observed through the radial energy profile, jet substructure, and fragmentation functions;

3. Jet centroid deflection due to soft multiple scattering or scattering from quasi-particles in the QGP, observed as medium-induced acoplanarity in coincidence measurements;

4. Response of the QGP medium to deposited energy, observed as energy and momentum flow over a large range in phase space relative to the jet axis.

This many-pronged approach to quantifying jet quenching is a valuable opportunity: it must provide a consistent pic-
ture of jet quenching, thereby constraining significantly our understanding of its underlying processes.

Theoretically, interactions of high-energy partons with the QGP are broadly categorised as elastic interactions between the propagating parton and the QGP constituents (2 → 2 scatterings), and inelastic scattering, or medium-induced gluon radiation events (2 → 3 processes). These medium-induced processes are interleaved with the spontaneous splittings of the parton shower process that also occur in pp and other elementary collisions without a QCD medium (vacuum emissions).

At high-$p_T$, radiative processes are the dominant energy loss mechanism. This has been explored for instance in the BDMPS-Z framework [66,545], which shows that interfer- ence between scattering and emission processes generate a characteristic dependence of the energy loss on the square of the in-medium path length. Various calculational approaches have been developed: multiple soft scattering approximation (BDMPS) [66,67]; opacity expansion (GLV) [62]; thermal- field theory (AMY) [546]; higher twist (HT) [547,548]; and Soft-Collinear Effective Theory with Glauber gluons (SCETg) [549–552]. While these different formalisms give numerically different results, their relationship is largely understood. In some cases explicit correspondence between the calculations has been established [542,553], for example by evaluating the all-order opacity expansion [554,555].

Each such formulation has its own natural parameter char- acterising the strength of the jet-medium interaction, the Debye screening mass, or the effective value of the strong coupling. Different formalisms can be compared via the transport coefficient $\hat{q}$, which is the mean squared transverse momentum transfer $q_\perp$ per unit path length. In models, $\hat{q}$ is calculated as the ratio of $\langle q_\perp^2 \rangle$ and mean free path $\lambda$, or by averaging the scattering rate $d\Gamma/dq_\perp dz = \rho d\sigma/dq_\perp$,

$$ \hat{q} = \frac{\langle q_{\perp}^2 \rangle}{\lambda} = \int d^2 \mathbf{q}_{\perp} q_{\perp}^2 \frac{d\Gamma}{d\mathbf{q}_{\perp} dz}. \qquad (16) $$

Theoretical calculations indicate that the following char- acteristics are expected for medium-induced radiative energy loss:

- **Path length dependence:** in the BDMPS formalism, for short path lengths the medium-induced energy loss varies linearly with $\hat{q}$ and quadratically with path length, which arises from the LPM effect [66], $\langle \Delta E \rangle \propto \alpha_S \hat{q} L^2$. Sim- ilar dependencies are found for other formalisms [553, 554].

- **Colour-charge dependence:** the energy loss is propor- tional to the colour factor of the propagating parton. Energy loss is expected to be 9/4 times larger for glu- ons than for quarks.

- **Quark mass dependence:** the phase space for gluon emission is limited to the region where the gluon does

not outrun the charge source, implying a minimum emis- sion angle for gluons (dead-cone effect, see Sect. 6.3). Consequently, charm and bottom quarks are expected to lose less energy than light quarks at moderate momen- tum, i.e. at a momentum less than a few times the quark mass.

While analytical calculations are required to connect mea- surements to the fundamental underlying theory, such calcu- lations do not provide a straightforward way to implement kinematic limits and they have to be worked out specif- ically for each observable. Monte Carlo event generators such as JEWEL [64], JETSCAPE [556] and the Hybrid Model [557,558] provide a complementary set of tools, gen- erating full events to which any analysis algorithm can be applied.

Further detailed discussion of theoretical approaches to jet quenching and their implementation in numerical modeling can be found in Ref. [543] and references therein. Extraction of the in-medium jet transport coefficient $\hat{q}$ from the com- parison of models and data can be found in Refs. [559–562].

ALICE has reported jet measurements using both charged- particle tracking and cluster energy from the highly granu- lar Electromagnetic Calorimeter (EMCal) [164,165,563], as well as jet measurements of charged-particle jets [564–572]. For coincidence measurements between a trigger hadron and recoil jets, ALICE employs a semi-inclusive approach. In these measurements the large uncorrelated jet background at low-$p_T^{jet}$ and large jet radius $R$ is suppressed by using a new observable, $\Delta_{recoil}$, which is the difference of two such trigger-normalised distributions in exclusive intervals in hadron trigger $p_{T,\text{trig}}$ ($\text{TT}_{\text{sig}}$ and $\text{TT}_{\text{ref}}$) [568]:

$$ \Delta_{\text{recoil}} = \frac{1}{N_{\text{trig}}^{\text{AA}}} \frac{d^3 N_{\text{jet}}^{\text{AA}}}{d p_{\text{T,jet}}^{\text{ch}} d \Delta \varphi d \Delta \eta} \Big|_{p_{\text{T,trig}} \in \text{TT}_{\text{Sig}}} - c_{\text{ref}} \frac{1}{N_{\text{trig}}^{\text{AA}}} \frac{d^3 N_{\text{jet}}^{\text{AA}}}{d p_{\text{T,jet}}^{\text{ch}} d \Delta \varphi d \Delta \eta} \Big|_{p_{\text{T,trig}} \in \text{TT}_{\text{Ref}}}, \qquad (17) $$

where $N_{\text{trig}}^{\text{AA}}$ is the number of trigger hadrons, $p_{T,jet}^{\text{ch}}$ is the recoil jet $p_T$, $\Delta\varphi$ is the azimuthal separation between trigger hadron and recoil jet axis, and $c_{\text{ref}}$ is a constant determined from data to account for finite area effects [568]. This data- driven suppression of uncorrelated jet yield enables measure- ments in central Pb–Pb collisions of recoil jets with large $R$ and low-$p_T^{jet}$, without fragmentation bias.

### 2.4.1 Heavy-quark transport and diffusion measurements

In this section ALICE measurements of heavy-quark diffu- sion in the QGP and their comparison to theoretical calcula-
tions are presented. Theoretical concepts and experimental
techniques are sketched in Sect. 2.4.

Figure 43 shows the nuclear modification factor $R_{AA}$ and the elliptic-flow parameter $v_2$ as a function of $p_T$ for prompt D mesons in central (0–10%) and semi-central (30–50%) Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [435,573]. These measurements are compatible with previous results by the ALICE and CMS Collaborations at the same $\sqrt{s_{NN}}$ [528,529,574–576].

At low- and intermediate-$p_T$, the $p_T$-differential yields and azimuthal distributions of hadrons carrying heavy quarks are expected to be governed by collisional interactions within the quark–gluon plasma. In order to test this expectation the measurements are compared in Fig. 43 with various model calculations based on charm-quark transport in a hydrodynamically expanding QGP (TAMU [492], LIDO [577], POWLANG [578–580], PHSD [581], MC@sHQ [582], Catania [583,584], LBT [585,586], LGR [587], and DAB-MOD [588]). All models are qualitatively in agreement with the data, although tension is observed in particular at low-$p_T$.

