Heavyflavour and quarkonium production in the LHC era: from proton–proton to heavyion collisions
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Abstract
This report reviews the study of open heavyflavour and quarkonium production in highenergy hadronic collisions, as tools to investigate fundamental aspects of Quantum Chromodynamics, from the proton and nucleus structure at high energy to deconfinement and the properties of the Quark–Gluon Plasma. Emphasis is given to the lessons learnt from LHC Run 1 results, which are reviewed in a global picture with the results from SPS and RHIC at lower energies, as well as to the questions to be addressed in the future. The report covers heavy flavour and quarkonium production in proton–proton, proton–nucleus and nucleus–nucleus collisions. This includes discussion of the effects of hot and cold strongly interacting matter, quarkonium photoproduction in nucleus–nucleus collisions and perspectives on the study of heavy flavour and quarkonium with upgrades of existing experiments and new experiments. The report results from the activity of the SaporeGravis network of the I3 Hadron Physics programme of the European Union 7\(\mathrm{th}\) Framework Programme.
1 Introduction
Heavyflavour hadrons, containing open or hidden charm and beauty flavour, are among the most important tools for the study of Quantum Chromodynamics (QCD) in highenergy hadronic collisions, from the production mechanisms in proton–proton collisions (pp) and their modification in proton–nucleus collisions (p–A) to the investigation of the properties of the hot and dense strongly interacting Quark–Gluon Plasma (QGP) in nucleus–nucleus collisions (AA).
Heavyflavour production in pp collisions provides important tests of our understanding of various aspects of QCD. The heavyquark mass acts as a long distance cutoff so that the partonic hardscattering process can be calculated in the framework of perturbative QCD down to low transverse momenta (\(p_{\mathrm {T}}\)). When the heavyquark pair forms a quarkonium bound state, this process is nonperturbative as it involves long distances and soft momentum scales. Therefore, the detailed study of heavyflavour production and the comparison to experimental data provides an important testing ground for both perturbative and nonperturbative aspects of QCD calculations.
In nucleus–nucleus collisions, open and hidden heavyflavour production constitutes a sensitive probe of the hot strongly interacting medium, because hardscattering processes take place in the early stage of the collision on a time scale that is in general shorter than the QGP thermalisation time. Disentangling the mediuminduced effects and relating them to its properties requires an accurate study of the socalled cold nuclear matter (CNM) effects, which modify the production of heavy quarks in nuclear collisions with respect to proton–proton collisions. CNM effects, which can be measured in proton–nucleus interactions, include: the modification of the effective partonic luminosity in nuclei (which can be described using nuclearmodified parton densities), due to saturation of the parton kinematics phase space; the multiple scattering of partons in the nucleus before and after the hard scattering; the absorption or breakup of quarkonium states, and the interaction with other particles produced in the collision (denoted as comovers).
The nuclear modification of the parton distribution functions can also be studied, in a very clean environment, using quarkonium photoproduction in ultraperipheral nucleus–nucleus collisions, in which a photon from the coherent electromagnetic field of an accelerated nucleus interacts with the coherent gluon field of the other nucleus or with the gluon field of a single nucleon in the other nucleus.
During their propagation through the QGP produced in highenergy nucleus–nucleus collisions, heavy quarks interact with the constituents of this medium and lose a part of their momentum, thus being able to reveal some of the QGP properties. QCD energy loss is expected to occur via both inelastic (radiative energy loss, via mediuminduced gluon radiation) and elastic (collisional energy loss) processes. Energy loss is expected to depend on the parton colourcharge and mass. Therefore, charm and beauty quarks provide important tools to investigate the energyloss mechanisms, in addition to the QGP properties. Furthermore, low\(p_{\mathrm {T}}\) heavy quarks could participate, through their interactions with the medium, in the collective expansion of the system and possibly reach thermal equilibrium with its constituents.
In nucleus–nucleus collisions, quarkonium production is expected to be significantly suppressed as a consequence of the colour screening of the force that binds the \({c\overline{c}}\) (\({b\overline{b}}\)) state. In this scenario, quarkonium suppression should occur sequentially, according to the binding energy of each state. As a consequence, the inmedium dissociation probability of these states are expected to provide an estimate of the initial temperature reached in the collisions. At high centreofmass energy, a new production mechanism could be at work in the case of charmonium: the abundance of c and \(\overline{c}\) quarks might lead to charmonium production by (re)combination of these quarks. An observation of the recombination of heavy quarks would therefore directly point to the existence of a deconfined QGP.
The first run of the Large Hadron Collider (LHC), from 2009 to 2013, has provided a wealth of measurements in pp collisions with unprecedented centreofmass energies \(\sqrt{s}\) from 2.76 to 8 TeV, in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV per nucleon–nucleon interaction, in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 2.76\) TeV, as well as in photoninduced collisions. In the case of heavyion collisions, with respect to the experimental programmes at SPS and RHIC, the LHC programme has not only extended by more than one order of magnitude the range of explored collision energies, but it has also largely enriched the studies of heavyflavour production, with a multitude of new observables and improved precision. Both these aspects were made possible by the energy increase, on the one hand, and by the excellent performance of the LHC and the experiments, on the other hand.
This report results from the activity of the SaporeGravis network^{1} of the I3 Hadron Physics programme of the European Union \(7\mathrm{th}\) FP. The network was structured in working groups, that are reflected in the structure of this review, and it focussed on supporting and strengthening the interactions between the experimental and theoretical communities. This goal was, in particular, pursued by organising two large workshops, in Nantes (France)^{2} in December 2013 and in Padova (Italy)^{3} in December 2014.
The report is structured in eight sections. Sections 2, 3, 4, 5 and 6 review, respectively: heavyflavour and quarkonium production in proton–proton collisions, the cold nuclear matter effects on heavyflavour and quarkonium production in proton–nucleus collisions, the QGP effects on open heavyflavour production in nucleus–nucleus collisions, the QGP effects on quarkonium production in nucleus–nucleus collisions, and the production of charmonium in photoninduced collisions. Sect. 7 presents an outlook of future heavyflavour studies with the LHC and RHIC detector upgrades and with new experiments. A short summary concludes the report in Sect. 8.
2 Heavy flavour and quarkonium production in proton–proton collisions
2.1 Production mechanisms of open and hidden heavyflavour in proton–proton collisions
2.1.1 Openheavyflavour production
Openheavyflavour production in hadronic collisions provides important tests of our understanding of various aspects of Quantum Chromodynamics (QCD). First of all, the heavyquark mass (\(m_Q\)) acts as a long distance cutoff so that this process can be calculated in the framework of perturbative QCD down to low \(p_{\mathrm {T}}\) and it is possible to compute the total cross section by integrating over \(p_{\mathrm {T}}\). Second, the presence of multiple hard scales (\(m_Q\), \(p_{\mathrm {T}}\)) allows us to study the perturbation series in different kinematic regions (\(p_{\mathrm {T}} < m_Q\), \(p_{\mathrm {T}} \sim m_Q\), \(p_{\mathrm {T}} \gg m_Q\)). Multiple hard scales are also present in other collider processes of high interest such as weak boson production, Higgs boson production and many cases of physics Beyond the Standard Model. Therefore, the detailed study of heavyflavour production and the comparison to experimental data provides an important testing ground for the theoretical ideas that deal with this class of problems.
On the phenomenological side, the (differential) cross section for openheavyflavour production is sensitive to the gluon and the heavyquark content in the nucleon, so that LHC data in \(\mathrm pp\) and p–Pb collisions can provide valuable constraints on these partondistribution functions (PDFs) inside the proton and the lead nucleus, respectively. In addition, these cross sections in \(\mathrm pp\) and p–A collisions establish the baseline for the study of heavyquark production in heavyion collisions. This aspect is a central point in heavyion physics since the suppression of heavy quarks at large \(p_{\mathrm {T}}\) is an important signal of the QGP (see Sect. 4). Finally, let us also mention that a solid understanding of opencharm production is needed in cosmicray and neutrino astrophysics [1]. In the following, we will focus on \(\mathrm pp\) collisions and review the different theoretical approaches to openheavyflavour production.
In Eq. (1), a sum over all possible subprocesses \(i+j \rightarrow Q + X\) is understood, where i, j are the active partons in the proton: \(i,j \in \{q,\overline{q}=(u,\overline{u}, d, \overline{d}, s, \overline{s}), g\}\) for a FFNS with three active flavours (3FFNS) usable for both charm and beauty production, and \(i,j \in \{q,\overline{q}=(u,\overline{u}, d, \overline{d}, s, \overline{s}, c, \overline{c}), g\}\) in the case of four active flavours (4FFNS) often used for beauty production. In the latter case, the charm quark is also an active parton (for \(\mu _F > m_c\)) and the charmquark mass is neglected in the hardscattering cross section \(\mathrm {d}{\tilde{\sigma }}\) whereas the beauty quark mass \(m_b\) is retained. At the leading order (LO) in \(\alpha _S\), there are only two subprocesses which contribute: (i) \(q+\overline{q} \rightarrow Q+ \overline{Q}\), (ii) \(g + g \rightarrow Q+ \overline{Q}\). At the nexttoleading order (NLO), the virtual oneloop corrections to these \(2\rightarrow 2\) processes have to be included in addition to the following \(2\rightarrow 3\) processes: (i) \(q+\overline{q} \rightarrow Q+ \overline{Q} + g\), (ii) \(g + g \rightarrow Q+ \overline{Q} + g\), (iii) \(g+q\rightarrow q + Q+ \overline{Q}\) and \(g+\overline{q}\rightarrow \overline{q} + Q+ \overline{Q}\). Complete NLO calculations of the integrated/total cross section and of oneparticle inclusive distributions were performed in the late 80s [2, 3, 4, 5]. These calculations form also the basis for more differential observables/codes [6] (where the phase space of the second heavy quark has not been integrated out) allowing us to study the correlations between the heavy quarks – sometimes referred to as NLO MNR. They are also an important ingredient to the other theories discussed below (FONLL, GMVFNS, POWHEG, MC@NLO).
The typical range of applicability of the FFNS at NLO is roughly \(0 \le p_{\mathrm {T}} \lesssim 5 \times m_Q\). A representative comparison with data has been made for \(\mathrm B^{+}\) production in [7] where it is clear that the predictions of the FFNS at NLO using the branching fraction \(B(b \rightarrow \mathrm{B}) = 39.8~\%\) starts to overshoot the Tevatron data for \(p_{\mathrm {T}} \gtrsim 15~~\text {GeV}/c\) even considering the theoretical uncertainties evaluated by varying the renormalisation and factorisation scales by factors of 2 and 1/2 around the default value^{4} \(\mu _F =\mu _R =m_\mathrm{T}\) with \(m_\mathrm{T}=\sqrt{m_Q^2+p_{\mathrm {T}} ^2}\).
The same conclusions about the range of applicability of the FFNS apply at the LHC where the heavyquark production is dominated by the ggchannel (see, e.g. Figure 3 in [7]) over the \({q\overline{q}}\) one. As can be seen, the uncertainty at NLO due to the scale choice is very large (about a factor of two). For the case of top pair production, complete NNLO calculations are now available for both the total cross section [8] and, most recently, the differential distributions [9]. To make progress, it will be crucial to have NNLO predictions for charm and beauty production as well.
 In the PerturbativeFragmentation Functions (PFF) approach [11], the FF \(D_k^H(z,\mu _F')\) is given by a convolution of a PFF accounting for the fragmentation of the parton k into the heavy quark Q, \(D_k^Q(z,\mu _F')\), with a scaleindependent FF \(D_Q^H(z)\) describing the hadronisation of the heavy quark into the hadron H:The PFFs resum the finalstate collinear logarithms of the heavyquark mass. Their scaledependence is governed by the DGLAP evolution equations and the boundary conditions for the PFFs at the initial scale are calculable in the perturbation theory. On the other hand, the scaleindependent FF is a nonperturbative object (in the case of heavylightflavoured hadrons) which is assumed to be universal. It is usually determined by a fit to \(e^{+}e^{} \) data, although approaches exist in the literature which attempt to compute these functions. It is reasonable to identify the scaleindependent fragmentation function in Eq. (3) with the one in Eq. (5). This function describing the hadronisation process involves longdistance physics and might be modified in the presence of a QGP, whereas the PFFs (or the unresummed collinear logarithms \(\ln p_{\mathrm {T}} ^2/m_Q^2\) in the FFNS) involve only shortdistance physics and are the same in \(\mathrm pp\), p–A, and AA collisions.$$\begin{aligned} D_k^H(z,\mu _F') = D_k^Q(z,\mu _F') \otimes D_Q^H(z). \end{aligned}$$(5)