The models differ in their implementation of the interac-
tion between charm quarks with the medium (Boltzmann vs.
Langevin transport, inclusion of collisional and/or radiative
interactions), nuclear PDFs and shadowing, bulk evolution
of the medium (ideal or viscous hydrodynamics, Boltzmann
quasiparticles or parton transport, off-shell parton transport),
charm hadronisation (fragmentation and/or recombination
or in-medium string formation), and hadronic scattering. In
particular, in the TAMU, POWLANG, PHSD, and Catania
calculations the interactions between the charm quarks and
the medium constituents are modelled with collisional pro-
cesses only, while the MC@sHQ, LBT, LIDO, DAB-MOD,
and LGR models consider radiative processes as well. All the
models include the hadronisation of the charm quark via coa-
lescence, in addition to the fragmentation mechanism. With
the exception of DAB-MOD, all the models employ nuclear
PDFs in the calculation of the initial $p_T$ distributions of charm
quarks in order to include initial-state effects. Initial-state
event-by-event fluctuations are included in the POWLANG,
LIDO, PHSD, MC@sHQ, LBT, and DAB-MOD models. In
order to disentangle the effects of elastic and radiative inter-
actions of the heavy quarks and of hadronisation, calculations
of the LIDO and LGR models are compared for configura-
tions including all physics processes with those excluding
radiative processes or hadronisation via coalescence. Fig-
ure 44 shows the comparison of these model variants with
the data from Fig. 43.

The predictions for prompt D-meson $R_{AA}$ and $v_2$ in the
range $p_T < 4-5$ GeV are similar for LGR and LIDO, with
and without radiative processes. They describe reasonably
well the $R_{AA}$ and slightly underestimate the $v_2$. This indicates
that elastic collisions are the predominant interaction process
in that $p_T$ range. Similar conclusions are drawn from mod-

els that only contain collisional processes (TAMU, PHSD,
Catania, POWLANG).

In this $p_T$ range, which is dominated by elastic collisions,
charm quarks may acquire radial and anisotropic azimuthal
flow, due to multiple interactions within the QGP medium.
Momentum exchanges between charm quarks and the heat
bath in such interactions typically have a magnitude similar
to the medium temperature, $q^2 \approx T^2$. This value is smaller
than the thermal momentum, due to the large charm-quark
mass ($m_{\text{charm}}$). Heavy quarks therefore experience Brow-
nian motion in the QGP, with a thermal relaxation time,
$\tau_{\text{charm}} = \tau_{\text{th}} m_{\text{charm}}/T$, that is significantly longer than the
thermalisation time of the bulk medium ($\tau_{\text{th}}$). In other words,
heavy-flavour hadrons probe the equilibration process. At
high $p_T$, the LGR model underestimates the energy loss if
radiative processes are not included. The role of radiative
processes in describing heavy quark production and flow at
higher $p_T$ is discussed in Sect. 2.4.2.

Hadronisation effects must also be taken into account for modelling D-meson $R_{AA}$ and $v_2$. Jet fragmentation is expected to dominate D-meson production at very high- $p_T$, while at intermediate-$p_T$ (~ few GeV), recombination effects are expected to play a significant role (see Sect. 2.3). In the TAMU model, hadronisation at intermediate-$p_T$ is implemented as a process in which heavy quarks interact with light quarks in the QGP, leading to formation and dissociation of broad bound hadronic states whose rates are governed by the Boltzmann equation [492]. In the POWLANG model, a heavy quark propagates stochastically in the fireball, which simultaneously expands and cools. Once the heavy quark is in a fluid cell whose temperature is below the decoupling temperature $T_{\text{dec}}$ of the medium, it is forced to hadronise with a light quark extracted from the thermal momentum distribution corresponding to the temperature $T_{\text{dec}}$ [578].

In the recombination process, the transverse momentum
of the generated D meson is typically larger than that of
the initiating charm quark. Hadronisation via recombination
therefore corresponds to a hardening of the D-meson $p_T$ dis-
tribution relative to jet fragmentation in vacuum, generating a
characteristic peak in $R_{AA}$ at $p_T \approx 2$ GeV. This peak results
from the interplay between coalescence, whose probability
relative to fragmentation decreases with increasing $p_T$, the
shape of the $p_T$-differential spectra of light and heavy quarks
at hadronisation, and the radial flow. A similar effect occurs
for azimuthally asymmetric flow of heavy-flavour hadrons
formed via recombination: the hadron inherits the flow of
both the heavy and the light quarks. Further discussion of
coalescence effects in charm-quark hadronisation via com-
parison of the abundances of different heavy-flavour hadron
species in pp and heavy-ion collisions is reported in Sect. 2.3.

As discussed above, the heavy-quark spatial diffusion
coefficient $D_s$ does not depend significantly on the heavy-
quark mass and is therefore a property of the QGP medium.
<figure><img src="image_53.png" /><figcaption>**Fig. 43** D-meson production in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [435,573]. (Left) $R_{AA}$ in 0–10% collisions; (right) $v_2$ in 30–50% collisions. Model calculations implement charm-quark transport in a hydrodynamically expanding QGP (see text for references)</figcaption></figure>

<figure><img src="image_54.png" /><figcaption>**Fig. 44** Data from Fig. 43 compared with calculations from the LIDO [577] and LGR [587] models using several variations of charm-quark interactions and hadronisation; see text for details</figcaption></figure>

The coefficient $D_s$, which is calculable within the framework of lattice QCD [589–591], can be determined by comparison of model calculations with data, via a comprehensive treatment of charm-quark transport in the QGP and hadronisation (see [432] for a recent review).

Experimental data from the LHC and theoretical efforts have enabled significant progress in the determination of $D_s$. Recent studies utilised ALICE measurements of elliptic and triangular flow of D mesons [573] as well as $R_{AA}$ and $v_2$ measurements of prompt D mesons [435]. Models that agree with the data at the level $\chi^2/ndf < 2$ yield a

value $1.5 < 2\pi D_s(T)T < 4.5$. The corresponding charm quark relaxation time $\tau_{\text{charm}} = (m_{\text{charm}}/T)D_s(T)$ is in the range $3 < \tau_{\text{charm}} < 9$ fm/c with $T = T_c$ and $m_{\text{charm}} = 1.5$ GeV/c$^2$. A similar range for $D_s$ was obtained for Au–Au collisions at $\sqrt{s_{NN}}$ = 200 GeV, based on measurements by the STAR Collaboration of $v_2$ and $R_{AA}$ of D$^0$ mesons [592]. These values are similar in magnitude to the estimated lifetime of the QGP at the LHC of about 10 fm/c [239], indicating that the charm quark may thermalise completely in the medium. The extracted value for $D_s$ is likewise consistent with calculations based on lattice QCD, which yield val-
ues in the range $2 < 2\pi D_s(T)T < 6$ [589–591], and up to an order of magnitude smaller than values predicted by pQCD calculations at leading order [69,519,593]. This provides significant evidence in the heavy-quark sector in favor of the formation of a strongly coupled QGP being produced in heavy-ion collisions at the LHC (see Sect. 2.7 for more details).

Future high-precision measurements of heavy-flavour hadron production with the upgraded ALICE detector in Run 3 down to very low transverse momenta are expected to set more stringent constraints on the production of charm quarks and their interactions in the QGP [139].

### 2.4.2 Jet quenching measurements

In this section ALICE measurements of jet quenching are presented, whose physics motivation and theoretical tools are discussed above.

#### 2.4.2.1 Energy loss

**High-$p_T$ inclusive hadron production and radiative energy loss.**

Radiative energy loss is expected to be the dominant mechanism responsible for the suppression of inclusive particle production at high-$p_T$. Figure 45, top panel, shows the nuclear modification factor $R_{AA}$ as a function of transverse momentum for charged particles at $\sqrt{s_{NN}} = 5.02$ TeV for central and peripheral Pb–Pb collisions, and p–Pb collisions. In p–Pb collisions, particle production approximately follows $N_{coll}$ scaling, with $R_{pPb}$ within 10% of unity in the region dominated by hard processes, $p_T > 2$ GeV. In central Pb–Pb collisions a peak is seen at intermediate transverse momentum, with yield suppression of a factor five observed at high-$p_T$. The peak at 2–3 GeV for charged hadrons is partly due to collective radial flow and enhancement in the baryon-to-meson yield ratio, as discussed in the next paragraph. The value of $R_{AA}$ has a minimum value of 0.18 at $p_T \approx 6$ GeV, and then increases for $p_T > 10$ GeV. The increase suggests that energy loss as observed by measuring high-$p_T$ hadrons is not proportional to the parent parton energy, but closer to constant [597]. In peripheral Pb–Pb collisions (the class 70–80% is reported in the figure) $R_{AA}$ is about 30% below unity at $p_T \approx 6$ GeV and increases to unity at high-$p_T$. The interpretation of this measurement will be discussed at the end of the section.