In the Binnewies–Kniehl–Kramer (BKK) approach [12, 13, 14], the FFs are not split up into a perturbative and a nonperturbative piece. Instead, boundary conditions at an initial scale \(\mu _F' \simeq m_Q\) are determined from \(e^{+}e^{}\) data for the full nonperturbative FFs, \(D_k^H(z,\mu _F')\), in complete analogy with the treatment of FFs into light hadrons (pions, kaons). These boundary conditions are again evolved to larger scales \(\mu _F'\) with the help of the DGLAP equations.
The condition \(\mathrm {d}\sigma _\mathrm{FONLL} \rightarrow d\sigma _\mathrm{RS}\) for \(p_{\mathrm {T}} \gg m_Q\) implies that the matching function \(G(m_Q,p_T)\) has to approach unity in this limit. Furthermore, in the limit of small transverse momenta \(\mathrm {d}\sigma _\mathrm{FONLL}\) has to approach the fixedorder calculation \(\mathrm {d}\sigma _\mathrm{FO}\). This can be achieved by demanding that \(G(m_Q,p_T)\rightarrow 0\) for \(p_{\mathrm {T}} \rightarrow 0\), which effectively suppresses the contribution from the divergent bquark initiated contributions in \(\mathrm {d}\sigma _{RS}\). In the FONLL, the interpolating function is chosen to be \(G(m_Q,p_{\mathrm {T}}) = p_{\mathrm {T}} ^2/(p_{\mathrm {T}} ^2 + a^2 m_Q^2)\) where the constant is set to \(a=5\) on phenomenological grounds. In this language the GMVFNS is given by \(\mathrm {d}\sigma _\mathrm{GMVFNS} = \mathrm {d}\sigma _\mathrm{FO} + \mathrm {d}\sigma _\mathrm{RS}  \mathrm {d}\sigma _\mathrm{FOM0}\), i.e. no interpolating factor is used.
Other differences concern the nonperturbative input. In particular, the FONLL scheme uses fragmentation functions in the PFF formalism whereas the GMVFNS uses fragmentation functions which are determined in the zspace in the BKK approach.
Monte Carlo generators The GMVFNS and FONLL calculations are mostly analytic and provide a precise description of the inclusive production of a heavy hadron or its decay products at NLO\(+\)NLL accuracy. Compared to this, generalpurpose MonteCarlo generators like PYTHIA [45] or HERWIG [46] allow for a more complete description of the hadronic final state but only work at LO\(+\)LL accuracy. However, in the past decade, NLO Monte Carlo generators have been developed using the MC@NLO [47] and POWHEG [48] methods for a consistent matching of NLO calculations with parton showers. They, therefore, have all the strengths of Monte Carlo generators, which allow for a complete modelling of the hadronic final state (parton showering, hadronisation, decay, detector response), while, at the same time, the NLO accuracy in the hard scattering is kept and the soft/collinear regimes are resummed at the LL accuracy. A comparison of POWHEG NLO Monte Carlo predictions for heavyquark production in \(\mathrm pp\) collisions at the LHC with the ones from the GMVFNS and FONLL can be found in [49].
2.1.2 Quarkoniumproduction mechanism
The theoretical study of quarkoniumproduction processes involves both pertubative and nonperturbative aspects of QCD. On one side, the production of the heavyquark pair, \({Q\overline{Q}}\), which will subsequently form the quarkonium, is expected to be perturbative since it involves momentum transfers at least as large as the mass of the considered heavy quark, as for openheavyflavour production discussed in the previous section. On the other side, the evolution of the \({Q\overline{Q}}\) pair into the physical quarkonium state is nonperturbative, over long distances, with typical momentum scales such as the momentum of the heavyquarks in the boundstate rest frame, \(m_Q v\) and their binding energy \(m_Q v^2\), v being the typical velocity of the heavy quark or antiquark in the quarkonium rest frame (\(v^2\sim 0.3\) for the charmonium and 0.1 for the bottomonium).
In nearly all the models or production mechanisms discussed nowadays, the idea of a factorisation between the pair production and its binding is introduced. Different approaches differ essentially in the treatment of the hadronisation, although some may also introduce new ingredients in the description of the heavyquarkpair production. In the following, we briefly describe three of them which can be distinguished in their treatment of the nonperturbative part: the ColourEvaporation Model (CEM), the ColourSinglet Model (CSM), the ColourOctet Mechanism (COM), the latter two being encompassed in an effective theory referred to as NonRelativistic QCD (NRQCD).
The ColourEvaporation Model (CEM) This approach is in line with the principle of quark–hadron duality [50, 51]. As such, the production cross section of quarkonia is expected to be directly connected to that to produce a \({Q\overline{Q}}\) pair in an invariantmass region where its hadronisation into a quarkonium is possible, that is between the kinematical threshold to produce a quark pair, \(2m_Q\), and that to create the lightest openheavyflavour hadron pair, \(2m_{H}\).
The cross section to produce a given quarkonium state is then supposed to be obtained after a multiplication by a phenomenological factor \(F_\mathcal{Q}\) related to a processindependent probability that the pair eventually hadronises into this state. One assumes that a number of nonperturbativegluon emissions occur once the \(Q \overline{Q}\) pair is produced and that the quantum state of the pair at its hadronisation is essentially decorrelated – at least colourwise – with that at its production. From the reasonable assumption [52] that one ninth – one coloursinglet \({Q\overline{Q}}\) configuration out of nine possible – of the pairs in the suitable kinematical region hadronises in a quarkonium, a simple statistical counting [52] was proposed based on the spin \(J_\mathcal{Q}\) of the quarkonium \(\mathcal{Q}\), \(F_\mathcal{Q}= {1}/{9} \times {(2 J_\mathcal{Q} +1)}/{\sum _i (2 J_i +1)}\), where the sum over i runs over all the charmonium states below the open heavyflavour threshold. It was shown to reasonably account for existing \(\mathrm {J}/\psi \) hadroproduction data of the late 1990s and, in fact, is comparable to the fit value in [53].
Possibly due to its high predictive power, the CSM has faced several phenomenological issues although it accounts reasonably well for the bulk of hadroproduction data from RHIC to LHC energies [65, 66, 67], \(e^{+}e^{}\) data at B factories [68, 69, 70] and photoproduction data at HERA [71]. Taking into account NLO – one loop – corrections and approximate NNLO contributions (dubbed NNLO\(^\star \) in the following) has reduced the most patent discrepancies in particular for \(p_{\mathrm {T}}\) up to a couple of \(m_\mathcal{Q}\) [72, 73, 74, 75]. A full NNLO computation (i.e. at \(\alpha ^5_s\)) is, however, needed to confirm this trend.
It is, however, true that the CSM is affected by infrared divergences in the case of Pwave decay at NLO, which were earlier regulated by an ad hoc binding energy [76]. These can nevertheless be rigorously cured [77] in the more general framework of NRQCD which we discuss now and which introduce the concept of colouroctet states.
The ColourOctet Mechanism (COM) and NRQCD Based on the effective theory NRQCD [78], one can express in a more rigorous way the hadronisation probability of a heavyquark pair into a quarkonium via longdistance matrix elements (LDMEs). In addition to the usual expansion in powers of \(\alpha _s\), NRQCD further introduces an expansion in v. It is then natural to account for the effect of higherFock states (in v) where the \({Q\overline{Q}}\) pair is in an octet state with a different angularmomentum and spin states – the sole consideration of the leading Fock state (in v) amounts to the CSM, which is thus a priori the leading NRQCD contribution (in v). However, this opens the possibility for nonperturbative transitions between these coloured states and the physical meson. One of the virtues of this is the consideration of \(^3S_1^{[8]}\) states in Pwave productions, whose contributions cancel the aforementioned divergences in the CSM. The necessity for such a cancellation does not, however, fix the relative importance of these contributions. In this precise case, it depends on a nonphysical scale \(\mu _\Lambda \).
Three groups (Hamburg [80], IHEP [81] and PKU [82]) have, in recent years, carried out a number of NLO studies^{6} of cross section fits to determine the NRQCD LDMEs. A full description of the differences between these analyses is beyond the scope of this review, it is, however, important to stress that they somehow contradict each other in their results as regards the polarisation observables. In particular, in the case of the \(\mathrm {J}/\psi \), the studies of the Hamburg group, which is the only one to fit low \(p_{\mathrm {T}}\) data from hadroproduction, electroproduction and \(e^{+}e^{}\) collisions at B factories, predict a strong transverse polarised yield at variance with the experimental data.
Theory prospects Although NRQCD is 20 years old, there does not exist yet a complete proof of factorisation, in particular, in the case of hadroproduction. A discussion of the difficulties in establishing NRQCD factorisation can be found in [64]. A first step was achieved in 2005 by the demonstration [84, 85] that, in the large\(p_{\mathrm {T}}\) region where a description in terms of fragmentation functions is justified, the infrared poles at NNLO could be absorbed in the NRQCD LDMEs, provided that the NRQCD production operators were modified to include nonabelian phases.
As mentioned above, it seems that the mere expansion of the hard matrix elements in \(\alpha _s\) is probably not optimal since higher QCD corrections receive contributions which are enhanced by powers of \(p_{\mathrm {T}}/m_\mathcal{Q}\). It may therefore be expedient to organise the quarkoniumproduction cross section in powers of \(p_{\mathrm {T}}/m_\mathcal{Q}\) before performing the \(\alpha _s\)expansion of the shortdistance coefficients for the \({Q\overline{Q}}\) production. This is sometimes referred to as the fragmentationfunction approach (see [86, 87]) which offers new perspectives in the theoretical description of quarkonium hadroproduction especially at mid and large \(p_{\mathrm {T}}\). Complementary information could also be obtained from similar studies based on Soft Collinear Effective Theory (SCET); see [88].
2.2 Recent cross section measurements at hadron colliders
 1.
Fully reconstruction of exclusive decays, such as \(\mathrm{B}^0 \rightarrow \mathrm {J}/\psi \, \mathrm{K^0_S}\) or \(\mathrm {D}^{0} \rightarrow \mathrm{K}^{} \, \pi ^{+}\).
 2.
Selection of specific (semi)inclusive decays. For beauty production, one looks for a specific particle, for example \(\mathrm {J}/\psi \), and imposes it to point to a secondary vertex displaced a few hundred^{7} \(\upmu \)m from the primary vertex. Such displaced or nonprompt mesons are therefore supposed to come from bdecay only.
 3.
Detection of leptons from these decays. This can be done (i) by subtracting a cocktail of known/measured sources (photon conversions, Dalitz decays of \(\pi ^0\) and \(\eta \) in the case of electrons, light hadron, Drell–Yan pair, \(\mathrm {J}/\psi \),...) to the lepton yield. Alternatively, the photon conversion and Dalitz decay contribution can be evaluated via an invariantmass analysis of the \(e^{+}e^{}\) pairs. (ii) By selecting displaced leptons with a track pointing to a secondary vertex separated by few hundred \(\upmu \)m from the primary vertex.
 4.
Reconstruction of \(c \) and \(b \)jets. Once a jet is reconstructed, a variety of reconstructed objects, such as tracks, vertices and identified leptons, are used to distinguish between jets from light or from heavy flavour. A review of \(b \)tagging methods used by the CMS Collaboration can be found in [96].
Hiddenheavyflavour, i.e. quarkonia, are also analysed through their decay products. The triplet Swaves are the most studied since they decay up to a few per cent of the time in dileptons. This is the case for \(\mathrm {J}/\psi \), \(\psi \text {(2S)} \), \(\Upsilon \text {(1S)} \), \(\Upsilon \text {(2S)} \), \(\Upsilon \text {(3S)} \). The triplet Pwaves, such as the \(\chi _c \) and \(\chi _b \), are usually reconstructed via their radiative decays into a triplet Swave. For other states, such as the singlet Swave, studies are far more complex. The very first inclusive hadroproduction study of \(\eta _c\) was just carried out this year in the \({\mathrm{p}\overline{\mathrm{p}}} \) decay channel by the LHCb Collaboration [98].
2.2.1 Leptons from heavyflavour decays
The first openheavyflavour measurements in heavyion collisions were performed by exploiting heavyflavour decay leptons at RHIC by the PHENIX and STAR Collaborations. These were done both in \(\mathrm pp\) and AA collisions [112, 113, 114, 115, 116]. At the LHC, the ATLAS and ALICE Collaborations have also performed such studies in heavyion collisions [117, 118, 119, 120, 121]. A selection of the \(p_{\mathrm {T}}\)differential production cross sections of heavyflavour decay leptons in \(\mathrm pp\) collisions at different rapidities and energies is presented in Fig. 2. The measurements are reported together with calculations of FONLL [44, 99] for \(\sqrt{s}\) \(=\) 0.2 and 2.76 \(\text {TeV}\), GMVFNS [15, 16] and \(k_{\mathrm {T}}\)factorisation [122] at \(\sqrt{s}\) \(=\) 2.76 \(\text {TeV}\). The POWHEG predictions [49], not shown in this figure, show a remarkable agreement with the FONLL ones. The differential cross sections of heavyflavourdecay leptons are well described by pQCD calculations.
In addition, leptons from open charm and beauty production can be separated out via: (i) a cut on the lepton impact parameter, i.e. the distance between the origin of the lepton and the collision primary vertex, (ii) a fit of the lepton impactparameter distribution using templates of the different contributions to the inclusive spectra, (iii) studies of the azimuthal angular correlations between heavyflavour decay leptons and charged hadrons (see e.g. [107, 123]). These measurements are also described by pQCD calculations.
2.2.2 Open charm
Figure 3 presents a selection of the Dmeson measurements compared to pQCD calculations. The \(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \) and \(\mathrm {D}^{*+} \) \(\mathrm{d}\sigma /\mathrm{d}p_{\mathrm {T}} \) are reproduced by the theoretical calculations within uncertainties. Yet, FONLL [44, 99] and POWHEG [48] predictions tend to underestimate the data, whereas GMVFNS [15, 16] calculations tend to overestimate the data (see Figures 3 and 4 in [49]). At low \(p_{\mathrm {T}}\), where the quark mass effects are important, the FONLL and POWHEG predictions show a better agreement with data. At intermediate to high \(p_{\mathrm {T}}\), where the quark mass effects are less important, all the FONLL, POWHEG, GMVFNS and \(k_{\mathrm {T}}\)factorisation calculations agree with data. The agreement among the FONLL and POWHEG calculations is better for heavyflavour decay leptons than for charmed mesons, which seems to be related to the larger influence of the fragmentation model on the latter. The \(\mathrm {D}^{+}_{s} \) \(p_{\mathrm {T}} \)differential cross section is compared to calculations in Fig. 3c. The \(\mathrm {D}^{+}_{s} \) measurements are also reproduced by FONLL, GMVFNS and \(k_{\mathrm {T}}\)factorisation predictions, but POWHEG calculations predict a lower production cross section than data.
2.2.3 Open beauty
Openbeauty production is usually measured by looking for \(b \)jets or for beauty hadrons via their hadronic decays, similarly to D mesons. They have been traditionally studied at the \(e^{+}e^{} \) Bfactories (see e.g. [134, 135]), where, despite the small \(b \)quark production cross section, the large luminosity allows for precise measurements, such as those of the CKM matrix. Yet, heavier states like the \(\mathrm {B}_s\), \(\mathrm {B}_c\) or \(\Lambda _c\) cannot be produced at the Bfactories. They are, however, studied at Tevatron and at the LHC hadron colliders. The higher collision energy increases their production cross section, although their reconstruction is more difficult at hadron colliders due to the larger combinatorics compared to the \(e^{+}e^{} \) environment. It should also be kept in mind that the experiments optimised to study the high\(p_{\mathrm {T}} \) probes, like top production, are not as good for low\(p_{\mathrm {T}} \) measurements, and often require the usage of dedicated triggers.
As discussed in the Sect. 2.1.1, predictions for openbeauty cross sections rely on the fragmentation functions derived from fits to \(e^{+}e^{} \) data [35, 138]. A high accuracy on the \(e^{+}e^{} \) measurements and on the fragmentation function parametrisations is required to calculate the \(b \)hadron production cross section at hadron colliders. \(b \)jet measurements have the advantage to be the least dependent on the \(b \)quark fragmentation characteristics.
Studies of openbeauty production have also been performed in exclusive channels at Tevatron and at the LHC, e.g. in the case of \(\mathrm {B}^{\pm }\), \(\mathrm {B}^{0}\) and \(\mathrm {B}^0_s\) [139, 140, 142, 143, 144, 145, 146, 147, 148]. As example, Fig. 6 presents the \(\mathrm {B}^{+}\) \(p_{\mathrm {T}} \) and y differential cross section in \(\mathrm pp\) collisions at \(\sqrt{s}\) \(=\) 7 \(\text {TeV}\) compared to theory predictions [139, 140]. PYTHIA (D6T tune), that has LO \(+\) LL accuracy, does not provide a good description of the data. This could be explained by the choice of \(m_b\) and by the fact that for \(p_{\mathrm {T}} \simeq m_b\), NLO and resummation effects become important, which are, in part, accounted for in FONLL [44, 99] or MC@NLO. POWHEG and MC@NLO calculations are quoted with an uncertainty of the order of 20–40 %, from \(m_b\) and the renormalisation and factorisation scales, and describe the data within uncertainties. The FONLL prediction provides a good description of the measurements within uncertainties.
2.2.4 Prompt charmonium
In this section, we show and discuss a selection of experimental measurements of prompt charmonium production at RHIC and LHC energies. We thus focus here on the production channels which do not involve beauty decays; these were discussed in the Sect. 2.2.3.
Historically, promptly produced \(\mathrm {J}/\psi \) and \(\psi \text {(2S)} \) have always been studied in the dilepton channels. Except for the PHENIX, STAR and ALICE experiments, the recent studies in fact only considered dimuons which offer a better signaloverbackground ratio and a purer triggering. There are many recent experimental studies. In Fig. 9, we show only two of these. First we show \(\mathrm{d}\sigma /\mathrm{d}p_{\mathrm {T}} \) for prompt \(\mathrm {J}/\psi \) at \(\sqrt{s}\) \(=\) 7\(~\text {GeV}\) as measured by LHCb compared to a few predictions for the prompt yield from the CEM and from NRQCD at NLO^{8} as well as the direct yield^{9} compared to a NNLO\(^\star \) CS evaluation. Our point here is to emphasise the precision of the data and to illustrate that at low and mid \(p_{\mathrm {T}}\)– which is the region where heavyion studies are carried out – none of the models can simply be ruled out owing to their theoretical uncertainties (heavyquark mass, scales, nonperturbative parameters, unknown QCD and relativistic corrections, ...). Second, we show the fraction of \(\mathrm {J}/\psi \) from \(b \) decay for y close to 0 at \(\sqrt{s}\) \(=\) 7 \(\text {TeV}\) as function of \(p_{\mathrm {T}}\) as measured by ALICE [108], ATLAS [170] and CMS [171]. At low \(p_{\mathrm {T}}\), the difference between the inclusive and prompt yield should not exceed 10 % – from the determination of the \(\sigma _{{b\overline{b}}}\), it is expected to be a few percent at RHIC energies [111]. It, however, steadily grows with \(p_{\mathrm {T}}\). At the highest \(p_{\mathrm {T}}\) reached at the LHC, the majority of the inclusive \(\mathrm {J}/\psi \) is from \(b \) decays. At \(p_{\mathrm {T}} \simeq \) 10 \(~\text {GeV}\), which could be reached in future quarkonium measurements in Pb–Pb collisions, it is already three times higher than at low \(p_{\mathrm {T}}\): 1 \(\mathrm {J}/\psi \) out of three comes from \(b \) decays.
Ultimately the best channel to look at all \(n=1\) charmonium yields at once is that of baryonantibaryon decay. Indeed, all \(n=1\) charmonia can decay in this channel with a similar branching ratio, which is small, i.e. on the order of \(10^{3}\). LHCb is a pioneer in such a study with the first measurement of \(\mathrm {J}/\psi \) into \({\mathrm{p}\overline{\mathrm{p}}} \), made along that of the \(\eta _c\). The latter case is the first measurement of the inclusive production of the charmonium ground state. It indubitably opens a new era in the study of quarkonia at colliders. The resulting cross section is shown in Fig. 10c and was shown to bring about constraints [180, 183, 184] on the existing global fits of NRQCD LDMEs by virtue of heavyquark spin symmetry (HQSS) which is an essential property of NRQCD. As for now, it seems that the CS contributions to \(\eta _c\) are large – if not dominant – in the region covered by the LHCb data and the different CO have to cancel each others not to overshoot the measured yield. The canonical channel used to study \(\chi _{c1,2}\) production at hadron colliders corresponds to the studies involving P waves decaying into \(\mathrm {J}/\psi \) and a photon. Very recently the measurement of \(\chi _{c0}\) relative yield was performed by LHCb [185] despite the very small branching ratio \(\chi _{c0} \rightarrow \mathrm {J}/\psi +\gamma \) of the order of one percent, that is 30 (20) times smaller than that of \(\chi _{c1}\) (\(\chi _{c2}\)). LHCb found that \(\sigma (\chi _{c0})/\sigma (\chi _{c2})\) is compatible with unity for \(p_{\mathrm {T}} > \)4\(~\text {GeV}/c\), in striking contradiction with statistical counting, 1/5.
Fig. 11a shows the typical size of the feeddown fraction of the \(\chi _c\) and \(\psi \text {(2S)} \) into \(\mathrm {J}/\psi \) at low and high \(p_{\mathrm {T}}\), which are different. One should therefore expect differences in these fraction between \(p_{\mathrm {T}}\)integrated yields and yields measured at \(p_{\mathrm {T}} = 10~\text {GeV}/c\) and above. Figure 11b shows the ratio of the \(\chi _{c2}\) over \(\chi _{c1}\) yields as measured^{11} at the LHC by LHCb, CMS and at the Tevatron by CDF. On the experimental side, the usage of the conversion method to detect the photon becomes an advantage. LHCb is able to carry out measurements down to \(p_{\mathrm {T}}\) as small as 2\(~\text {GeV}/c\), where the ratio seems to strongly increase. This increase is in line with the Landau–Yang theorem according to which \(\chi _{c1}\) production from collinear and onshell gluons at LO is forbidden. Such an increase appears in the LO NRQCD band, less in the NLO NRQCD one. At larger \(p_{\mathrm {T}}\), such a measurement helps to fix the value of the NRQCD LDMEs (see the pioneering study of Ma et al. [189]). As we just discussed, once the photon reconstruction efficiencies and acceptance are known, one can derive the \(\chi _c\) feeddown fractions which are of paramount importance to interpret inclusive \(\mathrm {J}/\psi \) results. One can of course also derive absolute cross section measurements which are interesting to understand the production mechanism of the Pwave quarkonia per se; these may not be the same as that of Swave quarkonia. Figure 11c shows the \(p_{\mathrm {T}}\) dependence of the yield of the \(\chi _{c1}\) measured by ATLAS (under the hypothesis of an isotropic decay), which is compared to predictions from the LO CSM,^{12} NLO NRQCD and \(k_{\mathrm {T}}\) factorisation. The NLO NRQCD predictions, whose parameters have been fitted to reproduce the Tevatron measurement, is in good agreement with the data. Similar cross sections have been measured for the \(\chi _{c2}\).
2.2.5 Bottomonium
Fig. 12a shows the rapidity dependence of the \(\Upsilon \text {(1S)}\) yield from two complementary measurements, one at forward rapidities by LHCb and the other at central rapidities by CMS (multiplied by the expected fraction of direct \(\Upsilon \text {(1S)}\) as discussed below). These data are in line with the CS expectations; at least, they do not show an evident need for CO contributions, nor they exclude their presence. As for the charmonia, the understanding of their production mechanism for mid and high \(p_{\mathrm {T}}\) is a challenge. Figure 12b shows a typical comparison with five theory bands. In general, LHC data are much more precise than theory. It is not clear that pushing the measurement to higher \(p_{\mathrm {T}}\) would provide striking evidence in favour of one or another mechanism – associatedproduction channels, which we discuss in Sect. 2.4, are probably more promising. Figure 12c shows ratios of different Swave bottomonium yields. These are clearly not constant as one might anticipate following the idea of the CEM. Simple mass effects through feeddown decays can induce an increase of these ratios [74, 199], but these are likely not sufficient to explain the observed trend if all the direct yields have the same \(p_{\mathrm {T}}\) dependence. The \(\chi _b\) feeddown, which we discuss in the following, can also affect these ratios.
2.2.6 \(\mathrm{B}_c\) and multiplecharm baryons
After a discovery phase during which the measurement of the mass and the lifetime of the \(\mathrm{B}_c\) was the priority, the first measurement of the \(p_{\mathrm {T}}\) and y spectra of promptly produced \(\mathrm{B}_c^{+}\) was carried out by the LHCb Collaboration [210]. Unfortunately, as for now, the branching \(\mathrm{B}_c^{+} \rightarrow \mathrm {J}/\psi \ \pi ^{+}\) is not yet known. This precludes the extraction of \(\sigma _{\mathrm{pp}\rightarrow \mathrm{B}_c^{+} +X}\) and the comparison with the existing theoretical predictions [213, 214, 215, 216, 217, 218, 219, 220]. Aside from this normalisation issue, the \(p_{\mathrm {T}}\) and y spectra are well reproduced by the theory (see a comparison in Fig. 15 with BCVEGPY [211, 212], which is based on NRQCD where the CS contribution is dominant).
Searches for doubly charmed baryons are being carried out (see e.g. [221]) on the existing data sample collected in \(\mathrm pp\) collisions at 7 and 8 \(\text {TeV}\). As for now, no analysis could confirm the signals seen by the fixedtarget experiment SELEX at Fermilab [222, 223].
2.3 Quarkoniumpolarisation studies
Measurements of quarkonium polarisation can shed more light on the longstanding puzzle of the quarkonium hadroproduction. Various models of the quarkonium production, described in the previous Sect. 2.1.2, are in reasonable agreement with the cross section measurements but they usually fail to describe the measured polarisation.
We have collected in this section all results of polarisation measurements performed by different experiments at different collision energies \(\sqrt{s_{\mathrm{NN}}}\) and in different kinematic regions. The results for \(\mathrm {J}/\psi \) and \(\psi \text {(2S)}\) can be found in Tables 1 and 2 for \(\mathrm pp\) and p–A collisions. Since there is no known mechanism that would change quarkonium polarisation from proton–proton to proton–nucleus collisions, results from p–A collisions are also shown in this section. Tables 3, 4 and 5 gather the results for, respectively, the \(\Upsilon \text {(1S)}\), \(\Upsilon \text {(2S)}\) and \(\Upsilon \text {(3S)}\) in \(\mathrm pp\) collisions.
World existing data for \(\mathrm {J}/\psi \) polarisation in \(\mathrm pp\) and p–A collisions
\(\sqrt{s_{\mathrm{NN}}}\) (GeV)  Colliding system  Experiment  \(y\) range  \(p_{\mathrm {T}}\) range (GeV/c)  Feeddown  Fit  Measured parameter(s)  Observed trend 

200  \(\mathrm pp\)  PHENIX [238]  \(\vert y \vert <\) 0.35  0–5  B: \(2\div 15~\%\) [239], \(\chi _c\): \(23\div 41~\%\) [240], \(\psi \text {(2S)}\): \(5\div 20~\%\) [240]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX & GJ  \(\lambda _{\theta }\) values from slightly positive (consistent with 0) to negative as \(p_{\mathrm {T}}\) increases 
200  \(\mathrm pp\)  STAR [241]  \(\vert y \vert <\) 1  2–6  B: \(2\div 15~\%\) [239], \(\chi _c\): \(23\div 41~\%\) [240], \(\psi \text {(2S)}\): \(5\div 20~\%\) [240]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) values from slightly positive (consistent with 0) to negative as \(p_{\mathrm {T}}\) increases 
1800  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [242]  \(\vert y \vert <\) 0.6  4–20  \(\chi _c\): \(25\div 35~\%\) [186], \(\psi \text {(2S)}\): \(10\div 25~\%\) [243]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  Small positive \(\lambda _{\theta }\) at smaller \(p_{\mathrm {T}}\) then for \(p_{\mathrm {T}}\) \(>\) 12 \(~\text {GeV}\) trend towards negative values 
1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [244]  \(\vert y \vert <\) 0.6  5–30  \(\chi _c\): \(25\div 35~\%\) [186], \(\psi \text {(2S)}\): \(10\div 25\%\) [243]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) values from 0 to negative as \(p_{\mathrm {T}}\) increases 
7000  \(\mathrm pp\)  ALICE [245]  2.5–4.0  2–8  B: \(10\div 30~\%\) [108], \(\chi _c\): \(15\div 30~\%\) [246], \(\psi \text {(2S)}\): \(8\div 20~\%\) [199]  1D  \(\lambda _{\theta }\), \(\lambda _{\phi }\) vs \(p_{\mathrm {T}}\) in HX & CS  \(\lambda _{\theta }\) and \(\lambda _{\phi }\) consistent with 0, with a possible hint for a longitudinal polarisation at low \(p_{\mathrm {T}}\) in the HX frame 
7000  \(\mathrm pp\)  LHCb [229]  2.0–4.5  2–15  \(\chi _c\): \(15\div 30~\%\) [246], \(\psi \text {(2S)}\): \(8\div 25~\%\) [247]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\) vs \(p_{\mathrm {T}}\) and y in HX & CS  \(\lambda _{\phi }\) and \(\lambda _{\theta \phi }\) consistent with 0 in the HX frame and \(\lambda _{\theta }\) (\(\lambda _{\theta } = \) \(0.145\) \(\pm \) 0.027) shows small longitudinal polarisation; \(\tilde{\lambda }\) in agreement in the HX and CS frames 
7000  \(\mathrm pp\)  CMS [232]  \(\vert y \vert <\) 1.2  14–70  \(\chi _c\): \(25\div 35~\%\), \(\psi \text {(2S)}\): \(15\div 20~\%\) [171]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) and in 2 \(\vert y \vert \) bins in HX,CS,PX  In the three frames, no evidence of large \(\lambda _{\theta }\) anywhere; \(\tilde{\lambda }\) in good agreement in all reference frames 
17.2  p–A  NA60 [236]  0.28–0.78  –  B: \(2\div 8~\%\), \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\), \(\lambda _{\phi }\) vs. \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) and \(\lambda _{\phi }\) consistent with 0; slight increase of the \(\lambda _{\theta }\) value with increasing \(p_{\mathrm {T}}\), no \(p_{\mathrm {T}}\) dependence \(\lambda _{\phi }\) 
27.4  p–A  NA60 [236]  \(\)0.17 to 0.33  –  B: \(2\div 8~\%\), \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\), \(\lambda _{\phi }\) vs \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) and \(\lambda _{\phi }\) consistent with 0, no \(p_{\mathrm {T}}\) dependence observed 
31.5  \(p\mathrm {Be}\)  E672/E706 [250]  \(x_{F}\): 0.00–0.60  0–10  B: \(2\div 8~\%\), \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\) in GJ  \(\lambda _{\theta } = \) 0.01 \(\pm \) 0.12 \(\pm \) 0.09, consistent with no polarisation 
38.8  \(p\mathrm {Be}\)  E672/E706 [250]  \(x_{F}\): 0.00–0.60  0–10  B: \(2\div 8~\%\), \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\) in GJ  \(\lambda _{\theta } = \) \(0.11\) \(\pm \) 0.12 \(\pm \) 0.09, consistent with no polarisation 
38.8  \(p\mathrm {Si}\)  E771 [251]  \(x_{F}\): \(0.05\) to 0.25  0–3.5  B: \(2\div 8~\%\), \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\) in GJ  \(\lambda _{\theta } = \) \(0.09\) \(\pm \) 0.12, consistent with no polarisation 
38.8  \(p\mathrm {Cu}\)  E866/NuSea [252]  \(x_{F}\): 0.25–0.9  0–4  \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(7\div 10~\%\) [249]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) and \(x_{F}\) in CS  \(\lambda _{\theta }\) values from small positive to negative with increasing \(x_{F}\), no significant \(p_{\mathrm {T}}\) dependence observed 
41.6  \(p\mathrm {C,W}\)  HERAB [253]  \(x_{F}\): \(0.34\) to 0.14  0–5.4  \(\chi _c\): \(25\div 40~\%\) [248], \(\psi \text {(2S)}\): \(5\div 15~\%\) [254]  1D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\) vs \(p_{\mathrm {T}}\) and \(x_{F}\) in HX, CS, GJ  \(\lambda _{\theta }\) and \(\lambda _{\phi }\) \(<0\), \(\lambda _{\theta }\) (\(\lambda _{\phi }\)) decrease (increase) with increasing \(p_{\mathrm {T}}\); no strong \(x_{F}\) dependence; for \(p_{\mathrm {T}} > 1\) \(~\text {GeV}/c\), \(\lambda _{\theta ,\phi }\) depends on the frame: \(\lambda ^\mathrm{CS}_{\theta }>\lambda ^\mathrm{HX}_{\theta }\simeq 0\) and \(\lambda ^\mathrm{HX} _{\phi } \ne 0\) 
World existing data for \(\psi \text {(2S)}\) polarisation in \(\mathrm pp\) collisions
\(\sqrt{s}\) (GeV)  Colliding system  Experiment  \(y\) range  \(p_{\mathrm {T}}\) range (GeV/c)  Feeddown  Fit  Measured parameter(s)  Observed trend 

1800  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [242]  \(\vert y \vert <\) 0.6  5.5–20  None  1D  \(\lambda _{\theta }\) vs \(p_{T}\) in HX  \(\lambda _{\theta }\) consistent with 0 
1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [244]  \(\vert y \vert <\) 0.6  5–30  None  1D  \(\lambda _{\theta }\) vs \(p_{T}\) in HX  \(\lambda _{\theta }\) values vs \(p_{\mathrm {T}}\) go from slightly positive to small negative 
7000  \(\mathrm pp\)  LHCb [230]  2.0–4.5  3.5–15  None  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs. \(p_{\mathrm {T}}\) and 3 \(\vert y \vert \) bins in HX, CS  No significant polarisation found, with an indication of small longitudinal polarisation – \(\tilde{\lambda }\) is negative with no strong \(p_{\mathrm {T}}\) and y dependence 
7000  \(\mathrm pp\)  CMS [232]  \(\vert y \vert <\) 1.5  14–50  None  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs. \(p_{\mathrm {T}}\) and in 3 \(\vert y \vert \) bins in HX, CS, PX  Non of the three reference frames show evidence of large transverse or longitudinal polarisation, in the whole measured kinematic range; \(\tilde{\lambda }\) in good agreement in all reference frames 
World existing data for \(\Upsilon \text {(1S)}\) polarisation in \(\mathrm pp\) collisions
\(\sqrt{s}\) (GeV)  Colliding system  Experiment  \(y\) range  \(p_{\mathrm {T}}\) range (GeV/c)  Feeddown  Fit  Measured parameter(s)  Observed trend 

1800  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [255]  \(\vert y \vert <\) 0.4  0–20  \(\Upsilon \text {(2S)}\): \(6\div 18~\%\) [209], \(\Upsilon \text {(3S)}\): \(0.4\div 1.4~\%\) [209], \(\chi _{b}\): \(30\div 45~\%\) [209]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) consistent with 0, no significant \(p_{\mathrm {T}}\) dependence 
1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [256]  \(\vert y \vert <\) 0.6  0–40  \(\Upsilon \text {(2S)}\): \(6\div 18~\%\) [209], \(\Upsilon \text {(3S)}\): \(0.4\div 1.4~\%\) [209], \(\chi _{b}\): \(30\div 45~\%\) [209]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) in HX, CS  The angular distribution found to be nearly isotropic 
1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  D0 [257]  \(\vert y \vert <\) 0.4  0–20  \(\Upsilon \text {(2S)}\): \(6\div 18~\%\) [209], \(\Upsilon \text {(3S)}\): \(0.4\div 1.4~\%\) [209], \(\chi _{b}\): \(30\div 45~\%\) [209]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  Significant negative \(\lambda _{\theta }\) at low \(p_{\mathrm {T}}\) with decreasing magnitude as \(p_{\mathrm {T}}\) increases 
7000  \(\mathrm pp\)  CMS [233]  \(\vert y \vert <\) 1.2  10–50  \(\Upsilon \text {(2S)}\): \(10\div 15~\%\), \(\Upsilon \text {(3S)}\): \(0.5\div 3~\%\) [198], \(\chi _{b}\): \(25\div 40~\%\) [203]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) and in 2 \(\vert y \vert \) bins in HX, CS, PX  No evidence of large transverse or longitudinal polarisation in the whole kinematic range in the three reference frames 
World existing data for \(\Upsilon \text {(2S)}\) polarisation in \(\mathrm pp\) collisions
\(\sqrt{s}\) (GeV)  Colliding system  Experiment  \(y\) range  \(p_{\mathrm {T}}\) range (GeV/c)  Feeddown  Fit  Measured parameter(s)  Observed trend 

1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  D0 [257]  \(\vert y \vert <\) 0.4  0–20  \(\Upsilon \text {(3S)}\): \(2.5\div 5~\%\) [198], \(\chi _{b}\): \(25\div 35~\%\) [203]  1D  \(\lambda _{\theta }\) vs \(p_{\mathrm {T}}\) in HX  \(\lambda _{\theta }\) consistent with zero at low \(p_{\mathrm {T}}\), trend towards strong transverse polarisation at \(p_{\mathrm {T}} >5\) \(~\text {GeV}/c\) 
1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [256]  \(\vert y \vert <\) 0.6  0–40  \(\Upsilon \text {(3S)}\): \(2.5\div 5~\%\) [198], \(\chi _{b}\): \(25\div 35~\%\) [203]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) in HX, CS  The angular distribution found to be nearly isotropic 
7000  \(\mathrm pp\)  CMS [233]  \(\vert y \vert <\) 1.2  10–50  \(\Upsilon \text {(3S)}\): \(2.5\div 5~\%\) [198], \(\chi _{b}\): \(25\div 35~\%\) [203]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) and in 2 \(\vert y \vert \) bins in HX, CS, PX  No evidence of large transverse or longitudinal polarisation in whole kinematic range in the three reference frames 
World existing data for \(\Upsilon \text {(3S)}\) polarisation in \(\mathrm pp\) collisions
\(\sqrt{s}\) (GeV)  Colliding system  Experiment  \(y\) range  \(p_{\mathrm {T}}\) range (GeV/c)  Feeddown  Fit  Measured parameter(s)  Observed trend 