Figure 45, bottom panel, shows $R_{AA}$ for identified light-flavour mesons and protons (ALICE [505]), and isolated photons (CMS [596]). The peak in the yields in the range $2 < p_T < 6$ GeV is most pronounced for protons, with maximum value ordered by mass. This ordering corresponds to the expected effect of radial flow (see Sect. 2.2), but inclusive yields in this region may also be affected by initial-state nuclear effects, in particular shadowing. Figure 45 shows

<figure><img src="image_55.png" /></figure>

<figure><img src="image_56.png" /><figcaption>**Fig. 45** Nuclear modification factor $R_{AA}$ for charged hadrons [594] in central and peripheral Pb–Pb collisions and NSD p–Pb collisions (top), and for various particle species in central Pb–Pb collisions [402,505, 595] (bottom). Isolated photons measured by CMS are also included for comparison [596]</figcaption></figure>

that all hadron species exhibit the same suppression for $p_T > 8$ GeV/c. This observation, combined with identified particle measurements in small systems [459], indicate that hadron formation in jets occurs outside of the QGP in this kinematic range.

In contrast to charged hadrons, which are jet fragments, electroweak particles do not carry colour charge and do not interact in the QGP. Indeed, as shown in Fig. 45, bottom panel, the inclusive production yield of isolated photons at high-$p_T$ is compatible with that expected from binary-scaled pp collisions within uncertainties ($R_{AA} \approx 1$). Their production rate is however affected by the initial state of the collision, notably nuclear modification of PDFs (see Sect. 4).
Similar trends are also reported for Z bosons measured at midrapidity [598,599], while at forward rapidity the $R_{AA}$ deviates from unity due to shadowing effects [600]. On the other hand, the $R_{AA}$ of $W^+$ and $W^-$ also differs from unity due to isospin effects and, depending on the rapidity coverage, shadowing effects [601,602] (see also Sect. 4).

For a more comprehensive understanding of the mechanisms responsible for parton energy loss in the QGP, jet quenching effects are compared for inclusive charged particles, which are mostly produced by gluon jets, and for prompt and non-prompt open charm mesons, which originate from charm and beauty quarks. In addition, model comparisons are made to explore the dependence of energy loss on colour charge (gluon vs. quark) and quark mass. Both the normalised yield $R_{AA}$ and the azimuthal anisotropy $v_2$ are compared to model calculations, to clarify the role of effects such as in-medium path length and density fluctuations.

Figure 46 shows the $p_T$-dependence of $R_{AA}$ (upper panels) and $v_2$ (lower panels) in the 0–10% and 30–50% Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV, for charged hadrons and pions, electrons from beauty-hadron decays, and for prompt and non-prompt D mesons. It can be seen in the figure that at high-$p_T$, the production yields of light and heavy-flavour hadrons are suppressed by a similar amount. At $p_T < 8$ GeV/c, a number of different effects comes into play, as discussed above and in Sect. 2.4.

In the high-$p_T$ region, $R_{AA}$ increases slowly with $p_T$ and the observed suppression is similar for all species, although the relative contributions of elastic and inelastic energy loss mechanisms, as well as the difference between gluon and quark quenching, have a different impact on the suppression pattern of the different hadron species.

To further explore path-length and fluctuation effects on energy loss, model calculations are compared with $p_T$-differential $v_2$ measurements in semi-central collisions. Figure 46 shows results from two different energy loss model calculations that provide results for $R_{AA}$ and $v_2$ of both light and heavy flavours: CUJET 3.1 [604–606] and DREENA-A [603]. Both models include radiative energy loss using the opacity expansion framework and collisional energy loss, coupled to a full hydrodynamic description of the medium density profile as a function of time.

For radiative energy loss in a homogeneous medium, a quadratic dependence of the energy loss on the path length is expected for short path lengths due to formation time effects [62,66,67,607] (Sect. 2.4), while collisional energy loss depends linearly on the path length in the medium. In the first studies with RHIC data, it was found that the calculated azimuthal anisotropy at high-$p_T$ was smaller than the observed $v_2$, even with quadratic path-length dependence of the energy loss [560,608]. This triggered detailed investigations of the models which showed that the predicted $v_2$ is highly sensitive to details of the medium density evolution

[609] and the effects of density fluctuations [610–612]. It was recently realised that the pre-equilibrium phase may also play an important role. Specifically, in most energy loss models, it is assumed that the QGP density builds up during the very early times and energy loss is small or negligible during this time, and it was found that the predicted azimuthal asymmetry depends strongly on the starting time of the evolution [613]. This dependence has also been explored within the DREENA model [603].

With these considerations in mind, the predictions of CUJET3.1 and DREENA-A are compared to the measurements in the high-$p_T$ region, where the approximations used in the models are valid. In this regime, the models describe the measurements shown in Fig. 46 reasonably well, though tension is observed between the data and the DREENA-A calculation, which predicts larger suppression in central collisions. The observed $R_{AA}$ and $v_2$ are in agreement with the expected behaviour, including the expected mass dependence due to the dead-cone effect [40,63,521]. Furthermore, the predicted $v_2$ of beauty hadrons (measured via non-prompt $D^0$ and leptons) is much smaller than that of charm and light-flavour hadrons. Also at low-$p_T$ the $v_2$ of electrons from beauty-hadron decays is smaller than the one of electrons originating from charm-hadron decays, consistent with the expected longer beauty quark relaxation time in the QGP.

Several other groups are exploring different approximations to energy loss calculations, such as the multiple-soft scattering approach [561,608], hard thermal loop calculations [546,614], higher-twist calculations [547,548], and soft collinear effective theory approaches [615]. Work is currently ongoing to extend the formalism beyond the eikonal limit [616,617]. These calculations lead to similar conclusions as those presented here.

The quark-mass dependence of parton energy loss can be tested by measurements of prompt D mesons, that originate from charm quarks, and non-prompt D or $J/\psi$ mesons from decays of B mesons, that originate from beauty quarks. Figure 47, left panel, compares $R_{AA}$ of prompt and non-prompt D mesons [435,541], indicating that the latter have a larger $R_{AA}$ at intermediate-$p_T$ of 5–10 GeV/c. The right panel of the same figure shows the centrality dependence of high-$p_T$ $R_{AA}$ for prompt D mesons and non-prompt $J/\psi$ mesons in Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [619]. The kinematic ranges ($8 < p_T < 16$ GeV for D, $6.5 < p_T < 30$ GeV for $J/\psi$) were chosen such that the average momenta of the B mesons that decay into $J/\psi$ are similar to the D-meson momenta; the difference in rapidity interval width is expected to have negligible effect. The different magnitudes of suppression indicate larger in-medium energy loss of the lighter charm quarks compared to beauty quarks. The figure also shows results of calculations which include the dead-cone effect [620], which are in reasonable agreement with the data. The dotted lines show the calculation for non-prompt
**Fig. 46** Comparisons of the $p_T$-differential $R_{AA}$ (upper panels) and $v_2$ (lower panels) measured for charged pions [290,402] and hadrons, prompt D mesons [435,573], non-prompt $D^0$ mesons ($R_{AA}$ only [541]), and electrons from beauty-hadron decays ($v_2$ only [533]) in the 10% most central ($R_{AA}$) and 30–50% semi-central ($v_2$) Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV with calculations using the DREENA-A [603] (left panels) and CUJET3.1 [604] (right panels) models. A global normalisation uncertainty of 3% on $R_{AA}$ is not shown in the figure

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /></figure>

**Fig. 47** (Left) $R_{AA}$ for prompt and non-prompt D mesons as a function of $p_T$ in the 10% most central Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV [435,541]. (Right) $R_{AA}$ for prompt D mesons [618]

<figure><img src="image_59.png" /></figure>

and non-prompt $J/\psi$ mesons (CMS [619]) as a function of the collision centrality for Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, compared to a model calculation with mass-dependent energy loss [620]
<figure><img src="image_60.png" /><figcaption>**Fig. 48** (Top) Measurement of the $p_T$-differential $R_{AA}$ for muons and electrons from heavy-flavour hadron decays at forward ($2.5 < y < 4$) and midrapidity ($|y| < 0.8$), respectively, in 0–10% Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [534,537], with comparison to MC@sHQ+EPOS2 calculations [582,621]. Statistical (vertical bars) and systematic (open boxes) uncertainties are shown. The filled boxes centered at $R_{AA} = 1$ represent the normalisation uncertainty. (Bottom) $p_T$-differential $R_{AA}$ of muons from heavy-flavour hadron decays is compared with MC@sHQ+EPOS2 calculations with pure collisional energy loss (dashed lines) and a combination of collisional and radiative energy loss</figcaption></figure>

J/ψ mesons with the same initial momentum distribution and fragmentation function for beauty quarks but using the charm instead of the beauty-quark mass for the calculation of the energy loss. This results in larger suppression, demonstrating the mass dependence of radiative energy loss directly. As similar study is reported in Ref. [541], showing that models can achieve a larger $R_{AA}$ for non-prompt D⁰ mesons with respect to prompt ones at $p_T$ of 5–10 GeV/c only if they include mass-dependent energy loss. It can thus be concluded that this observed $R_{AA}$ hierarchy is indeed consistent with the expected mass dependence of radiative energy loss.

ALICE has also studied the rapidity dependence of in-medium heavy-quark energy loss using muons from heavy-

flavour hadron decays in the rapidity interval $2.5 < y < 4$. Figure 48, top panel, shows the $p_T$-differential $R_{AA}$ of muons ($2.5 < y < 4$) [537] and electrons ($|y| < 0.8$) [534] from heavy-flavour hadron decays in 0–10% central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV. The magnitude of suppression is similar at forward and midrapidity, within uncertainties. This shows that heavy quarks experience in-medium energy loss over a wide rapidity interval. The similar magnitude of suppression does not necessarily correspond to similar energy loss, however, since the shape of $p_T$-spectrum also influences the suppression [622]. The calculation based on the MC@sHQ+EPOS2 transport model [582,621] agrees with data at both central and forward rapidity. Figure 48, bottom panel, compares the measured $R_{AA}$ of forward muons from heavy-flavour hadron decay to calculations based on two configurations of the MC@sHQ+EPOS2: (i) energy loss from both medium-induced gluon radiation and collisional processes, or (ii) collisional energy loss only. Both calculations agree with the data, though the configuration with both collisional and radiative energy loss is closer to the data for $p_T > 10$ GeV, indicating the importance of radiative processes in the high-$p_T$ region.

**Interplay of inclusive jets and hadrons.** As discussed in Sect. 2.4, high-$p_T$ hadrons and reconstructed jets explore different aspects of jet quenching: hadrons are sensitive principally to energy loss in the hardest branch of the jet shower, while jets, which subtend an area approximately $\pi R^2$ for jet resolution parameter $R$, are sensitive more broadly to modification of the shower. A comprehensive understanding of jet quenching requires measurements of both high-$p_T$ hadrons and reconstructed jets, with the latter spanning significant range in $p_T^{\text{jet}}$ and $R$. In this section, measurements of inclusive jets after a statistical subtraction of the underlying event are discussed and compared to the inclusive hadron measurements presented in Sect. 2.4.2.

Figure 49 shows $R_{AA}$ in central Pb–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV for inclusive jets with $R = 0.4$, together with charged hadrons. Jet $R_{AA}$ exhibits larger suppression than hadrons at the same $p_T$, with the ALICE jet spectrum extending down to $p_T = 60$ GeV/c. At higher $p_T$ the ALICE [563] and ATLAS [624] jet data are consistent, and show slowly increasing $R_{AA}$ with increasing $p_T$.

In general, a reconstructed jet will catch a fraction of medium-induced radiation, and the inclusive $p_T$-spectrum of jets is significantly harder than that of hadrons. Both factors suggest that $R_{AA}$ for inclusive jets will be larger than that for inclusive hadrons. It is to be noted that, while the opposite is observed in the $p_T$ region where the measurements of hadrons and jets overlap, such direct comparison is not meaningful since hadrons and jets at any given $p_T$ originate from different parton energies. For a proper interpretation of this observation, the inclusive hadron population must be
<figure><img src="image_61.png" /><figcaption>**Fig. 49** Measurement of $R_{AA}$ for charged hadrons (ALICE [594] and CMS [623]) and jets (ALICE [563] and ATLAS [624]) in central Pb–Pb collisions. A global normalisation uncertainty of 3% on $R_{AA}$ is not shown in the figure</figcaption></figure>

mapped to that of jets, taking into account the bias imposed by selecting high-$p_T$ hadrons.

Model calculations indicate that high-$p_T$ hadrons are more likely to originate from narrow, hard fragmenting jets, which on average undergo fewer interactions with the QGP. These fewer interactions result in less energy loss [625–627]. Detailed model comparisons provide the opportunity to elucidate the relationship between hadron and jet $R_{AA}$, and the impact of $\hat{q}$ on those observables. Figure 50 presents comparisons of the measured $R_{AA}$ of charged hadrons (left) and reconstructed $R = 0.2$ jets (right) with JETSCAPE [562], JEWEL [64,628,629] and the hybrid model [557,630]. Jets with $R = 0.2$ are utilised for this comparison due to their higher precision and larger $p_T$ range.

JETSCAPE employs detailed hydrodynamic modelling of the QGP, together with a two-stage jet quenching calculation based on MATTER and LBT which use the Higher Twist formalism for virtuality-dependent jet-medium interactions. The LBT model also incorporates medium response to the deposited energy via the Boltzmann transport approach. Transition between MATTER (high virtuality) and LBT (low virtuality) occurs at a fixed parton virtuality $Q_{\text{switch}}$, which is determined from data [562]. As shown in Fig. 50, the JETSCAPE calculations predict slightly larger $R_{AA}$ for charged hadrons over the entire measured $p_T$ interval and broadly describes the jet measurements. Implementation of the $Q$-dependence of $\hat{q}$ was found to be critical to simultaneously describe both measurements. Without such a dependence, the energy loss at high-$p_T$ increases in the calculation, leading to an underprediction of the $R_{AA}$.

JEWEL overestimates the suppression for jet $R_{AA}$. This conclusion holds regardless of the inclusion of medium recoil, although the model with recoil tends to get closer

to the data. The hybrid model describes the trends of the jet $R_{AA}$, but exhibits some tension.

**Measurement of jet energy loss.** As discussed above, yield suppression measurements are only indirectly related to jet energy loss, since yield suppression depends on both the population-averaged energy loss and the shape of the spectrum. However, these effects can be disentangled for jet yield suppression measurements since reconstructed jets account for all correlated hadronic energy within the jet area (colloquially, within the “jet cone”), and jet yield suppression therefore must arise from energy transport out of the jet cone, i.e. jet energy loss. Several recent papers have carried out such a phenomenological extraction of energy loss, by converting yield suppression to the equivalent $p_T$-shift of the spectrum [568, 570, 631, 632].