1960  \({\mathrm{p}\overline{\mathrm{p}}} \)  CDF [256]  \(\vert y \vert <\) 0.6  0–40  \(\chi _{b}\): \(30\div 50~\%\) [203]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) in HX, CS  The angular distribution found to be nearly isotropic 
7000  \(\mathrm pp\)  CMS [233]  \(\vert y \vert <\) 1.2  10–50  \(\chi _{b}\): \(30\div 50~\%\) [203]  2D  \(\lambda _{\theta }\), \(\lambda _{\phi }\), \(\lambda _{\theta \phi }\), \(\tilde{\lambda }\) vs \(p_{\mathrm {T}}\) and in 2 \(\vert y \vert \) bins in HX, CS, PX  No evidence of large transverse or longitudinal polarisation in whole kinematic range in the three reference frames 
In spite of the frame dependence of \(\lambda _{\theta }, \lambda _{\phi }, \lambda _{\theta \phi }\), there exist some combinations which are frame invariant [224, 228]. An obvious one is the yield, another one is \(\tilde{\lambda } = (\lambda _{\theta } + 3 \lambda _{\phi }) / (1  \lambda _{\phi })\) [224]. As such, it can be used as a good crosscheck between measurements done in different reference frames. Different methods have been used to extract the polarisation parameter(s) from the angular dependence of the yields. In the following, we divide them into two groups: (i) 1D technique: fitting \(\cos \theta \) distribution with the angular distribution, Eq. (11), averaged over the azimuthal \(\phi \) angle, and fitting the \(\phi \) distribution, Eq. (11), averaged over the polar \(\theta \) angle (ii) 2D technique: fitting a twodimensional \(\cos \theta \) vs \(\phi \) distribution with the full angular distribution, Eq. (11).
Beyond the differences in the methods employed to extract these parameters, one should also take into consideration that some samples are cleaner than other ones, physicswise.^{14} Indeed, as we discussed in the previous section, a given quarkonium yield can come from different sources, some of which are not of specific interests for data–theory comparisons. The most obvious one is the nonprompt charmonium yield, which is expected to be the result of quite different mechanism that the prompt yield. Nowadays, the majority of the studies are carried out on a prompt sample thanks to a precise vertexing of the events. Yet, a further complication also comes from feeddown from the excited states in which case vertexing is of no help. As for now, no attempt of removing it from e.g. prompt \(\mathrm {J}/\psi \) and inclusive \(\Upsilon \text {(1S)}\) samples has been made owing its intrinsic complication. We have therefore found it important to specify what kind of feeddown could be expected in the analysed sample.
In view of this, Tables 1, 2, 3, 4 and 5 contain, in addition to the information on the colliding systems and the kinematical coverages, information on the fit technique and a short reminder of the expected feeddown. For each measurement, we also briefly summarise the observed trend. The vast majority of the experimental data do not show a significant quarkonium polarisation, neither polar nor azimuthal anisotropy. Yet, values as large as \(\pm 0.3\) are often not excluded either – given the experimental uncertainties. Despite these, a simultaneous description of both measured quarkonium cross sections and polarisations is still challenging for theoretical models of quarkonium hadroproduction.
As an example, we show in Fig. 16 the \(p_{\mathrm {T}}\)dependence of \(\lambda _{\theta }\) for prompt \(\mathrm {J}/\psi \) [229] (left panel) and \(\psi \text {(2S)}\) [230] (right panel) measured by LHCb at \(2.5<y<4.0\) in the helicity frame compared with a few theoretical predictions. NLO NRQCD calculations [80, 81, 82] show mostly positive or zero values of \(\lambda _{\theta }\) with a trend towards the transverse polarisation with increasing \(p_{\mathrm {T}}\), and a magnitude of the \(\lambda _{\theta }\) depending on the specific calculation and the kinematical region. On the other hand, NLO CSM models [59, 72] tend to predict an unpolarised yield at low \(p_{\mathrm {T}}\) and an increasingly longitudinal yield (\(\lambda _{\theta } < 0\)) for increasing \(p_{\mathrm {T}}\). None of these predictions correctly describes the measured \(\mathrm {J}/\psi \) and \(\psi \text {(2S)}\) \(\lambda _{\theta }\) parameters and their \(p_{\mathrm {T}}\) trends. The NLO NRQCD fits of the PKU group [180, 231], however, open the possibility for an unpolarised direct yield but at the cost of not describing the world existing data in ep and \(e^{+}e^{}\) collisions and data in \(\mathrm pp\) collisions for \(p_{\mathrm {T}} \le 5\) \(~\text {GeV}/c\).
In order to illustrate the recent progress in these delicate studies, let us stress that LHC experiments have performed measurements of the three polarisation parameters as well as in different reference frames. This has not always been the case before by lack of statistics and of motivation since it is difficult to predict theoretically azimuthal effects, e.g. \(\lambda _{\theta \phi }\). Figures 17a and b show CMS measurements of \(\lambda _{\theta }\), \(\lambda _{\phi }\) and \(\lambda _{\theta \phi }\), in the HX frame for \(\mathrm {J}/\psi \), \(\psi \text {(2S)}\) [232] and \(\Upsilon \text {(1S)}\), \(\Upsilon \text {(2S)}\), \(\Upsilon \text {(3S)}\) [233], respectively. CMS has also conducted polarisation measurements in the CS and PX frames, in addition to the HX frame and they could crosscheck their analysis by obtaining the consistency in \(\tilde{\lambda }\) in these three frames for different \(p_{\mathrm {T}}\) and y. As for most of the previous measurements, no evidence of a large transverse or longitudinal quarkonium polarisation is observed in any reference frame, and in the whole measured kinematic range.
2.4 New observables
Thanks to the large heavyflavour samples available at hadron colliders, studies of the production of open or hidden heavyflavour production in association with another particle (light or heavy hadrons, quarkonium, or vector boson) are possible. The cross section of these processes is heavily sensitive to the particle production mechanisms and can help distinguishing between them. In addition, these final states can also results from multiple parton–parton interactions (or doubleparton scatterings, DPS), where several hard partonparton interactions occur in the same event [258, 259, 260, 261]. Analogously, heavyflavourproduction dependence with the underlying event multiplicity brings information about their production mechanisms. A complete understanding of heavyflavour production in hadronic collisions is mandatory to interpret heavyflavour measurements in p–A and AA collisions, and disentangle cold (see Sect. 3) and hot (see Sects. 4 and 5) nuclear matter effects at play.
2.4.1 Production as a function of multiplicity
The correlation of open or hidden heavyflavour yields with charged particles produced in hadronic collisions can provide insight into their production mechanism and into the interplay between hard and soft mechanisms in particle production. In high energy hadronic collisions, multiple parton–parton interactions may also affect heavyflavour production [262, 263], in competition to a large amount of QCDradiation associated to hard processes. In addition to these initialstate effects, heavyflavour production could suffer from finalstate effects due to the high multiplicity environment produced in high energy \(\mathrm pp\) collisions [264, 265].
At the LHC, \(\mathrm {J}/\psi \) yields were measured as a function of the chargedparticle density at midrapidity by the ALICE Collaboration in \(\mathrm pp\) collisions at \(\sqrt{s}\) \(=\) 7 \(\text {TeV}\) [266]. Figure 18 shows the \(\mathrm {J}/\psi \) yields at forward rapidity, studied via the dimuondecay channel at \(2.5 < y < 4\), and at midrapidity, analysed in its dielectrondecay channel at \(y < 0.9\). The results at mid and forwardrapidities are compatible within the measurement uncertainties, indicating similar correlations over three units of rapidity and up to four times the average chargedparticle multiplicity. The relative \(\mathrm {J}/\psi \) yield increases with the relative chargedparticle multiplicity. This increase can be interpreted in terms of the hadronic activity accompanying \(\mathrm {J}/\psi \) production, as well as in terms of partonparton interactions, or in the percolation scenario [267].
Hidden and open heavyflavour production measurements as a function of the event activity were initiated during the LHC Run 1 leading to unexpected results with impact on our understanding of the production mechanisms and the interpretation of p–Pb and Pb–Pb results. Run 2 data, with the increased centreofmass energy of 13 \(\text {TeV}\) in \(\mathrm pp\) collisions and larger luminosities, will allow one to reach higher multiplicities and to perform \(p_{\mathrm {T}} \)differential studies of hidden and open heavyflavour hadron production.
2.4.2 Associated production
Heavyflavour azimuthal correlations in hadronic collisions allow for studies of heavyquark fragmentation and jet structure at different collision energies, which help to constrain Monte Carlo models, and to understand the different production processes for heavy flavour. Heavy quarks can originate from flavour creation, flavour excitation, and parton shower or fragmentation processes of a gluon or a light (anti)quark including gluon splitting [273]. These three different sources of the heavyflavour production are expected to lead to different correlations between heavy quark and antiquark, and so a measurement of the opening angle in azimuth (\(\Delta \phi \)) of two heavyflavour particles gives an access to different underlying production subprocesses. Azimuthal correlations arising from the flavour creation populate mostly the awayside (\(\Delta \phi \approx \pi \)), while the nearside region (\(\Delta \phi \approx 0\)) is sensitive to the presence of the flavour excitation and gluon splitting [273]. Since D–D and B–B correlation measurements are statistically demanding one can also look at angular correlations between heavyflavour particles with charged hadrons (e.g. D–h) and correlations between electrons from heavyflavour decays with charged (e.g. \(e_{HF}\)–h) or heavyflavour hadrons (e.g. \(e_{HF}\)–D).
In addition to providing information on the heavyflavour production mechanisms, the azimuthal correlations of heavyflavour hadrons with light particles allow one to extract the relative contribution of charm and beauty hadron decays to the heavyhadron yields. Due to the different decay kinematics, the azimuthal distribution of the particles produced from Bhadron decays presents a wider distribution at \(\Delta \phi \approx 0\) than the one for D decays. The \(e_{HF}\)–h angular correlations were measured at midrapidity in \(\mathrm pp\) collisions at \(\sqrt{s} = 200\) \(~\text {GeV}\) [111, 113, 115] and at \(\sqrt{s} = 2.76\) \(\text {TeV}\) [107]. Figure 22a presents the azimuthal correlation of \(e_{HF}\)–h at the LHC. PYTHIA calculations of the D and B decay contributions are also shown. The contribution of beauty decays to the heavyflavour electron yield increases with \(p_{\mathrm {T}} \) and is described by FONLL pQCD calculations, both at \(\sqrt{s} =200\) \(~\text {GeV}\) and at \(\sqrt{s} = 2.76\) \(\text {TeV}\) (see Fig. 22 [107, 115]). The beauty contribution to heavyflavour electron yields becomes as important as the charm one at \(p_{\mathrm {T}} \sim 5\) \(~\text {GeV}/c\). The results of \(e_{HF}\) – \(\mathrm {D}^{0} \) angular correlations at \(\sqrt{s} = 200\) \(~\text {GeV}\) are consistent with the \(e_{HF}\)–h ones [115]; see Fig. 22b.
 1.
 2.
open charm hadron plus a \(\mathrm {J}/\psi \) or another open charm hadron at LHCb [274],
 3.
 4.
open charm hadron plus a W boson at CMS [283],
 5.
\(\mathrm {J}/\psi \) and W production at ATLAS [284],
 6.
\(\mathrm {J}/\psi \) and Z production at ATLAS [285],
 7.
open beauty hadron or jet plus a Z boson at CDF [286] and D0 [287], and at ATLAS [288], CMS [289] and LHCb [290],
 8.
the search of production of \(\Upsilon (1S)\) associated with W or Z production at CDF [291],
 9.
the search of the exclusive decay of \(H^0\) into \(\mathrm {J}/\psi +\gamma \) and \(\Upsilon +\gamma \) [292].
This is also supported by the measurement of the ratio of the double and inclusive production cross sections, defined as \(R_{C_1 \, C_2}=\alpha ( \sigma _{C_1} \sigma _{C_2} / \sigma _{C_1C_2} )\), where \(\alpha = 1/4\) if \(C_1\) and \(C_2\) are identical and nonselfconjugate, \(\alpha = 1\) if \(C_1\) and \(C_2\) are different and either \(C_1\) or \(C_2\) is selfconjugate, and \(\alpha = 1/2\) otherwise. This quantity, which would be equal to \(\sigma _\mathrm{eff}\) in the case of a pure DPS yield, was evaluated by LHCb for the different aforementioned observed systems. These are plotted in Fig. 23 (right) and are compared, in the case of \(\mathrm {J}/\psi +\) charm, to the results obtained from multijet events at the Tevatron, displayed by a green shaded area in the figure. They point at values close to 15 mb.
However, double \(\mathrm {J}/\psi \) production has recently been studied by D0 [279] and CMS [280] respectively at large rapidity separations and large transverse momenta. As for now, the D0 [279] study is the only one which really separated out the double and singlepartonscattering contributions by using the yield dependence on the (pseudo)rapidity difference between the J\(/\psi \) pair, \(\Delta y\), an analysis which was first proposed in Ref. [294]. The DPS rapidityseparation spectrum is much broader and it dominates at large \(\Delta y\). D0 has obtained the result that, in the region where DPS should dominate, the extracted value of \(\sigma _\mathrm{eff}\) is on the order of 5 mb, that is significantly smaller than the values obtained with multijet events and \(\mathrm {J}/\psi +\) charm as just discussed. At small rapidity separations, the usual singlepartonscattering (SPS) contribution is found to be dominant and the yield is well accounted for by the CSM at NLO [306, 307, 308]. CO contributions are only expected to matter at very large transverse momenta, in particular at large values of the smaller \(p_{\mathrm {T}}\) of both \(p_{\mathrm {T}}\) of each \(\mathrm {J}/\psi \).
Such a small value of \(\sigma _\mathrm{eff}\) (meaning a large DPS yield) has been shown to be supported by the CMS measurement [280] at 7 TeV which overshoots by orders of magnitude the NLO SPS predictions at large transverse momenta. Indeed, adding the DPS yield obtained with \(\sigma _\mathrm{eff}=5\) mb solves [308] this apparent discrepancy first discussed in [307].
To summarise, the study of associated production of heavy quarks and heavy quarkonia has really taken off with the advent of the LHC and the analysis of the complete data sample taken at the Tevatron. There is no doubt that forthcoming studies will provide much more new information – and probably also puzzles – on the production of these particles. It is also probable that some of these observables at LHC energies are dominated by DPS contributions and, in such a case, specific nuclear dependences should be observed in proton–nucleus and nucleus–nucleus collisions (see e.g. [310, 311]).
2.5 Summary and outlook

Heavyflavour decay lepton \(p_{\mathrm {T}}\) and \(y\)differential production cross sections are well described by pQCD calculations.

Dmeson \(p_{\mathrm {T}}\)differential cross sections are well described by pQCD calculations within uncertainties. FONLL and POWHEG central calculations tend to underestimate the data, whereas GMVFNS tends to overestimate it. The \(\Lambda _c^{+}\) \(p_{\mathrm {T}}\)differential cross section was measured up to 8\(~\text {GeV}/c\) and is well described by GMVFNS.

The \(p_{\mathrm {T}}\)differential cross section of charmonia from beauty decays (nonprompt \(\mathrm {J}/\psi \), \(\psi \text {(2S)}\), \(\eta _c\), \(\chi _{c1}\) and \(\chi _{c2}\)) at low to intermediate \(p_{\mathrm {T}}\) is well described by pQCD calculations. At high \(p_{\mathrm {T}}\) the predictions tend to overestimate the data. \(p_{\mathrm {T}}\) and \(y\)differential cross section measurements were performed for exclusive decays: \(\mathrm {B}^{\pm }\), \(\mathrm {B}^{0}\) and \(\mathrm {B}^0_s\). bjet cross section measurements are well described by pQCD calculations taking into account the matching between NLO calculations and parton showers.

The \(\mathrm{B}_c^{+}\) \(p_{\mathrm {T}}\) and \(y\)differential cross section was for the first time measured at the LHC and it is well reproduced by theory.

Prompt \(\mathrm {J}/\psi \) and \(\psi \text {(2S)}\) differential cross sections were measured, none of the tested models can be ruled out due to large theoretical uncertainties.

\(\Upsilon \text {(1S)}\) differential cross section description remains a challenge at mid and high \(p_{\mathrm {T}}\), LHC data being more precise than theory.

Quarkonium polarisation studies were performed in various reference frames for \(\mathrm {J}/\psi \), \(\psi \text {(2S)}\) and \(\Upsilon \). At present, none of the models can describe all observed features.

Inclusive \(\mathrm {J}/\psi \) (at central and forward rapidity), prompt D meson and nonprompt \(\mathrm {J}/\psi \) (at central rapidity) yields were measured at \(\sqrt{s}\) \(=\) 7 \(\text {TeV}\) versus chargedparticle multiplicity. Heavyflavour yields increase as a function of chargedparticle multiplicity at midrapidity; Dmeson results present a fasterthanlinear increase at the highest multiplicities. Possible interpretations of these results are the contribution of multipleparton interactions or the event activity accompanying heavyflavour hadrons. The increase of the prompt Dmeson yields is qualitatively reproduced by an hydrodynamic calculation with the EPOS event generator and the percolation scenario. The \(\Upsilon \) measurement at \(\sqrt{s}\) \(=\) 2.76 \(\text {TeV}\) also presents an increase with chargedparticle multiplicity but the decrease of the fraction of the \(\Upsilon \text {(nS)}\) to the \(\Upsilon \text {(1S)}\) state is at present not understood.