Figure 51, left panel, shows the ratio of $p_T$-distributions of the $\Delta_{\text{recoil}}$ observable (Eq. 17), for a semi-inclusive hadron+jet measurement in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV compared to that for pp collisions calculated by PYTHIA [568]. A $p_T$-independent yield suppression of 0.6 is observed for central Pb–Pb collisions. Both $\Delta_{\text{recoil}}$ spectra are well approximated over the measured $p_T$ range by an exponential function $\sim e^{-p_T/b}$ with $b \sim 16$ GeV, so that a single number for the $p_T$ shift, $(8 \pm 2)$ GeV, characterises the measured suppression [568]. JETSCAPE calculations reproduce the data at low-$p_T$ but show a rising trend with increasing $p_T$ that is not seen in the data.

A similar analysis was carried out for high multiplicity p–Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV [570]) (see Sect. 3.5), giving a limit of 0.4 GeV at 90% CL for medium-induced energy transport to angles greater than $R = 0.4$ for such collisions. A similar $p_T$-shift analysis was likewise carried out for jet measurements in central Au–Au collisions at $\sqrt{s_{NN}} = 200$ GeV, giving shift values of $(3.3 \pm 0.3 \pm 0.7)$ GeV for an inclusive jet population [632] and $(5.0 \pm 0.5 \pm 1.2)$ GeV for a semi-inclusive recoil jet population [631], both with $R = 0.4$. These values suggest that energy loss due to jet quenching is larger in central nucleus–nucleus collisions at the LHC than at RHIC, though with limited significance. Future measurements with improved systematic uncertainties and extending over a broader $p_T$ range, together with theoretical calculations, will provide more insight into this comparison.

Figure 51, right panel, shows the ratio of $\Delta_{\text{recoil}}$ distributions for recoil jets with $R = 0.2$ and 0.5, separately for Pb–Pb and pp collisions. The value of this ratio reflects the transverse jet shape and can be modified by medium-induced energy transport. However, these ratios for Pb–Pb collisions and for pp collisions as calculated using PYTHIA are consistent within uncertainties, suggesting that the medium-
<figure><img src="image_62.png" /><figcaption>**Fig. 50** ALICE measurements of $R_{AA}$ for charged hadrons [594] and jets [563] in central Pb–Pb collisions compared to calculations from JETSCAPE, JEWEL, and the hybrid model. A global normalisation uncertainty of 3% on $R_{AA}$ is not shown in the figure</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Fig. 51** Semi-inclusive hadron+jet coincidence measurement for Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [568] compared to JETSCAPE expectations. The observable $\Delta_{recoil}$ is defined in Eq. 17. The notation “$TT\{a, b\}$” refers to a trigger hadron within the interval $a < p_T < b$, and $\Delta\phi$ refers to the azimuthal separation between trigger hadron and</figcaption></figure>

recoil jet. (Left) Ratio of $\Delta_{recoil}$ distribution to that for pp collisions calculated with the PYTHIA6 event generator. (Right) Ratio of $\Delta_{recoil}$ distributions for recoil jets with $R = 0.2$ and 0.5, measured for central Pb–Pb and calculated with PYTHIA for pp collisions

induced lost energy due to quenching is predominantly trans-ported to angles greater than 0.5 rad.

**Jet quenching in peripheral Pb–Pb collisions.** Jet energy loss in a static medium is expected to vary parametrically as $L^2$, where L is the in-medium path length [66,553,554]. Detailed modeling is required to account for additional effects due to the spatial distribution and orientation of propagating partons, and the dynamical expansion of the QGP. A valuable tool to test and constrain such calculations is the measurement of jet quenching observables with variation in system size. Jet quenching measurements in small collision systems (pp, p–Pb) are discussed in Sect. 3.5; to date, no significant signals of jet quenching have been observed in such systems. In contrast, inclusive yield measurements in peripheral Pb–Pb collisions exhibit values of $R_{AA}$ less than unity, as shown in Fig. 45 (top) for the 70–80% centrality class. This measurement suggests significant jet quenching

effects even in the most peripheral (large impact parame-ter) Pb–Pb collisions (see also Ref. [633]). However, most of the suppression observed in this centrality class and in more peripheral collisions can be ascribed to biases caused by the event selection and collision geometry. In essence, for peripheral collisions selected using charged-particle multi-plicity, the nucleon–nucleon impact parameter distribution is biased towards larger values, leading to a lower yield of hard processes. A PYTHIA-based prediction of this effect [173] describes a measurement of $R_{AA}$ in fine peripheral centrality classes carried out by ALICE [633]. This could explain the unintuitive observation that $R_{AA}$ is below unity in peripheral Pb–Pb, but equal to unity in minimum-bias p–Pb collisions despite similar charged-particle multiplicities.
### 2.4.2.2 Jet substructure modification

Modifications to the internal structure of jets can be studied with jet-substructure observables – defined by first clustering a jet, and then constructing an observable as a function of the properties of the constituents of that jet [571,572,630,634–638]. Jet substructure observables can be constructed to be sensitive to specific regions of jet radiation phase space in a way that is theoretically calculable from first principles [639–643], and can target limited regions of phase space to explore specific jet quenching mechanisms that cannot be resolved using jet $p_T$ measurements alone [558,630,644–651]. Questions addressed by jet substructure measurements in nucleus-nucleus collisions include the strength of the jet-medium coupling, the rate of medium-induced emissions, and constraints on medium properties such as coherence scales and the nature of the QGP degrees of freedom.

There is a close connection between substructure modifications due to quenching and in-medium scattering and deflection measurements: deflection of jet constituents from in-medium interactions can be seen with jet substructure observables, and modification of a branch of a jet that has split must generate acoplanarity of the reconstructed jet centroid.

The fact that jet substructure observables depend on the detailed distribution of jet constituents brings an experimental challenge, however, since the underlying event can dramatically distort the reconstructed observables [569,571] (see Sect. 2.4). Therefore, precise measurements of jet substructure observables utilise background subtraction techniques that remove constituents of the jet event-by-event, such as constituent subtraction [652,653] and subtraction in moment space [654]. Each observable must be verified individually to ensure that it is robust to the bias introduced by such procedures. Jet substructure observables which tag a specific substructure object, such as groomed jet observables, face an additional challenge in that the underlying event can cause the object to be incorrectly identified [655]. This underscores the importance of selecting observables that can both be calculated theoretically and measured experimentally.

ALICE jet substructure measurements are carried out using charged-particle jets, to take advantage of the precise spatial resolution down to small angular scales within the jet core (Sect. 2.4). These measurements are unfolded for detector and background effects in two dimensions ($p_T$ and the substructure observable), enabling direct comparison to theoretical jet quenching calculations.

**Groomed jet substructure.** Jet substructure can be used as a starting point to understand jet quenching by constructing observables that isolate the perturbative part of the substructure. In this approach, jet grooming algorithms such as Soft Drop (SD) [640,641,656] are applied to remove

soft, wide-angle radiation and identify a single hard “splitting”. Monte Carlo event generators suggest that groomed jet splittings are correlated to parton shower splittings in vacuum [641,657,658]. In heavy-ion collisions this is complicated by both medium and background effects, but recent theoretical studies show that this correlation of the time structure and splittings remains [657], and groomed jet splittings may provide a handle on the space-time evolution of jet propagation through the QGP [646].