Measurements of the azimuthal correlations between charm (beauty) and anticharm (antibeauty) point to the importance of the near production via the gluon splitting mechanism in addition to the backtoback production.

\(\mathrm {J}/\psi \) plus open charm and double open charm hadron production cross section measurements suggest a nonnegligible contribution of doubleparton scatterings to double charm production. Measurements of vector boson production in association with a \(\mathrm {J}/\psi \) provide further constraints to the model calculations.
3 Cold nuclear matter effects on heavy flavour and quarkonium production in proton–nucleus collisions
Characterising the hot and dense medium produced in heavyion (AA) collisions requires a quantitative understanding of the effects induced by the presence of nuclei in the initialstate, the socalled cold nuclear matter (CNM) effects. These effects can be studied in proton–nucleus (p–A) or deuteron–nucleus (d–A) collisions.^{15}
This section starts (Sect. 3.1) with a brief introduction to the physics of CNM effects on heavy flavour and with a compilation of available p–A data. Next, the different theoretical approaches are discussed in Sect. 3.2, before a review of recent RHIC and LHC experimental results in Sect. 3.3. Afterwards, the extrapolation of CNM effects from p–A to AA collisions is discussed in Sect. 3.4, from both the theoretical and the experimental points of view. Finally, Sect. 3.5 includes a summary and a discussion of shortterm perspectives.
3.1 Heavy flavour in p–A collisions

Modification of the effective partonic luminosity in colliding nuclei, with respect to colliding protons. This effect is due to the different dynamics of partons within free protons with respect those in nucleons, mainly as a consequence of the larger resulting density of partons. These effects depend on x and on the scale of the parton–parton interaction \(Q^2\) (the square of the fourmomentum transfer). In collinearly factorised pQCD calculations the nuclear effects on the parton dynamics are described in terms of nuclearmodified PDFs (hereafter indicated as nPDF). Quite schematically three regimes can be identified for the nPDF to PDF ratio of parton flavour i, \(R_i(x,Q^2)\), depending on the values of x: a depletion (\(R_i<1\)) – often referred to as shadowing and related to phasespace saturation – at small \(x \lesssim 10^{2}\), a possible enhancement \(R_i>1\) (antishadowing) at intermediate values \(10^{2} \lesssim x \lesssim 10^{1}\), and the EMC effect, a depletion taking place at large \(x \gtrsim 10^{1}\). The \(R_i(x,Q^2)\) parametrisations are determined from a global fit analyses of lepton–nucleus and proton–nucleus data (see Sect. 3.2.2).

The physics of parton saturation at small x can also be described within the Colour Glass Condensate (CGC) theoretical framework. Unlike the nPDF approach, which uses DGLAP linear evolution equations, the CGC framework is based on the Balitsky–Kovchegov or JIMWLK nonlinear evolution equations (see Sect. 3.2.3).

Multiple scattering of partons in the nucleus before and/or after the hard scattering, leading to parton energyloss (either radiative or collisional) and transverse momentum broadening (known as the Cronin effect). In most approaches (see Sect. 3.2.4) it is characterised by the transport coefficient of cold nuclear matter, \(\hat{q}\).

Finalstate inelastic interaction, or nuclear absorption, of \({Q\overline{Q}}\) bound states when passing through the nucleus. The important parameter of these calculations is the “absorption” (or breakup) cross section \(\sigma _\mathrm {abs}\), namely the inelastic cross section of a heavyquarkonium state with a nucleon.

On top of the above genuine CNM effects, the large set of particles (partons or hadrons) produced in p–A collisions at high energy may be responsible for a modification of open heavy flavour or quarkonium production. It is still highly debated whether this set of particles could form a “medium” with some degree of collectivity. If this was the case, this medium could impart a flow to heavyflavour hadrons. Moreover, heavy quarkonia can be dissociated by comovers, i.e., the partons or hadrons produced in the collision in the vicinity of the heavyquarkonium state (see Sect. 3.2.5).
The typical range for the momentum fractions probed is therefore a function of both the acceptance of the detector (rapidity coverage) and the nature of the particles produced and their associated energy scale. Moreover, assuming different underlying partonic production processes can end up in average values of x that may differ from one another.
Available p–A data in collider: the probes, the colliding system, \(\sqrt{s_{\mathrm{NN}}} \), the kinematic range (with \(y \) the rapidity in the centreofmass frame), the observables (as a function of variables) are given as well as the references
Probes  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (TeV)  \(y\)  Observables (variables)  References 

PHENIX  
\(\mathrm{HF} \rightarrow e^{\pm }\)  d–Au  0.2  \(y <0.35\)  \(R_{\mathrm {dAu}}\) (\(p_{\mathrm {T}}\),\(\mathrm {N_{coll}}\)), \(\langle p_{\mathrm {T}} ^2\rangle \)  [313] 
\(\mathrm{HF} \rightarrow \mu ^{\pm }\)  \(1.4<y <2\)  \(R_{\mathrm {dAu}}\) (\(\mathrm {N_{coll}}\),\(p_{\mathrm {T}}\))  [314]  
\({b\overline{b}}\)  \(y <0.5\)  \(\sigma (y)\)  [315]  
\(e^\pm ,\mu ^\pm \)  \(y <0.5\) & \(1.4<y <2.1\)  \(\Delta \phi \), \(J_\mathrm{dAu}\)  [114]  
\(\mathrm {J}/\psi \)  \(2.2<y <2.4\)  \(R_{\mathrm {dAu}}\), \(R_{\mathrm {CP}}\) (\(\mathrm {N_{coll}}\),\(y\),\(x_2\),\(x_{\mathrm {F}}\),\(p_{\mathrm {T}}\)), \(\alpha \)  
\(2.2<y <2.2\)  \(R_{\mathrm {dAu}}\) (\(p_{\mathrm {T}}\),\(y\),\(\mathrm {N_{coll}}\)), \(\langle p_{\mathrm {T}} ^2\rangle \)  [319]  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\), \(\chi _c\)  \(y <0.35\)  \(R_{\mathrm {dAu}}\) (\(\mathrm {N_{coll}}\)), double ratio  [320]  
\(\Upsilon \)  \(1.2<y <2.2\)  \(R_{\mathrm {dAu}}\) (\(y\),\(x_2\),\(x_{\mathrm {F}}\)), \(\alpha \)  [321]  
STAR  
\(\mathrm {D}^{0}\), \(\mathrm{HF} \rightarrow e^{\pm }\)  d–Au  0.2  \(y <1\)  Yield(\(y\),\(p_{\mathrm {T}}\))  [322] 
\(\Upsilon \)  \(y <1\)  \(\sigma \), \(R_{\mathrm {dAu}}\) (\(y\),\(x_{\mathrm {F}}\)), \(\alpha \)  [323]  
ALICE  
D  p–Pb  5.02  \(0.96<y <0.04\)  \(\sigma \), \(R_{\mathrm {pPb}}\) (\(p_{\mathrm {T}}\),\(y\))  [324] 
\(\mathrm {J}/\psi \)  \(4.96<y <2.96\) & \(2.03<y <3.53\)  \(\sigma \), \(R_{\mathrm {pPb}}\) (\(y\)), \(R_{\mathrm {FB}}\)  [325]  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  \(\sigma \), \(R_{\mathrm {pPb}}\) (\(y\),\(p_{\mathrm {T}}\)), double ratio  [326]  
\(\mathrm {J}/\psi \)  & \(1.37<y <0.43\)  \(\sigma \)(\(y\),\(p_{\mathrm {T}}\)), \(R_{\mathrm {pPb}}\) (\(y\),\(p_{\mathrm {T}}\)), [\(R_{\mathrm {pPb}}\) (+\(y\))\(\cdot \) \(R_{\mathrm {pPb}}\) (\(y\))] (\(p_{\mathrm {T}}\))  [327]  
\(\Upsilon \text {(1S)}\), \(\Upsilon \text {(2S)}\)  \(\sigma \), \(R_{\mathrm {pPb}}\) (\(y\)), \(R_{\mathrm {FB}}\), ratio  [328]  
ATLAS  
\(\mathrm {J}/\psi \) (from B)  p–Pb  5.02  \(2.87< y <1.94\)  \(\sigma (y,p_{\mathrm {T}})\), ratio(y,\(p_{\mathrm {T}}\)), \(R_{\mathrm {FB}}\) (y,\(p_{\mathrm {T}}\))  [329] 
CMS  
\(\Upsilon \text {(nS)}\)  p–Pb  5.02  \(y <1.93\)  Double ratio (\(E_\mathrm{T}^{\eta >4},N_\mathrm{tracks}^{\eta <2.4})\)  [268] 
LHCb  
\(\mathrm {J}/\psi \) (from B)  p–Pb  5.02  \(5.0<y <2.5\) & \(1.5<y<4.0\)  \(\sigma (p_{\mathrm {T}},y)\), \(R_{\mathrm {pPb}}\) (\(y\)), \(R_{\mathrm {FB}}\) (\(y\),\(p_{\mathrm {T}}\))  [330] 
\(\Upsilon \text {(nS)}\)  \(\sigma (y)\), ratio(\(y\)), \(R_{\mathrm {pPb}}\) (\(y\)), \(R_{\mathrm {FB}}\)  [331] 
Available p–A data in fixed target: the probes, the target, \(\sqrt{s_{\mathrm{NN}}} \), the kinematic range (with \(y \) the rapidity in the centreofmass frame), the observables (as a function of variables) are given as well as the references. The superscript letter a (\(^\mathrm{a}\)) means that a cut on \(\cos \theta _\mathrm{CS}<0.5\) is applied in the analysis, where \(\theta _\mathrm{CS}\) is the decay muon angle in the Collins–Soper frame. Feynmanx variable \(x_{\mathrm {F}} =\frac{2p_\mathrm{L,CM}}{\sqrt{s_\mathrm{NN}}}\), where \(p_\mathrm{L,CM}\) is the longitudinal momentum of the partonic system in the CM frame, is connected to the momentum fraction variables by \(x_{\mathrm {F}} \approx x_1x_2\), in the limit \(p_\mathrm{T} \ll p\)
Probes  Target  \(\sqrt{s_{\mathrm{NN}}}\) (GeV)  \(y\) (or \(x_{\mathrm {F}}\))  Observables (variables)  References 

NA3  
\(\mathrm {J}/\psi \)  H\(_2\), Pt  16.8–27.4  \(0<x_{\mathrm {F}} <0.9\)  \(\sigma (x_{\mathrm {F}},p_{\mathrm {T}})\)  
NA38  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  Cu, U  19.4  \(0.2<y<1.1\)  \(\sigma (E_\mathrm{T},A)\), \(\langle p_{\mathrm {T}} ^{(2)}\rangle (\epsilon )\), ratio(\(\epsilon ,E_\mathrm{T},A,L)\)  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\), \({c\overline{c}}\)  W  19.4  \(0<y<1\)  Ratio(\(\epsilon \)), \(\sigma _{{c\overline{c}}}(p_\mathrm{lab})\)  [337] 
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  C, Al, Cu, W  29.1  \(0.4<y <0.6^\mathrm{a}\)  \(\sigma (A)\) and ratio(A)  [338] 
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\), DY  O, S  19.4 (29.1)  \(0(0.4)<y <1(0.6)\)  \(\sigma (A,L)\), ratio(A, L)  [339] 
NA38/NA50  
\({c\overline{c}}\)  Al, Cu, Ag, W  29.1  \(0.52<y <0.48^\mathrm{a}\)  \(\sigma _{{c\overline{c}}}\)  [97] 
NA50  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\), DY  Be, Al, Cu, Ag, W  29.1  \(0.4<y <0.6\)  \(\sigma (A)\), ratio\((A,E_\mathrm{T},L)\), \(\sigma _\mathrm {abs} \)  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  \(0.1<x_{\mathrm {F}} <0.1^\mathrm{a}\)  \(\sigma (A,L)\), \(\sigma _\mathrm {abs}\) (\(x_{\mathrm {F}} \))  [342]  
\(\Upsilon \), DY  \(0.5<y <0.5^\mathrm{a}\)  \(\sigma (A)\), \(\langle p_{\mathrm {T}} ^2\rangle (L)\), \(\langle p_{\mathrm {T}} \rangle \)  [343]  
NA60  
\(\mathrm {J}/\psi \)  Be, Al, Cu, In, W, Pb, U  17.3 (27.5)  \(0.3(0.2)<y<0.8(0.3)^\mathrm{a}\)  \(\sigma \), \(\sigma _\mathrm {abs}\), ratio(L), \(\alpha (x_{\mathrm {F}},x_2)\)  [344] 
E772  
\(\mathrm {J}/\psi \),\(\psi \text {(2S)}\)  H\(_2\), C, Ca, W  38.8  \(0.1<x_{\mathrm {F}} <0.7\)  Ratio(A,\(x_{\mathrm {F}}\),\(p_{\mathrm {T}}\)), \(\alpha (x_{\mathrm {F}},x_2,p_{\mathrm {T}})\)  [345] 
\(\Upsilon \)  \(0.15<x_{\mathrm {F}} <0.5\)  \(\sigma (p_{\mathrm {T}},x_{\mathrm {F}})\), ratio(A), \(\alpha (x_{\mathrm {F}},x_2,p_{\mathrm {T}})\)  [346]  
E789  
\(\mathrm {D}^{0}\)  Be, Au  38.8  \(0<x_{\mathrm {F}} <0.08\)  \(\sigma (p_{\mathrm {T}})\), \(\alpha (x_{\mathrm {F}},p_{\mathrm {T}})\), ratio  [347] 
\({b\overline{b}}\)  \(0<x_{\mathrm {F}} <0.1\)  \(\sigma (x_{\mathrm {F}},p_{\mathrm {T}})\)  [348]  
\(\mathrm {J}/\psi \)  Be, Cu  38.8  \(0.3<x_{\mathrm {F}} <0.95\)  \(\sigma (x_{\mathrm {F}})\), \(\alpha (x_{\mathrm {F}})\)  [349] 
\(\mathrm {J}/\psi \),\(\psi \text {(2S)}\)  Be, Au  \(0.03<x_{\mathrm {F}} <0.15\)  \(\sigma (p_{\mathrm {T}},x_{\mathrm {F}},y)\), ratio(\(p_{\mathrm {T}}\),\(x_{\mathrm {F}}\))  [350]  
\(\mathrm {J}/\psi \)  Be, C, W  \(0.1<x_{\mathrm {F}} <0.1\)  \(\alpha \) (\(x_{\mathrm {F}}\),\(x_\mathrm{target}\),\(p_{\mathrm {T}}\))  [351]  
E866/NuSea  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  Be, Fe, W  38.8  \(0.1<x_{\mathrm {F}} <0.93\)  \(\alpha \) (\(p_{\mathrm {T}}\),\(x_{\mathrm {F}}\))  [352] 
\(\mathrm {J}/\psi \)  Cu  \(0.3<x_{\mathrm {F}} <0.9\)  \(\lambda _\theta (p_{\mathrm {T}},x_{\mathrm {F}})\)  [252]  
\(\Upsilon \text {(nS)}\), DY  \(0<x_{\mathrm {F}} <0.6\)  \(\lambda _\theta (p_{\mathrm {T}},x_{\mathrm {F}})\)  [353]  
HERAB  
D  C, Ti, W  41.6  \(0.15<x_{\mathrm {F}} <0.05\)  \(\sigma (x_{\mathrm {F}},p_{\mathrm {T}} ^2)\)  [354] 
\({b\overline{b}}\), \(\mathrm {J}/\psi \)  \(0.35<x_{\mathrm {F}} <0.15\)  \(\sigma \), ratio  [355]  
\({b\overline{b}}\)  \(0.3<x_{\mathrm {F}} <0.15\)  \(\sigma \)  [356]  
\({b\overline{b}}\)  C, Ti  \(0.25<x_{\mathrm {F}} <0.15\)  \(\sigma \)  [357]  
\(\mathrm {J}/\psi \)  C, Ti, W  \(0.225<x_{\mathrm {F}} <0.075\)  \(\sigma (A,y)\)  [358]  
\(0.34<x_{\mathrm {F}} <0.14\)  \(\langle p_{\mathrm {T}} ^2\rangle (A)\), \(\alpha (p_{\mathrm {T}},x_{\mathrm {F}})\)  [359]  
C, W  \(0.34<x_{\mathrm {F}} <0.14\)  \(\lambda _\theta ,\lambda _\phi ,\lambda _{\theta \phi }(p_{\mathrm {T}},x_{\mathrm {F}})\)  [253]  
\(\mathrm {J}/\psi \), \(\psi \text {(2S)}\)  C, Ti, W  \(0.35<x_{\mathrm {F}} <0.1\)  Ratio\((x_{\mathrm {F}},p_{\mathrm {T}},A)\), \(\alpha ^\prime \)\(\alpha \)(\(x_{\mathrm {F}}\))  [254]  
\(\mathrm {J}/\psi \), \(\chi _c\)  Ratio(\(x_{\mathrm {F}}\),\(p_{\mathrm {T}}\))  [360]  
\(\Upsilon \)  C, Ti, W  \(0.6<x_{\mathrm {F}} <0.15\)  \(\sigma (y)\)  [361] 
In LHC Run 1 p–Pb collisions, protons have an energy of 4 TeV and the Pb nuclei an energy \(Z/A(4 \mathrm{TeV})=1.58\) TeV (\(Z=82\), \(A=208\)), leading to \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV and a relative velocity of the CM with respect to the laboratory frame \(\beta =0.435\) in the direction of the proton beam. The rapidity of any particle in the CM frame is thus shifted, \(y=y_{\mathrm {lab}}0.465\). Applying those experimental conditions to heavyflavour probes such as D and B mesons and quarkonia, and according to Eqs. (18) and (19), leads to a large coverage of \(x_2\) from \(10^{5}\) for the D meson at forward rapidity, to 0.5 for 10 GeV/c \(\Upsilon \) at backward rapidity, as reported in Fig. 25.
3.2 Theoretical models for CNM effects
We discuss in this section various theoretical approaches to treat CNM effects, with emphasis on heavyquark and quarkonium production at the LHC.
3.2.1 Typical time scales

The typical time to produce a heavyquark pair \({Q\overline{Q}}\), sometimes referred to as the coherence time, which is of the order of \(\tau _c \sim 1/m_{{Q\overline{Q}}} \lesssim 0.1\) fm/c in the \({Q\overline{Q}}\) rest frame. In the rest frame of the target nucleus, however, this coherence time, \(t_{c}=E_{{Q\overline{Q}}}/m^2_{{Q\overline{Q}}}\) (where \(E_{{Q\overline{Q}}}\) is the \({Q\overline{Q}}\) energy in the nucleus rest frame), can be larger than the nuclear size, leading to shadowing effects due to the destructive interferences from the scattering on different nucleons.

The time needed to produce the quarkonium state, also known as the formation time, is much larger than the coherence time. It corresponds to the time interval taken by the \({Q\overline{Q}}\) pair to develop the quarkonium wave function. Using the uncertainty principle, it should be related to the mass splitting between the 1S and 2S states [362], i.e. \(\tau _\mathrm{f}\sim (m_{2S}m_{1S} )^{1}\sim \)0.3–0.4 fm/c. Because of the Lorentz boost, this formation time in the nucleus rest frame, \(t_\mathrm{f}\), becomes much larger than the nuclear size at the LHC. Consequently the quarkonium state is produced far outside the nucleus and should not be sensitive to nuclear absorption. The time to produce a heavyquark hadron is longer than for quarkonium production, of the order of \({\Lambda _\mathrm{QCD}}^{1} \simeq 1\) fm /c in its rest frame.

Another important time scale is the typical time needed for the \({Q\overline{Q}}\) pair to neutralise its colour. In the colour singlet model, this process occurs through the emission of a perturbative gluon and should thus occur in a time comparable to \(\tau _{c}\). In the colour octet model (or colour evaporation model), colour neutralisation happens through a soft process, i.e. on “long” time scales, typically of the order the quarkonium formation time \(\tau _\mathrm{f}\).
3.2.2 Nuclear PDFs
The modification of parton densities in nuclei affects the yields of heavyquark and quarkonium production. In this section, the effects of nPDF on \(\mathrm {J}/\psi \) and \(\Upsilon \) production in p–Pb collisions at the LHC are first presented. The production of open beauty (through its decay into nonprompt \(\mathrm {J}/\psi \)) is then discussed.
The values of the central charm quark mass and scale parameters are \(m_c = 1.27 \pm 0.09\) GeV/\(c^2\), \(\mu _F/m_c = 2.10 ^{+2.55}_{0.85}\), and \(\mu _R/m_c = 1.60 ^{+0.11}_{0.12}\) [169]. The normalisation \(F_\Phi \) is obtained for the central set, \((m_c,\mu _F/m_c, \mu _R/m_c) = (1.27 \, \mathrm{GeV}/c^2, 2.1,1.6)\). The calculations for the estimation of the mass and scale uncertainties are multiplied by the same value of \(F_\Phi \) to obtain the \(\mathrm {J}/\psi \) uncertainty band [169]. \(\Upsilon \) production is calculated in the same manner, with the central result obtained for \((m_b,\mu _F/m_b, \mu _R/m_b) = (4.65 \pm 0.09 \, \mathrm{GeV}/c^2, 1.4^{+0.77}_{0.49},1.1^{+0.22}_{0.20})\) [365]. In the NLO calculations of the rapidity and \(p_{\mathrm {T}} \) dependence, instead of \(m_Q\), the transverse mass, \(m_{\mathrm {T}} \), is used with \(m_{\mathrm {T}} = \sqrt{m_Q^2 + p_{\mathrm {T}} ^2}\) where \(p_{\mathrm {T}} ^2 = 0.5\,(p_{\mathrm{T}_Q}^2 + p_{\mathrm{T}_{\overline{Q}}}^2)\). All the calculations are NLO in the total cross section and assume that the intrinsic \(k_{\mathrm {T}} \) broadening is the same in \(\mathrm pp\) as in p–Pb.
Figure 26 (left) shows the uncertainty in the shadowing effect on \(\mathrm {J}/\psi \) due to the variations in the 30 EPS09 NLO sets [364] (dashed red) as well as those due to the mass and scale uncertainties (dashed blue) calculated with the EPS09 NLO central set. The uncertainty band calculated in the CEM at LO with the EPS09 LO sets [364] is shown for comparison. It is clear that the LO results, represented by the smooth magenta curves in Fig. 26, exhibit a larger shadowing effect. This difference between the LO results, also shown in Ref. [363], and the NLO calculations arises because the gluon distributions in the proton that the EPS09 LO and NLO gluon shadowing parametrisations are based on CTEQ61L and CTEQ6M, respectively, which behave very differently at low x and moderate values of the factorisation scale [364]. If one uses instead the nDS or nDSg parametrisations [367], based on the GRV98 LO and NLO proton PDFs, the LO and NLO results differ by only a few percent. The right panel shows the same calculation for \(\Upsilon \) production. Here the difference between the LO and NLO calculations is reduced because the mass scale, and hence the factorisation scale, is larger. The x values probed are also correspondingly larger.
The \(p_{\mathrm {T}} \) dependence of the nPDF effects at forward rapidity for \(\mathrm {J}/\psi \) and \(\Upsilon \) has also been computed in Ref. [366]. There is no LO comparison because the \(p_{\mathrm {T}} \) dependence cannot be calculated in the LO CEM. The effect is rather mild and \(R_{\mathrm {pPb}}\) increases slowly with \(p_{\mathrm {T}} \), from roughly \(R_{\mathrm {pPb}} \simeq 0.7\)–0.9 for \(\mathrm {J}/\psi \) at low \(p_{\mathrm {T}} \) to \(R_{\mathrm {pPb}} \simeq 1\) at \(p_{\mathrm {T}} = 20\) GeV/c. There is little difference between the \(\mathrm {J}/\psi \) and \(\Upsilon \) results for \(R_{\mathrm {pPb}} (p_{\mathrm {T}})\) because, for \(p_{\mathrm {T}} \) above a few GeV / c, the \(p_{\mathrm {T}} \) scale dominates over the mass scale. The nPDF effects are somewhat similar for open heavy flavour as a function of \(p_{\mathrm {T}}\), yet the effects (estimated using EPS09 NLO) tend to go away faster with \(p_{\mathrm {T}}\) due to the different production dynamics between quarkonium and open heavy flavour.
Nonprompt \(\mathrm {J}/\psi \) production The nPDF effects on nonprompt \(\mathrm {J}/\psi \) (coming from B decays) has been investigated by Ferreiro et al. in [368]. Contrary to the more complex case of bottomonium production, it is sufficient to rely on LO calculations [369] to deal with openbeauty production data integrated in \(p_{\mathrm {T}} \) as those of LHCb [330]. Indeed such computations are sufficient to describe the low\(p_{\mathrm {T}} \) cross section up to (1–2) \(m_b\), where the bulk of the yield lies.
The nPDF effects on nonprompt \(\mathrm {J}/\psi \) have been evaluated using two parametrisations,^{16} namely EPS09 LO [364]^{17} and nDSg LO [367]. In addition to the choice of the nPDFs, one also has to fix the value of the factorisation scale \(\mu _F\), which is set to \(\mu _F=\sqrt{m_Q^2+p_{\mathrm {T}} ^2}\). One can also consider the spatial dependence of the nPDFs, either by simply assuming an inhomogeneous shadowing proportional to the local density [370, 371] or extracting it from a fit [372]. These effects would then translate into a nontrivial centrality (or impact parameter b) dependence of the nuclear modification factor. To this end, it is ideal to rely on a Glauber MonteCarlo which does not factorise the different nuclear effects (such as JIN [373] which is used to study the nuclear matter effects on quarkonium production both at RHIC [374, 375] and LHC [376, 377] energies).
3.2.3 Saturation in the colour glass condensate approach
In the large\(N_c\) approximation (where \(N_c=3\) is the number of colours in QCD), the multipoint functions reduce to a product of two dipole amplitudes in the fundamental representation and the evolution equation for the dipole has a closed form, called the Balitsky–Kovchegov (BK) equation. The BK equation with running coupling corrections (rcBK) is today widely exploited for phenomenological studies of saturation, and its numerical solution for \(x<x_0=0.01\) is constrained with HERA DIS data and has been applied to hadronic reactions successfully [382]. Nuclear dependence is taken into account here in the initial condition for the rcBK equation by setting larger initial saturation scales, \(Q_{s,A}^2(x_0)\) (below which gluon distribution in a nucleus starts to saturate) depending on the nuclear thickness.
References [378, 379] show the evaluation of heavy quark production applying the CGC framework in the large\(N_c\) approximation with the numerical solution of the rcBK equation. In hadronisation processes, the colour evaporation model (CEM) is used for \(\mathrm {J}/\psi \) (\(\Upsilon \)) and the vacuum fragmentation function for D meson production, assuming that the hadronisation occurs outside the target as mentioned in Sect. 3.2.1. The rapidity dependence of the nuclear modification factor \(R_{\mathrm {pA}} (y)\) of \(\mathrm {J}/\psi \) is one of the significant observables to investigate the saturation effect and the CGCbased model reproduced the RHIC data by setting \(Q_{s,A}^2(x_0)=(4  6) Q_{s,p}^2(x_0)\). Extrapolation to the LHC energy predicted a stronger suppression, reflecting stronger saturation effects at the smaller values of x (Fig. 28). Quarkonium suppression in this framework also includes the multiple scattering effects on the quark pair traversing the dense target. The comparison with experimental results will shown in Sect. 3.3.
Several improvements to this approach can be performed. The CGC expression for the heavy quark production is derived at LO in the eikonal approximation for the colour sources. The NLO extension should be investigated to be consistent with the use of the rcBK equation. Furthermore, for quarkonium production, colour channel dependence of the hadronisation process will be important and brings in a new multipoint function, which is simply ignored in CEM. Finally, using a similar approach but with an improved treatment of the nuclear geometry and a different parametrisation of the dipole cross section, Ducloué et al. [383] showed that the \(\mathrm {J}/\psi \) suppression in p–Pb collisions was less pronounced.
More recently, attempts to compute quarkonium production in \(\mathrm pp\) and p–A collisions have been made by implementing smallx evolution and multiple scattering effects in the NRQCD formalism [93]. Depending on which NRQCD channel dominates the \(\mathrm {J}/\psi \) production cross section in p–Pb collisions at the LHC, the \(\mathrm {J}/\psi \) suppression predicted in this formalism may agree with the current ALICE and LHCb measurements [384].
3.2.4 Multiple scattering and energy loss
In this section various approaches of parton multiple scattering in nuclei are discussed. These effects include \({Q\overline{Q}}\) propagation in nuclei, initial and finalstate energy loss, and coherent energy loss.
\({Q\overline{Q}}\) propagation and attenuation in nuclei This section summarises the approach by Kopeliovich et al. [385, 386]. At LHC energies, the coherence time, \(t_c\), for the production of charm quarks exceeds the typical nuclear size, \(t_c\gg R_\mathrm{A}\). As a consequence, all the production amplitudes from different bound nucleons are in phase. In terms of the dipole description this means that Lorentz time delay “freezes” the \({c\overline{c}}\) dipole separation during propagation through the nucleus, which simplifies calculations compared with the pathintegral technique, required at lower energies [362, 387, 388].
At this point one should emphasise that attenuation of \({c\overline{c}} \) dipoles in nuclear matter is a source of nuclear suppression of \(\mathrm {J}/\psi \), although it is often not included in model calculations. Moreover, independently of model details, the general features of dipole interactions are: (i) the dipole cross section studied in detail at HERA, which is proportional to the dipole size squared (of the order of \(1/m_c^2\)) and to the gluon density, (ii) the rise of the dipole cross section (and therefore the magnitude of the nuclear suppression) coming from the observed steep rise of the gluon density at small x. The observed energy independence of nuclear suppression of \(\mathrm {J}/\psi \) is incompatible with these features, and the only solution would be the presence of a nuclear enhancement mechanism rising with energy. Indeed, such a mechanism was proposed in [389] and developed in [388]. It comes from new possibilities, compared to a proton target, for \(\mathrm {J}/\psi \) production due to multiple colour exchange interaction of a \({c\overline{c}} \) in the nuclear matter, e.g. the relative contribution of double interaction is enhanced in nuclei as \(A^{1/3}\) and rises with energy proportionally to the dipole cross section [389]. Numerical evaluation of this effect is under way [390]. This approach for charmonium production cannot be simply extrapolated from p–A to AA collisions [385]. The latter case includes new effects of double colour filtering and a boosted saturation scale [385].
As the heavy quark introduces a new mass scale, the dependence of CNM corrections on this scale and their relative significance needs to be reassessed in light of the experimental data.
For the case of quarkonium production, a large uncertainty arises form the fact that the Cronin effect is not understood [83], nor have there been attempts to fit it in this approach. Consequently, for \(\mathrm {J}/\psi \) and \(\Upsilon \) results with only CNM energy loss are shown. Due to the uncertainties in the magnitude of the Cronin effect and the magnitude of the cold nuclear matter energy loss, the nuclear modification for open heavy flavour can show either small enhancement and small suppression in the region of \(p_{\mathrm {T}} \sim \) few GeV / c. The uncertainties in the magnitude of \(\Delta E / E\) can be quite significant [393]. Motivated by other multiple parton scattering effects, such as the Cronin and the coherent power corrections, which are both compatible with possibly smaller transport parameters of cold QCD matter we also consider an energy loss that is \(35~\%\) smaller than the one from using the parameters above. The results for quarkonium modification in p–A collisions is then presented as a band. The left panel of Fig. 30 shows theoretical predictions for \(\Upsilon \) \(R_{\mathrm {dAu}} \) at RHIC [83]. The right panel of Fig. 30 shows theoretical predictions for \(\mathrm {J}/\psi \) \(R_{\mathrm {pPb}} \) at the LHC [83] that will be compared to data in Sect. 3.3.
Coherent energy loss Another approach of parton energy loss in cold nuclear matter has been suggested by Arleo et al. in Refs. [394, 395, 396, 397, 398]. A few years ago it was emphasised that the mediuminduced radiative energy loss \(\Delta E\) of a highenergy gluon crossing a nuclear medium and being scattered to small angle is proportional to the gluon energy E [394, 397]. The behaviour \(\Delta E \propto E\) arises from soft gluon radiation which is fully coherent over the medium. Coherent energy loss is expected in all situations where the hard partonic process looks like forward scattering of an incoming parton to an outgoing compact and colourful system of partons [398]. In the case of \(\mathrm {J}/\psi \) hadroproduction at low \(p_{\mathrm {T}} \lesssim m_{\mathrm {J}/\psi }\), viewed in the target rest frame as the scattering of an incoming gluon to an outgoing colour octet \({c\overline{c}} \) pair,^{18} such an energy loss provides a successful description of \(\mathrm {J}/\psi \) nuclear suppression in p–A as compared to \(\mathrm pp\) collisions, from fixedtarget (SPS, HERA, FNAL) to collider (RHIC, LHC) energies [395, 396].
In Refs. [395, 396], the \(\mathrm {J}/\psi \) differential cross section \(\mathrm {d}^2\sigma _\mathrm{pp}/\mathrm {d}y \, \mathrm {d}p_{\mathrm {T}} \) is determined from a fit of the \(\mathrm pp\) data, and \(\mathrm {d}^2\sigma _\mathrm{pA}/\mathrm {d}y \, \mathrm {d}p_{\mathrm {T}} \) is obtained by performing a shift in rapidity (and in \(p_{\mathrm {T}} \)) accounting for the energy loss \(\varepsilon \) with probability \(\mathcal{P}(\varepsilon )\) (and for the transverse broadening \(\Delta p_{\mathrm {T}} \)) incurred by the compact octet state propagating through the nucleus. Independent of the \(\mathrm pp\) production mechanism, the model is thus able to predict \(\mathrm {J}/\psi \) and \(\Upsilon \) nuclear suppression, \(R_{\mathrm {pA}} \), as a function of y, \(p_{\mathrm {T}} \) and centrality. The model depends on a single parameter \(\hat{q}_0\), which fully determines both the broadening \(\Delta p_{\mathrm {T}} \) and the energyloss probability distribution, \(\mathcal{P}(\varepsilon )\). It is determined by fitting the model calculations to the E866 measurements [352] in p–W collisions at \(\sqrt{s_{\mathrm{NN}}} =38.7\) GeV. The result of the fit, which yields \(\hat{q}_0=0.075\) GeV\(^2\)/fm, is shown in Fig. 31 (left) in comparison to the data.
In order to assess the uncertainties of the model predictions, the parameter entering the \(\mathrm pp\) data parametrisation is varied around its central value, as well as the magnitude of the transport coefficient from 0.07 to 0.09 GeV\(^2\)/fm [395]. The prescription for computing the model uncertainties can be found in [404]. The model predictions for \(\mathrm {J}/\psi \) and \(\Upsilon \) suppression in p–Pb collisions at the LHC as a function of rapidity are shown in Fig. 31 (right). The extrapolation of the model to AA collisions is discussed in Sect. 3.4.
3.2.5 Nuclear absorption
The \(\mathrm {J}/\psi \) absorption cross section, \(\sigma _\mathrm{abs}^{\mathrm {J}/\psi }\), was traditionally assumed to be independent of the production kinematics until measurements covering broad phasespace regions showed clear dependences of the nuclear effects on \(x_{\mathrm {F}} \) and \(p_{\mathrm {T}} \). It was further assumed to be independent of collision centreofmass energy, \(\sqrt{s_{\mathrm{NN}}} \), neglecting any nuclear effects on the parton distributions. However, \(\mathrm {J}/\psi \) production is sensitive to the gluon distribution in the nucleus and the fixedtarget measurements probe parton momentum fractions, x, in the possible antishadowing region. This effect may enhance the \(\mathrm {J}/\psi \) production rate at midrapidity and a larger absorption cross section would be required to match the data.
If one focuses on the behavior of \(\mathrm {J}/\psi \) production at \(x_{\mathrm {F}} \approx 0\), the absorption cross section is found to depend on \(\sqrt{s_{\mathrm{NN}}} \), essentially independent of the chosen nPDF parametrisation [399], as shown in Fig. 32 (left). The yellow band represents the uncertainty corresponding to an empirical powerlaw fit (solid curve) to all the data points analysed in [399] from measurements by NA3 [332], NA50 [341, 342], E866 [352], HERAB [359], NA60 [406] and PHENIX [407]. The extrapolation of the powerlaw fit in Fig. 32 (left) to the current LHC p–A energy leads to a vanishingly small cross section within the illustrated uncertainties.
Away from midrapidity, the extracted \(\sigma _\mathrm{abs}^{\mathrm {J}/\psi }\) grows with \(x_{\mathrm {F}} \) up to unrealistically large values, as shown in Fig. 32 (right). This seems to indicate that another mechanism, in addition to absorption and shadowing, such as initialstate energy loss, may be responsible for the \(\mathrm {J}/\psi \) suppression in the forward region (\(x_{\mathrm {F}} > 0.25\)). This confirms that the effective parameter \(\sigma _\mathrm{abs}^{\mathrm {J}/\psi }\) should not be interpreted as a genuine inelastic cross section. It seems that the rise starts closer to \(x_{\mathrm {F}} = 0\) for lower collision energies [64]. More recent analyses [408], using EPS09 [364], are in general agreement with the results of Ref. [399].
Despite different conclusions on the possible energy dependence of \(\sigma _\mathrm{abs}\) from fixedtarget experiments to RHIC energy in [405] and [399], one expects nuclear absorption effects to become negligible at the LHC since the quarkonium formation time becomes significantly larger than the nuclear size at all values of the rapidity. This is also confirmed by a more recent analysis. In Ref. [408], the authors show that the \(\mathrm {J}/\psi \) suppression seems to scale with the crossing time \(\tau _\mathrm{cross}\) (see Sect. 3.2.1), independently of the centreofmass energy, above a typical crossing time \(\tau _\mathrm{cross} \gtrsim 0.05\) fm/c. Below this scale, however, the lack of scaling indicates that nuclear absorption is probably not the dominant effect. Using the \(2\rightarrow 1\) kinematics, \(\tau _\mathrm{cross} \simeq 2 m_p\ L\ e^{y} / \sqrt{s_{\mathrm{NN}}} \), the condition \(\tau _\mathrm{cross} < 0.05\) fm/c would correspond to \(y >  3.8\) (using \(L_\mathrm{Pb}\simeq 3/4\ R_\mathrm{Pb} \simeq 5\) fm) at the LHC.
3.2.6 Summary of CNM models
Summary of the various models of CNM approaches discussed in the text and compared to data in Sect. 3.3. The main physical processes and ingredients used in each calculation are listed
Acronym  Production mechanism  Medium effects  Main parameters  References 