The SD grooming algorithm identifies a single splitting by re-clustering the constituents of a jet. The splitting is selected from the history of the re-clustering with a grooming condition, $z > z_{cut}\theta^\beta$, where $\beta$ and $z_{cut}$ are tunable parameters and $z$ is defined as the fraction of transverse momentum carried by the sub-leading prong,

$$z = \frac{p_{T, \text{subleading}}}{p_{T, \text{leading}} + p_{T, \text{subleading}}}, \qquad (18)$$

and the angle $\theta$ is defined as the angular distance between the two branches of the identified splitting,

$$\theta_g = \frac{R_g}{R} = \frac{\sqrt{\Delta y^2 + \Delta \varphi^2}}{R}, \qquad (19)$$

where $y$ is the rapidity, $\varphi$ is the azimuthal angle, and $R$ is the jet resolution parameter. The groomed splitting can be characterised by two kinematic observables: the groomed momentum fraction, $z_g$, and the groomed jet radius, $\theta_g$, with $z$ and $\theta$ for the groomed splitting as defined in Eqs. 18 and 19.

Several different physics mechanisms have been conjectured for modifications of the $z_g$ distribution due to jet quenching, such as medium-induced radiation being hard enough to pass the grooming condition and thereby enhancing the number of asymmetric splittings. Additionally, $z_g$ may be sensitive to effects due to the medium response, breaking of colour coherence, and modification to the DGLAP splitting function in the QGP [649,659–661].

Initial heavy-ion jet substructure measurements by CMS [636] and ALICE [572] indicated a suppression of symmetric splittings relative to asymmetric splittings in Pb-Pb collisions compared to pp collisions. However, further analysis revealed that this effect could arise from a cut on the sub-jet separation distance, and that there were additional background contributions from mistagged splittings arising from the underlying event [655]. More recently, ALICE has addressed these mistagging effects by using stronger grooming conditions and smaller $R$, and carried out measurements in more peripheral collisions. Figure 52, left panel, shows the measurement of $z_g$, which exhibits no significant modification of the $z_g$ distribution in Pb-Pb collisions compared to pp collisions. This is consistent within uncertainties with a variety of jet quenching models, also shown.

These analysis techniques also enable measurements of the angular distribution $\theta_g$, which has been predicted to
<figure><img src="image_64.png" /><figcaption>**Fig. 52** Jet $z_g$ (left) and $\theta_g$ (right) in 0–10% centrality for $R = 0.2$ charged-particle jets [662]. The ratio of the distributions in Pb–Pb and pp collisions is shown in the bottom panels and is compared to various jet quenching calculations</figcaption></figure>

<figure><img src="image_65.png" /><figcaption>**Fig. 53** (Left) Distribution of jet angularity for $R = 0.2$ charged-particle jets in 0–10% central Pb–Pb collisions compared to that for pp collisions calculated with the PYTHIA6 event generator [571]. (Right) Jet mass for $R = 0.4$ charged-particle jets in 0–10% central compared to</figcaption></figure>

PYTHIA calculations and various jet quenching models [569]. Systematic uncertainties have not been assigned to the PYTHIA calculations in both panels

be sensitive to the quark-gluon fraction, splitting formation time, and colour coherence [645,647]. In contrast to the $z_g$ distribution, Fig. 52 (right) shows a significant narrowing of the $\theta_g$ ($R_g$) distribution in central Pb–Pb collisions compared to pp collisions, which may arise from modification of the angular scale of jets in the quark-gluon plasma. This narrowing is consistent with models implementing (trans-

verse) incoherent interaction of the jet shower constituents with the medium, but also with medium-modified quark-to-gluon fractions and fully coherent energy loss; further measurements will be needed to characterise the mechanism underlying the narrowing. Taken together, these measurements suggest that the hard substructure of jets is consistent with (i) little-to-no modification of the momentum splitting,
and (ii) stronger suppression of jets with wide fragmentation patterns. This indicates that the medium has a significant resolving power for splittings with a particular dependence on the angular scale, preserving narrow jets or filtering out wider jets.

**Jet shapes.** Jet substructure observables can also be used to probe soft non-perturbative physics by studying the distribution of radiation inside the jet, without selecting on the hard structure. Such ungroomed observables can probe the interplay between the modification of the jet structure and the response of the medium to the jet propagating through it.

Jet shape is measured both by jet-by-jet functions of the jet constituent momentum, such as jet mass and width, and by inclusive and semi-inclusive measurements of intra- and inter-jet distributions, such as the ratio of jet yields measured with different R. The jet-by-jet jet shapes in particular are sensitive to the underlying soft physics. The first radial moment [663], or the angularity (girth or width), probes the radial energy profile of the jet. The jet mass, which is related to the second radial moment [663], captures the virtuality of the original parton that produced the jet and increases with increasing contribution of large-angle, typically soft particles [664]. These observables are complementary to measurements of the jet fragmentation using the longitudinal momentum fraction of the jet by ATLAS [634] and CMS [665] that suggest that the jet energy in heavy-ion collisions is transferred to soft particles and wider angles inside the jet. ATLAS and CMS have also investigated the radial dependence of charged particles within a jet, demonstrating enhancement of softer particles at larger angles [635,666] and enhanced yield of high-$p_T$ charged particles in the jet core [666].

ALICE has measured jet angularity for charged-particle jets with $R = 0.2$ in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [571]. The angularity is defined as

$$g = \sum_{i \in \text{jet}} \frac{p_{T,i}}{p_{T,\text{jet}}} \Delta R_{\text{jet},i}, \qquad (20)$$

where $p_{T,i}$ is the transverse momentum of the ith constituent and $\Delta R_{jet,i}$ is the distance in $(\eta, \phi)$ space between the ith constituent and the jet axis. Smaller values of $g$ correspond to jets with more collinear fragmentation. Figure 53, left panel, shows the jet-angularity distribution measured in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV compared to that for pp collisions calculated with the PYTHIA6 event generator (Perugia 2011 tune) [571]. The width is observed to be reduced in Pb–Pb collisions, which is consistent with jets in heavy-ion collisions being narrower or having harder fragments; this may arise for example from greater suppression of gluon jets relative to quark jets. This is qualitatively consistent with the observations in Fig. 52, right panel, showing narrowing of the groomed core of jets. This measurement is also

compared to calculations based on the Hybrid Model [558] and JETSCAPE [556] and favors an incoherent energy loss picture, similar to the measurement of the groomed jet radius. Comparison is also made to Hybrid Model calculations with and without medium recoil (referred to as a wake), with no difference observed between these alternative model components. This effect may arise because soft particles from the wake are generated at large angles to the jet centroid and therefore make little contribution to small radius jets.

ALICE has measured the jet mass for charged-particle jets with $R = 0.4$ in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV [569]. The jet mass is defined as

$$M = \sqrt{E^2 - p_T^2 - p_z^2}, \qquad (21)$$

where E is the energy of the jet, $p_T$ is the transverse and $p_z$ is the longitudinal momentum of the jet. Figure 53, right panel, shows the jet-mass distribution measured in central Pb–Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV, compared to model calculations [569]. The calculations include PYTHIA [667], which represents pp collisions without jet quenching, and models with medium-induced jet energy loss: Hybrid Model [558]; JEWEL [628,629]; and JETSCAPE [562]. The JEWEL and Hybrid Models are run both with and without medium recoil.

The jet mass is observed to be larger with recoils present, which is consistent with the picture that the recoils add large-angle soft particles to the jet. The jet mass is however overestimated by both the Hybrid model and JEWEL with medium recoil, and is underestimated without the medium recoil, except in the case of the Hybrid model with coherent energy loss and no wake. Rather, it is consistent within systematic uncertainties with the PYTHIA simulation, suggesting no significant modification of the jet mass in heavy-ion collisions. Alternatively, this could be due to partial cancellation of in-medium effects; specifically, momentum broadening could lead to energy loss outside the jet and a smaller mass, and medium response could add soft particles inside the jet and increase the mass [651]. Note that, in contrast, the jet angularity exhibits strong modification in Pb–Pb compared to pp collisions, though with smaller jet resolution parameter ($R = 0.2$ instead of $R = 0.4$), which is expected to be less sensitive to the medium response.