Open heavy flavour  
pQCD\(+\)EPS09 LO  pQCD LO  nPDF  4\(+\)1 EPS09 LO sets  [368] 
SAT  pQCD LO\(+\)CGC  Saturation  \(Q_{s,p}^2(x_0)\), \(Q_{s,A}^2(x_0)\)  [378] 
ELOSS  pQCD LO  E. loss, power cor., broa.  \(\epsilon _a\), \(\xi _d\), \(\mu ^2\), \(\lambda \)  [391] 
Quarkonia  
EXT\(+\)EKS98LO\(+\)ABS  Generic \(2\rightarrow 2\) LO  nPDF and absorption  EKS98 LO, \(\sigma _\mathrm {abs}\)  
EXT\(+\)EPS09 LO  Generic \(2\rightarrow 2\) LO  nPDF  4\(+\)1 EPS09 LO sets  
CEM\(+\)EPS09 NLO  CEM NLO  nPDF  30\(+\)1 EPS09 NLO sets  [363] 
SAT  CEM LO\(+\)CGC  Saturation  \(Q_{s,p}^2(x_0)\), \(Q_{s,A}^2(x_0)\)  [379] 
ELOSS  NRQCD LO  E. loss, power cor.  \(\epsilon _a\), \(\xi _d\), \(\mu ^2\), \(\lambda \)  [83] 
COH.ELOSS  \(\mathrm pp\) data  Coherent E. loss  \(\hat{q}\)  
KPS  dipole model  Dipole absorption  \(\sigma _{c\bar{c}}\) 
3.3 Recent RHIC and LHC results
In this section we summarise the recent measurements in p–A collisions at RHIC and at the LHC. Open heavyflavour results are described in Sect. 3.3.2 and hidden heavyflavour data in Sect. 3.3.3. As described in the previous section, in order to understand the role of the CNM effects, the interpretation of these measurements is commonly obtained by a comparison with measurements in \(\mathrm pp\) collisions at the same centreofmass energy as for p–A and in the same rapidity interval. At the LHC, so far it has not been possible to carry out \(\mathrm pp\) measurements at the same energy and rapidity as for p–Pb. In Sect. 3.3.1 the procedures to define the pp reference for \(R_\mathrm{pA}\) are described.
3.3.1 Reference for p–A measurements at the LHC
In the quarkonium analyses, different strategies have been adopted depending on the precision of the existing measurements. They are mainly based on phenomenological functions and are briefly described in the following.
At midrapidity in ALICE, the \(\mathrm {J}/\psi \) \(\mathrm pp\) integrated cross section reference has been obtained by performing an interpolation based on \(\mathrm {J}/\psi \) measurements at midrapidity in \(\mathrm pp\) collisions at \(\sqrt{s}\) \(=\) 200\(~\text {GeV}\) [411], 2.76 \(\text {TeV}\) [412] and 7 \(\text {TeV}\) [413], and in \({\mathrm{p}\overline{\mathrm{p}}} \) collisions at \(\sqrt{s}\) \(=\) 1.96 \(\text {TeV}\) [414]. Several functions (linear, power law and exponential) were used to parametrise the cross section dependence as a function of \(\sqrt{s}\). The interpolation leads to a total uncertainty of \(17~\%\) on the integrated cross section. The effect of the asymmetric rapidity coverage, due to the shift of the rapidity by 0.465 in the centreofmass system in p–Pb collisions at the LHC, was found to be negligible as compared to the overall uncertainty of the interpolation procedure. Then the same method as described in [415] was followed to obtain the \(p_{\mathrm {T}}\)dependent cross section. The method is based on the empirical observation that the \(\mathrm {J}/\psi \) cross sections measured at different energy and rapidity scale with \(p_{\mathrm {T}}\)/\(\langle p_{\mathrm {T}} \rangle \). The \(\langle p_{\mathrm {T}} \rangle \) value was evaluated at \(\sqrt{s}\) \(=\) 5.02 TeV by an interpolation of the \(\langle p_{\mathrm {T}} \rangle \) measured at midrapidity [411, 413, 414] using exponential, logarithmic and powerlaw functions.
At forward rapidity, a similar procedure for the \(\mathrm {J}/\psi \) cross section interpolation has been adopted by ALICE and LHCb and is described in [416]. In order to ease the treatment of the systematics correlated with energy, the interpolation was limited to results obtained with a single apparatus. The inclusive \(\mathrm {J}/\psi \) cross sections measured at 2.76 [412] and 7 \(\text {TeV}\) [199] were included in the ALICE procedure while the inclusive, prompt \(\mathrm {J}/\psi \) and \(\mathrm {J}/\psi \) from Bmesons cross sections measured at 2.76 [417], 7 [172] and 8 \(\text {TeV}\) [206] were considered in the LHCb one. The interpolation of the cross section with energy is based, as in the midrapidity case, on three empirical shapes (linear, power law and exponential). The resulting interpolated cross section for inclusive \(\mathrm {J}/\psi \) obtained by ALICE and LHCb in \(2.5<y <4\) at \(\sqrt{s}\) \(=\) 5.02 \(\text {TeV}\) were found to be in good agreement with a total uncertainty of \(\sim \)8 and \(\sim \)5 % for ALICE and LHCb, respectively. The interpolation in \(\sqrt{s}\) was also performed by ALICE independently for each \(p_{\mathrm {T}}\) interval and outside of the rapidity range of \(\mathrm pp\) data in order to cope with the p–Pb centreofmass rapidity shift. In that case an additional interpolation with rapidity was carried out by using several empirical functions (Gaussian, second and fourthorder polynomials).
In the case of the \(\Upsilon \) at forward rapidity, the interpolation procedure results also from a common approach by ALICE and LHCb and is described in [418]. It is based on LHCb measurements in \(\mathrm pp\) collisions at 2.76 \(\text {TeV}\) [419], 7 \(\text {TeV}\) [199] and 8 \(\text {TeV}\) [206]. Various phenomenological functions and/or the \(\sqrt{s}\)dependence of the CEM and FONLL models are used for the \(\sqrt{s}\)dependence of the cross section, similarly to the \(\mathrm {J}/\psi \) interpolation procedure at forward rapidity. This interpolation results in a systematic uncertainty that ranges from 8 to \(12~\%\) depending on the rapidity interval.
3.3.2 Open heavyflavour measurements
Heavyflavour decay leptons The production of heavyflavour decay leptons, i.e. leptons from charm and beauty decays, has been studied at RHIC and at LHC energies in d–Au and p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =200\) \(~\text {GeV}\) and 5.02 \(\text {TeV}\), respectively. The p–A measurements exploit the inclusive lepton \(p_{\mathrm {T}} \)spectrum, electrons at midrapidity (\(\eta <0.5\) for PHENIX, \(0<\eta <0.7\) for STAR and \(\eta <0.6\) for ALICE) and muons at forward rapidities (\(1.4<\eta <2.0\) for PHENIX and \(2.5<\eta <4.0\) for ALICE). The heavyflavour decay spectrum is determined by extracting the nonheavyflavour contribution to the inclusive lepton distribution. The photonic background sources are electrons from photon conversions in the detector material and \(\pi ^0\) and \(\eta \) Dalitz decays, which involve virtual photon conversion. The contribution of photon conversions is evaluated with the invariantmass method or via Montecarlo simulations. The Dalitz decays contribution can be determined considering the measured \(\pi ^0\) and \(\eta \) distributions. Background from light hadrons, hard processes (prompt photons and Drell–Yan) and quarkonia is determined with Montecarlo simulations, based, when possible, on the measured spectrum. STAR data is not corrected for the of \(\mathrm {J}/\psi \) decays contribution, which is nonnegligible at high \(p_{\mathrm {T}} \). Beauty decayelectron spectra can be obtained from the heavyflavour decayelectron spectra by a cut or fit of the lepton impactparameter distribution, i.e. the distance between the lepton track and the interaction vertex, or exploiting the lepton azimuthal correlation to heavy flavours or charged hadrons. For the latter see the last paragraph of this section.
Heavyflavour decay lepton \(R_{\mathrm {dAu}} \) measurements at midrapidity in minimumbias d–Au collisions at \(\sqrt{s_{\mathrm{NN}}} =200\) \(~\text {GeV}\) by STAR and PHENIX [313, 420] are consistent and suggest no modification of the multiplicityintegrated yields for \(1<p_{\mathrm {T}} <10~\text {GeV}/c \) within uncertainties. The \(p_{\mathrm {T}} \) dependence of \(R_{\mathrm {dAu}} \) on the multiplicity and the rapidity was studied by PHENIX [313, 314] and is reported in Fig. 33. It shows a mild dependence with the multiplicity at midrapidity. The results at forward and backward rapidities are similar for peripheral collisions, but evidence a strong deviation for the most central events. As shown in Fig. 34 and in [313], the measurements at forward rapidity are described both by the model of Vitev et al. [391, 392] – considering nPDFs, \(k_{\mathrm {T}} \) broadening and CNM energy loss – (ELOSS model described in Sect. 3.2.4) or by nPDFs alone. Data at backward rapidity cannot be described considering only the nPDFs, suggesting that other mechanisms are at work.
The preliminary results at LHC energies by the ALICE Collaboration [421] present \(R_{\mathrm {pPb}} \) multiplicityintegrated values close to unity at midrapidity, as observed at lower energies. The rapidity dependence of the multiplicityintegrated \(R_{\mathrm {pPb}} \) is also similar to that observed at RHIC. In contrast to RHIC, model calculations with nPDFs present a fair agreement with LHC data. The first preliminary measurements of the beautyhadron decayelectron \(R_{\mathrm {pPb}} \) at midrapidity by ALICE are consistent with unity within larger uncertainties [421].
The similar behaviour of RHIC and LHC heavyflavour decay lepton \(R_{\mathrm {pA}} \), within the large uncertainties, despite the different xBjorken ranges covered, suggests that nPDFs might not be the dominant effect in heavyflavour production. Additional mechanisms like \(k_\mathrm{T}\)broadening, initial or finalstate energy loss could be at play.
D mesons The \(p_{\mathrm {T}} \)differential production cross section of \(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \), \(\mathrm {D}^{*+} \) and \(\mathrm {D}^{+}_{s} \) in minimum bias p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV for \(y_\mathrm{lab}<0.5\) was published in [324] by ALICE. D mesons are reconstructed via their hadronic decays in different \(p_{\mathrm {T}} \) intervals from \(1~\text {GeV}/c \) up to \(24~\text {GeV}/c \). Prompt Dmeson yields are obtained by subtracting the contribution of secondaries from Bhadron decays, determined using pQCDbased estimates [125, 324]. No significant variation of the \(R_{\mathrm {pPb}} \) among the Dmeson species is observed within uncertainties. The multiplicityintegrated prompt D (average of \(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \) and \(\mathrm {D}^{*+} \)) meson \(R_{\mathrm {pPb}} \) is shown in Fig. 36 together with model calculations. \(R_{\mathrm {pPb}} \) is compatible with unity in the measurement \(p_{\mathrm {T}} \) interval, indicating smaller than 10–20 % nuclear effects for \(p_{\mathrm {T}} >2~\text {GeV}/c \). Data are described by calculations considering only initialstate effects: NLO pQCD estimates (MNR [6]) considering EPS09 nPDFs [364] or Colour Glass Condensate computations [378] (SAT model described in Sect. 3.2.3). Predictions including nPDFs, initial or finalstate energy loss and \(k_{\mathrm {T}} \)broadening [422] (ELOSS model discussed in Sect. 3.2.4) also describe the measurements.
Open beauty measurements The first measurements of the beauty production cross section in p–A collisions down to \(p_{\mathrm {T}} =0\) were carried out by LHCb in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV [330]. These results were achieved via the analysis of nonprompt \(\mathrm {J}/\psi \) mesons at large rapidities, \(2<y_{\mathrm {lab}} <4.5\). \(\mathrm {J}/\psi \) mesons were reconstructed by an invariant mass analysis of opposite sign muon pairs. The fraction of \(\mathrm {J}/\psi \) originating with beauty decays, or nonprompt \(\mathrm {J}/\psi \) fraction, was evaluated from a fit of the component of the pseudoproper decay time of the \(\mathrm {J}/\psi \) along the beam direction. The \(R_{\mathrm {pPb}} \) of nonprompt \(\mathrm {J}/\psi \) was computed considering as \(\mathrm pp\) reference an interpolation of the measurements performed in the same rapidity interval at \(\sqrt{s} =2.76\), 7 and 8 \(\text {TeV}\) (see Sect. 3.3.1). Figure 37 (left) reports the \(p_{\mathrm {T}} \)integrated \(R_{\mathrm {pPb}} \) as a function of rapidity, whereas Fig. 37 (right) presents the double ratio of the production cross section at positive and negative rapidities, \(R_{\mathrm {FB}} \), as a function of the \(\mathrm {J}/\psi \) transverse momentum. The \(p_{\mathrm {T}} \)integrated \(R_{\mathrm {pPb}} \) is close to unity in the backwardrapidity range, and shows a modest suppression in the forwardrapidity region. \(R_{\mathrm {FB}} \) is compatible with unity within the uncertainties in the measured \(p_{\mathrm {T}} \) interval, with values almost systematically smaller than unity. These results indicate a moderate rapidity asymmetry and are consistent with the \(R_{\mathrm {pPb}} \) ones. The results are in agreement with LO pQCD calculations including EPS09 or nDSg nuclear PDF parametrisations. The ATLAS Collaboration has also measured the \(R_{\mathrm {FB}} \) of nonprompt \(\mathrm {J}/\psi \) for \(8<p_{\mathrm {T}} <30~\text {GeV}/c \) and \(y <1.94\) [329]. These results are consistent with unity within experimental uncertainties and no significant \(p_{\mathrm {T}} \) or \(y \) dependence is observed within the measured kinematic ranges.
A preliminary measurement of the production of B mesons in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =5.02\) \(\text {TeV}\) was carried out by the CMS Collaboration [424, 425]. B\(^0\), B\(^{+}\) and B\(_s^0\) mesons are reconstructed via their decays to \(\mathrm {J}/\psi + \mathrm{K}\) or \(\phi \) at midrapidity for \(10<p_{\mathrm {T}} <60~\text {GeV}/c \). The \(\mathrm{d}\sigma /\mathrm{d}p_{\mathrm {T}} \) of B\(^0\), B\(^{+}\) and B\(_s^0\) are described within uncertainties by FONLL predictions scaled by the number of nucleons in the nucleus. B\(^{+}\) \(\mathrm{d}\sigma /\mathrm{d}y \) is also described by FONLL binaryscaled calculations, and presents no evidence of rapidity asymmetry within the measurement uncertainties. These results suggest that Bhadron production for \(p_{\mathrm {T}} >10~\text {GeV}/c \) is not affected, or mildly, by CNM effects.