Additionally, ALICE has measured the N-subjettiness jet shape distribution [668], which quantifies the degree to which a jet corresponds to an N-pronged substructure [669]. Specifically, measurement of the 2-subjettiness to 1-subjettiness ratio ($\tau_2/\tau_1$) suggests a relative reduction in the rate of 2-prong jets for central Pb–Pb collisions. However, the absence of strong modification of $\tau_2/\tau_1$ suggests that medium induced radiation is not sufficiently hard to produce an independent hard prong, similar to the case observed for $z_g$. The $\tau_2/\tau_1$ observable is only weakly correlated with other substructure observables discussed in this section, with a slightly stronger
correlation with $z_g$ than the other observables [670], and thus
adds independent information on in-medium jet substructure
modification.

Jet substructure observables contain some degree of sen-
sitivity to the same physics mechanisms as other jet observ-
ables, including ensemble-based measurements using inclu-
sive or semi-inclusive jet distributions at different jet R [564,
568,671]. Figure 51, right panel, shows one such ratio, for
semi-inclusive yields of jets recoiling from a high-$p_T$ hadron
for jet R = 0.2 and 0.5 [568]. No significant modification of
the intra-jet energy profile for different jet R is observed,
indicating no significant in-medium transfer of energy to
large angles. In addition, the theoretical descriptions of jet
substructure and (semi-)inclusive jet observables have sig-
nificant differences for jets in vacuum; for instance, the jet
shape (profile) requires significantly larger soft power cor-
rections than the inclusive jet cross section as a function of
R [672].

2.4.2.3 Jet multiple scattering and deflection

Medium-induced modification of the jet shower is studied
via intrajet shapes and substructure, which are sensitive to
the redistribution of jet momentum and constituents to wider
angles. Such effects will likewise generate a change in the
jet direction as a whole, which is explored in this section
via the semi-inclusive azimuthal angular distributions of jets
recoiling from a hadron trigger [568,631].

In vacuum, the width of the azimuthal distribution of
recoil jets in this observable arises largely from soft radiation
(Sudakov radiation [673]), while for jets in-medium, modifi-
cation of the vacuum angular distribution arises from (inelas-
tic) gluon emission and elastic scattering off the medium
constituents. Measurements which disentangle vacuum and
medium-induced azimuthal decorrelation effects provide
direct sensitivity to the transport coefficient $\hat{q}$. Since Sudakov
radiation dominates at high jet $p_T$ with respect to medium
effects [673], measurement of the azimuthal decorrelation at
low values of recoil $p_T^{\text{jet}}$ is desirable; however, measurements
at low-$p_T^{\text{jet}}$ in heavy-ion collisions are challenging because
of the large uncorrelated background, and new experimental
approaches are needed. The expected difference in the para-
metric dependence of jet energy loss and momentum broad-
ening on the medium path length L [674,675] also motivates
the simultaneous measurement of observables sensitive to
both energy loss and momentum broadening, to discriminate
between weakly- and strongly-coupled scenarios.

Measurement of the rate of jet scattering to large angles
with respect to the trigger axis may also provide evidence
of weakly-coupled degrees of freedom within the strongly-
coupled QGP ("quasi-particles"), analogous to the Ruther-
ford scattering experiment that revealed the existence of

<figure><img src="image_66.png" /><figcaption>**Fig. 54** ALICE measurement of h+jet acoplanarity in 0–10% Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [568] compared to an embedded PYTHIA+Pb–Pb reference and JETSCAPE calculations, in pp and Pb–Pb collisions for charged-particle recoil jets with R = 0.4 and 40 < $p_T^{\text{jet}}$ < 60 GeV. See text for discussion of JETSCAPE calculations</figcaption></figure>

the atomic nucleus [676,677]. The deflection of an ener-
getic quark projectile in the QGP is expected to be Gaus-
sian if the QGP is strongly coupled at all scales. However,
QCD is asymptotically free and thus weakly-coupled quark
and gluon quasi-particle degrees of freedom are expected to
emerge when the QGP is probed at sufficiently short dis-
tances. The scattering off point-like quasi-particles will lead
to a power-law tail in the momentum transfer (1/$k_T^4$), the
so-called Moliere scattering [676,677]. An excess of large-
angle deflections observed in Pb–Pb relative to pp collisions
would be a direct observation of such quasi-particles in the
QGP.

The large-angle scattering signal is expected to be small,
however, requiring high experimental sensitivity to observe
it. The $\Delta_{\text{recoil}}$ observable (Eq. 17) has several features crucial
for such a high-sensitivity search: (i) absolute normalisation,
rather than normalisation to the total number of pairs, mean-
ing that each data-point is independently measured; (ii) fully
data-driven correction for background; and (iii) full correc-
tion for the contribution of multiple partonic interactions,
which are uncorrelated with the trigger and, therefore, gen-
erate an azimuthally uniform background [568].

Figure 54 shows the first ALICE measurement of the azimuthal decorrelation in 0–10% Pb–Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV [568]. The horizontal axis represents the azimuthal angle $\Delta\varphi$ between a high-$p_T$ charged hadron and the recoiling jets. The vertical axis corresponds to the $\Delta_{\text{recoil}}$ observable (Eq. 17) measured differentially in $\Delta\varphi$. The widths of the distributions in Pb–Pb data and in the vacuum reference, which in this case is a PYTHIA calculation for pp collisions embedded into Pb–Pb data, are statistically
compatible. The absolute yield of the Pb–Pb distribution is seen to be smaller than that of the pp reference, indicating that the recoiling jet population is suppressed in Pb–Pb collisions (see also Fig. 51).

For the selected kinematic cuts and jet resolution *R*, the strong energy loss is therefore not accompanied by a medium-induced acoplanarity within the statistical and systematic limits of our measurement. The change of recoil jet yield at large angles with respect to the PYTHIA reference was also studied, and within the current experimental uncertainties no evidence of large-angle scattering was found.

The figure also shows the result of JETSCAPE calculations which have been smeared to account for instrumental effects and background fluctuations for comparison, since these data have not been unfolded for such effects. The JETSCAPE calculation matches the distribution measured in pp collisions well, and slightly overshoots the distribution measured in Pb–Pb collisions in the region near $\Delta\phi \sim \pi$. In order to compare shapes directly as a function of $\Delta\phi$, which could reveal quasi-particle effects, the JETSCAPE Pb–Pb distribution is also shown scaled to the same integral as the measured Pb–Pb distribution, which factors out the effect of an overall yield suppression. We note that the JETSCAPE calculation of jet quenching is based on the LBT model at low virtuality, which includes elastic scattering in the QGP. JETSCAPE therefore includes a specific implementation of Moliere scattering. The scaled JETSCAPE distribution reproduces the data well, likewise exhibiting no evidence of in-medium acoplanarity broadening or quasi-particle scattering.

The uncertainties in this measurement are dominated by the statistical uncertainty, so that future measurements with higher-statistics data sets will improve its precision. On the theoretical side, the parametrically different dependence in some models on in-medium path length of energy loss $\propto L^2$ and acoplanarity due to multiple scattering $\propto L$ suggest that simultaneous consideration of yield suppression and acoplanarity in data-to-model comparisons may provide significant discrimination of weakly- and strongly-coupled QGP [675]. Extensions of the analysis to jets recoiling from different triggers (e.g. high-$p_T$ photons/EW bosons) will provide complementary ways to study in-medium broadening.

### 2.4.3 Conclusions

**Heavy-quark diffusion.** Differential studies of the yield suppression and flow of D mesons highlight the predominance of elastic charm quark-QGP interactions for $p_T$ less than a few GeV. Heavy quarks in this range experience Brownian motion in the QGP, and therefore provide incisive probes of the equilibration process. This interaction is characterised by a spatial diffusion constant $D_s$. Its value, $1.5 < 2\pi D_s(T)T < 4.5$, is smaller than that determined from a

$p$QCD calculation for a gas of weakly-coupled quarks and gluons, providing clear evidence in the heavy-flavour sector for a strongly-coupled QGP. Recent lattice QCD calculations [678] give even slightly smaller values for $D_s$.

**Jet quenching: energy loss.** The magnitude of hadron suppression for $p_T > 8$ GeV in central Pb–Pb collisions is the same for a wide range of light-flavour hadrons, providing definitive evidence that jet quenching occurs at the partonic level. This is likewise strong evidence for the existence of a dense, extended QGP phase. Suppression measurements in the heavy-flavour sector at intermediate-$p_T$ indicate that beauty quarks lose less energy than charm quarks. These measurements are described by models that include mass-dependent elastic energy loss and a reduction of gluon radiation off heavy quarks. The latter mechanism is known as the QCD dead-cone effect and has been directly measured by ALICE for radiation in vacuum using pp collisions. Measurements of reconstructed jets provide unique insight into the jet–medium interaction. Since jets subtend finite area, such modifications reflect the spatial distribution of medium-induced energy loss and the medium response. Inclusive jet suppression has been observed up to a $p_T$ of several hundred GeV. Jet energy loss has been measured for the semi-inclusive distribution of jets recoiling from a hadron trigger to be $(8 \pm 2)$ GeV for central Pb–Pb collisions. This value is larger than that determined from similar analyses at RHIC [679], though the comparison currently has limited significance.

**Jet quenching: substructure modification.** The hard substructure of jets, measured using groomed observables, shows a strong angular modification – jets with wide substructure are suppressed – but weak momentum modification ($z_g$). Measurements of soft jet substructure, which are carried out using ungroomed observables, exhibit effects that are consistent with the suppression of wide-angle radiation due to quenching. Soft jet substructure also probes the dynamics of soft radiation in the jet, in particular the contribution of medium response, that cannot be calculated using purely perturbative techniques. Such groomed and ungroomed approaches are therefore complementary, and they show that the medium resolves hard substructure and promotes narrower spatial structures. Theoretical calculations suggest that this may be due to colour decoherence, i.e. the gradual loss of the coherence of radiation emitted by a colour antenna while propagating through a dense medium, or to modification of the quark-to-gluon fraction. The spacetime structure of jets also plays a role, since wider splittings may occur earlier and their products therefore traverse longer paths in the QGP.
*Jet quenching: acoplanarity.* The dijet acoplanarity, which has direct sensitivity to the transport coefficient $\hat{q}$ and large-angle scattering off quasiparticles, has been measured via hadron+jet correlations. No angular modifications have been observed within the currently statistics-dominated uncertainties. These measurements will be extended with larger data samples and kinematic coverage and this approach remains a promising tool to explore the nature of the QGP constituents.

## 2.5 Deconfinement and modification of the QCD force

Heavy quarkonia, the bound states of a $c\bar{c}$(charmonium) or $b\bar{b}$pair (bottomonium), have been the subject of intense studies since their discovery in the 1970s. The investigation, in the frame of QCD, of their production processes, of the rich spectroscopy of the various states, and of their decay modes is a lively field until today, and although great progress has been accomplished, a complete understanding of their properties is still to be reached (for a general review see [680]).

Quarkonium states also represent a very important tool for the study of the QGP and of its properties (for a recent review see [681]). It was early realised that the binding of the heavy-quark pair can be affected to various extents when quarkonia are immersed in a deconfined medium. The high density of free colour charges in the QGP leads to a screening of the QCD force and ultimately to the dissolution of the quarkonium [86]. This simple but profound idea has led to a wealth of theoretical and experimental studies that have revealed new and somewhat unexpected effects. Early studies focused on establishing a direct connection between the suppression of the quarkonium states and the temperature of the deconfined phase [682]. The rich spectroscopic structure of quarkonia, with binding energies varying from a few MeV ($\psi$(2S)) to more than 1 GeV (Υ(1S)), may lead to a “sequential suppression” with increasing temperature, with the more strongly bound states surviving up to a dissociation temperature $T_{diss} \sim 2T_{pc}$ and the weakly bound states melting at temperatures close to $T_{pc}$. In nuclear collisions, the temperature of the QGP can in principle be varied by selecting the centrality of the collision or its energy. If the “melting temperature” of each state could be precisely singled out by lattice QCD studies [683–687], quarkonium would represent an ideal thermometer for the medium.

The above considerations are valid in a static picture of the medium in which the quarkonium states are assumed to be immersed. When moving to a study of the dynamics of the bound states and their interaction with an evolving medium, several effects lead to a more complex description. In particular, the formation of the quarkonium states is a multi-stage process [20] (production of the $q\bar{q}$ pair and formation of the bound state) that spans over a time covering a significant fraction of the collision history. In addition, the quarkonium potential, as calculated at $T > 0$ in effective

field theories, has also an imaginary part, corresponding to the collisional damping of the states, which leads to a loss of correlation in the pair and consequently to an in-medium modification of the spectral functions [688]. Furthermore, in a system with a high multiplicity of heavy quarks, the effects related to the combination of uncorrelated pairs originated from different hard scattering processes, or the recombination of previously destroyed ones, can lead to a significant increase of the quarkonium yields [441,689], counterbalancing the suppression. If a partial or full kinetic equilibration of the deconfined heavy quarks in the medium [690] takes place, collective flow effects can be inherited by quarkonia produced in the (re)combination process.<fnref n="9" /> Finally, also in the hadronic stage of the collisions the yields might be altered, due to quarkonium-meson break-up effects which could play a role in particular for weakly bound states [691,692]. The theoretical treatment of quarkonium production in the hot QCD medium is multi-faceted, comprising statistical hadronisation, transport models, hydrodynamics and the recently emerging approach through quantum dynamics [681,693].

At the LHC start-up, quarkonium production in nuclear collisions, and in particular $J/\psi$ studies, had already been pursued for many years in the frame of QGP-related studies. Early experiments at SPS energies resulted in high precision results on the ratio between the $J/\psi$ and the Drell–Yan yields, with the latter used as a reference process because of its electromagnetic nature. In particular, the NA50 collaboration had shown a suppression effect of ~ 30% for central Pb–Pb collisions at $\sqrt{s_{NN}} = 17.3$ GeV [92,694], with respect to the expected size of cold nuclear matter effects, evaluated via p–A studies at the same centre-of-mass energy by the NA60 experiment [695]. The size of the suppression approximately corresponds to the feed-down to $J/\psi$ from $\chi_c$ and $\psi$(2S) states [696], leading to the interpretation of the measured effect as due to the melting of these weakly bound states in the medium [682]. Indeed a hierarchy between the $J/\psi$ and $\psi$(2S), with the latter experiencing stronger suppression, was observed [92,694,697]. The exact nature of this medium, either a deconfined phase or a dense hadron gas, sparked a considerable controversy, due to competing theoretical approaches being able to fairly reproduce the data [698–700].

The $J/\psi$ studies received a second considerable boost with the availability of RHIC data for Au–Au collisions at $\sqrt{s_{NN}} = 200$ GeV by PHENIX [701,702], with important contributions also from STAR [703–705]. At RHIC, the charmonium suppression was evaluated via the study of the nuclear modification factor, as the Drell–Yan contribution becomes increasingly negligible with respect to competing sources such as semileptonic heavy-quark decays.

<footnote n="9">⁹ The terms (re)combination and (re)generation are both used to describe this process.</footnote>