Heavyflavour azimuthal correlations As described in Sect. 2.4.2, heavyflavour particle production inherits the heavyquark pair correlation, bringing information on the production mechanisms. Heavyflavour production in p–A collisions is influenced by initial and/or finalstate effects. The modification of the PDFs or the saturation of the gluon wave function in the nucleus predict a reduction of the overall particle yields. The CGC formalism also predicts a broadening and suppression of the twoparticle awayside azimuthal correlations, more prominent at forward rapidities [427, 428, 429]. Energy loss or multiple scattering processes in the initial or final state are also expected to cause a depletion of the twoparticle correlation awayside yields [430]. These effects could also affect heavyflavour correlations in p–A collisions.
Heavyflavour decay electron (\(p_{\mathrm {T}} >0.5~\text {GeV}/c \), \(\eta <0.5\)) to heavyflavour decay muon (\(p_{\mathrm {T}} >1~\text {GeV}/c \), \(1.4<\eta <2.1\)) \(\Delta \phi \) azimuthal correlations have been studied by PHENIX in \(\mathrm pp\) and d–Au collisions at \(\sqrt{s} =200\) \(~\text {GeV}\) [114]. They exploit the forwardrapidity muon measurements in order to probe the lowx region in the gold nucleus. The analysis considers the angular correlations of all sign combinations of electron–muon pairs. The contribution from lightflavour decays and conversions is removed by subtracting the likesign yield from the unlikesign yield. Figure 38 presents the electron–muon heavyflavour decay \(\Delta \phi \) correlations. Model calculations are compared to data for \(\mathrm pp\) collisions; see Fig. 38 (left). Calculations from NLO generators seem to fit better the \(\Delta \phi \) distribution than LO simulations. The corresponding measurement in d–Au collision; see Fig. 38 (right), shows a reduction of the awayside peak as compared to \(\mathrm pp\) scaled data, indicating a modification of the charm kinematics due to CNM effects.
Preliminary results of Dhadron azimuthal correlations in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =5.02\) \(\text {TeV}\) were carried out by the ALICE Collaboration [276]. The measurement uncertainties do not allow a clear conclusion to be drawn on a possible modification of heavyquark azimuthal correlations with respect to pp collisions.
3.3.3 Quarkonium measurements
Quarkonia are mainly measured via their leptonic decay channels. In the PHENIX experiment, the Ring Imaging Cherenkov associated with the Electromagnetic Calorimeter (EMCAL) allows one to identify electrons at midrapidity (\(y < 0.35\)). In this rapidity range where the EMCAL can reconstruct the photons, \(\chi _c\) can also be measured from its decay channel to \(\mathrm {J}/\psi \) and photon. At backward and forward rapidity (\(1.2 < y < 2.2\)), two muon spectrometers allow for the reconstruction of quarkonia via their muonic decay channel. In the STAR experiment, quarkonia are reconstructed at midrapidity (\(y < 1\)) thanks to the electron identification and momentum measurements from the TPC. In the ALICE experiment, a TPC at midrapidity (\(y_{\mathrm {lab}}  < 0.9\)) is used for electron reconstruction and identification and a spectrometer at forward rapidity for muon reconstruction (\(2.5 < y_{\mathrm {lab}} < 4\)). The LHCb experiment is a forward spectrometer that allows for the quarkonium measurement via their muonic decay channel for \(2 < y_{\mathrm {lab}} < 4.5\). In the CMS experiment, quarkonia are reconstructed in a large range around midrapidity (\(y_{\mathrm {lab}}  < 2.4\)) via the muonic decay channel. In LHCb, CMS and in ALICE at midrapidity, the separation of prompt \(\mathrm {J}/\psi \) from inclusive \(\mathrm {J}/\psi \) exploits the long lifetime of b hadrons, with \(c\tau \) value of about 500 \(\upmu \)m, using the good resolution of the vertex detector.
Charmonium The nuclear modification factor for inclusive and/or prompt \(\mathrm {J}/\psi \) has been measured for a large range in rapidity and is shown in Fig. 39 for RHIC (left) and LHC (right). It should be emphasised that there are no \(\mathrm pp\) measurements at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 5.02 TeV at the LHC and the \(\mathrm pp\) cross section interpolation procedure described in Sect. 3.3.1 results in additional uncertainties.
The measurements from PHENIX [318] in d–Au collisions at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 200 \(~\text {GeV}\) cover four units of rapidity. The \(\mathrm {J}/\psi \) is suppressed with respect to binaryscaled \(\mathrm pp\) collisions in the full rapidity range with a suppression that can reach more than 40 % at \(y=2.3\). Inclusive \(\mathrm {J}/\psi \) includes a contribution from prompt \(\mathrm {J}/\psi \) (direct \(\mathrm {J}/\psi \) and excited charmonium states, \(\chi _c\) and \(\mathrm \psi (2S)\)) and a contribution from decays of B mesons. At RHIC energy, the contribution from Bmeson decays to the inclusive yield is expected to be small, of the order of \(3~\%\) [431], but has not been measured so far in d–Au collisions. The contribution from excited states such as \(\chi _c\) and \(\mathrm \psi (2S)\) has been measured at midrapidity [320] and is discussed later in this section. While the inclusive \(\mathrm {J}/\psi \) \(R_{\mathrm {dAu}}\) for \( y < 0.9\) is found to be \(0.77\pm 0.02\mathrm { (stat)}\pm 0.16\mathrm { (syst)}\), the correction from \(\chi _c\) and \(\mathrm \psi (2S)\) amounts to \(5~\%\) and leads to a feeddown corrected \(\mathrm {J}/\psi \) \(R_{\mathrm {dAu}}\) of \(0.81\pm 0.12\mathrm { (stat)}\pm 0.23\mathrm { (syst)}\).
At the LHC, the results for inclusive \(\mathrm {J}/\psi \) from ALICE [325, 327] and for prompt \(\mathrm {J}/\psi \) from LHCb [330] show a larger suppression of the \(\mathrm {J}/\psi \) production with respect to the binaryscaled \(\mathrm {J}/\psi \) production in \(\mathrm pp\) collisions at forward rapidity (40 % at \(y =3.5\)). In the backwardrapidity region the nuclear modification factor is slightly suppressed (prompt \(\mathrm {J}/\psi \) from LHCb) or enhanced (inclusive \(\mathrm {J}/\psi \) from ALICE) but within the uncertainties compatible with unity. ATLAS and LHCb have also measured the production of \(\mathrm {J}/\psi \) from B mesons [329, 330]: they contribute to the inclusive \(\mathrm {J}/\psi \) yield integrated over \(p_{\mathrm {T}}\) by 8 % at \(4 < y < 2.5\) and 12 % at \(1.5<y <4\) with an increase towards midrapidity and high \(p_{\mathrm {T}}\) region. At \(p_{\mathrm {T}} >8\) GeV/c, the fraction of B mesons can reach up to 34 % at midrapidity and up to 26 % in the backwardrapidity region covered by LHCb. In addition, the nuclear modification factor for \(\mathrm {J}/\psi \) from B mesons is above 0.8 when integrated over \(p_{\mathrm {T}}\), as shown in Fig. 37, At low \(p_{\mathrm {T}}\), a small effect from B mesons on inclusive \(\mathrm {J}/\psi \) measurements is therefore expected at LHC energy and this is confirmed by the comparison of prompt to inclusive \(\mathrm {J}/\psi \) that shows good agreement as seen in the right panel of Fig. 39.
It is interesting to check whether a simple scaling exists on \(\mathrm {J}/\psi \) suppression between RHIC and LHC. The effects of nPDF or saturation are expected to scale with the momentum fraction \(x_2\), independently of the centreofmass energy of the collision. It is also the case of nuclear absorption, since the \(\mathrm {J}/\psi \) formation time is proportional to the Lorentz factor \(\gamma \), which is uniquely related to \(x_2\), \(\gamma = m / (2 m_p\ x_2)\), assuming \(2\rightarrow 1\) kinematics for the production process. In order to test the possible \(x_2\) scaling expected in the case of nPDF and nuclear absorption theoretical approaches, the data from RHIC and LHC [318, 325, 327, 330] of Fig. 39 are shown together in Fig. 40 as a function of \(x_2 = \frac{m}{\sqrt{s_{\mathrm{NN}}}}\exp (y)\) where the low \(x_2\) values correspond to forwardrapidity data. Note that Eq. (19) for the calculation of \(x_2\), which refers to a \(2 \rightarrow 2\) partonic process, cannot be used since the \(\langle p_{\mathrm {T}} \rangle \) values for all the data points have not been measured. While at \(x_2 < 10^{2}\), the nuclear modification factors are compatible at RHIC and LHC energy within uncertainties, the data presents some tension with a \(x_2\) scaling at large \(x_2\).
The transverse momentum distribution of the nuclear modification factor is shown for different rapidity ranges in Fig. 41 for RHIC (left) and LHC (right) energies. At \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 200 \(~\text {GeV}\), the \(\mathrm {J}/\psi \) \(R_{\mathrm {dAu}}\) is suppressed at low \(p_{\mathrm {T}}\) and increases with \(p_{\mathrm {T}}\) for the full rapidity range. The mid and forward rapidity results show a similar behaviour: \(R_{\mathrm {dAu}}\) increases gradually with \(p_{\mathrm {T}}\) and is consistent with unity at \(p_{\mathrm {T}} \gtrsim 4\) \(~\text {GeV}/c\). At backward rapidity, \(R_{\mathrm {dAu}}\) increases rapidly to reach 1 at \(p_{\mathrm {T}} \approx 1.5\) \(~\text {GeV}/c\) and is above unity for \(p_{\mathrm {T}} > 2.5\) \(~\text {GeV}/c\). At \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 5.02 \(\text {TeV}\), a similar shape and amplitude is observed for \(R_{\mathrm {pPb}}\) at mid and forward rapidity: in that case it is consistent with unity at \(p_{\mathrm {T}} \gtrsim 5\) \(~\text {GeV}/c\). The backwardrapidity results are consistent throughout the full \(p_{\mathrm {T}}\) interval with unity.
In addition to the aforementioned models, the calculations based on the energyloss approach from [83] (ELOSS), valid for \(y \ge 0\) and \(p_{\mathrm {T}} > 3\) GeV/c, are also compared to the data. Among these models, only the COH.ELOSS and SAT model includes effects from initial or finalstate multiple scattering that may lead to a \(p_{\mathrm {T}}\) broadening. The \(p_{\mathrm {T}}\) dependence of \(R_{\mathrm {pA}}\) is correctly described by the CEM EPS09 NLO model except at backward rapidity and \(\sqrt{s_{\mathrm{NN}}} =200\) \(~\text {GeV}\). The model based on EXT EKS98 LO ABS with an absorption cross section of 4.2 mb describes the mid and forward rapidity results at \(\sqrt{s_{\mathrm{NN}}} =200\) \(~\text {GeV}\) but not the \(p_{\mathrm {T}}\) dependence at backward rapidity. A good agreement is reached at \(\sqrt{s_{\mathrm{NN}}} =5.02\) \(\text {TeV}\) with CEM EPS09 NLO and EXT EPS09 LO calculations. The ELOSS model describes correctly the \(p_{\mathrm {T}}\) dependence at mid and forward rapidity at both energies. The COH.ELOSS calculations describe correctly the data with, however, a steeper \(p_{\mathrm {T}}\) dependence at forward rapidity and \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV. Finally the SAT model gives a good description of the data at midrapidity at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV but does not describe the \(p_{\mathrm {T}}\) dependence at forward rapidity at \(\sqrt{s_{\mathrm{NN}}} =\)200 \(~\text {GeV}\) and overestimates the suppression at forward rapidity at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV.
It is also worth mentioning that the ratio \(R_{\mathrm {FB}}\) of the nuclear modification factors for a rapidity range symmetric with respect to \(y \sim 0\) has also been extracted as a function of rapidity and \(p_{\mathrm {T}}\) at \(\sqrt{s_{\mathrm{NN}}} =5.02\) TeV [325, 329, 330]. Despite the reduction of statistics (since the rapidity range is limited), the \(\mathrm pp\) cross section and its associated systematics cancels out in the ratio and results on \(R_{\mathrm {FB}}\) provides additional constraints to the models.
The centrality dependence of the \(\mathrm {J}/\psi \) nuclear modification factor has also been studied in ALICE [433, 434]. In these analyses, the event activity is determined from the energy measured along the beam line by the Zero Degree Neutron (ZN) calorimeter located in the nucleus direction. In the hybrid method described in [435], the centrality of the collision in each ZN energy event class is determined assuming that the chargedparticle multiplicity measured at midrapidity is proportional to the number of participants in the collision. In the \(\mathrm {J}/\psi \) case, the data is compatible, within uncertainties, to the binaryscaled \(\mathrm pp\) production for peripheral events at backward and forward rapidity. The \(\mathrm {J}/\psi \) production in p–Pb is, however, significantly modified towards central events. At backward rapidity the nuclear modification factor is compatible with unity at low \(p_{\mathrm {T}}\) and increases with \(p_{\mathrm {T}}\) reaching about 1.4 at \(p_{\mathrm {T}}\) \(\sim 7\) \(~\text {GeV}/c\). At forward rapidity the suppression of \(\mathrm {J}/\psi \) production shows an increase towards central events specially at low \(p_{\mathrm {T}}\). The \(\mathrm {J}/\psi \) production was also studied as a function of the relative chargedparticle multiplicity measured at midrapidity as was already done in \(\mathrm pp\) collisions [266]. The results show an increase with the relative multiplicity at backward and forward rapidity. At forward rapidity the multiplicity dependence becomes weaker than at backward rapidity for high relative multiplicities. In \(\mathrm pp\) collisions [266] this increase is interpreted in terms of the hadronic activity accompanying \(\mathrm {J}/\psi \) production, from contribution of multiple parton–parton interactions or in the partonpercolation scenario. In p–Pb collisions, in addition to the previous contributions, the cold nuclear matter effects should be considered when interpreting these results.
The binding energy of the excited charmonium states is significantly smaller than that of the ground state [437]: the \(\psi \text {(2S)}\) has the lowest binding energy (0.05 GeV), following by the \(\chi _c\) (0.20 GeV) and the \(\mathrm {J}/\psi \) (0.64 GeV). The excited charmonium states are then expected to be more sensitive to the nuclear environment as compared to the \(\mathrm {J}/\psi \). The relative suppression of the \(\psi \text {(2S)}\) to \(\mathrm {J}/\psi \) from earlier measurements at lower energy and at midrapidity [254, 341, 352] has been understood as a larger absorption of the \(\psi \text {(2S)}\) in the nucleus since, under these conditions, the crossing time \(\tau _\mathrm{cross}\) of the \({c\overline{c}}\) pair through the nucleus is larger than the charmonium formation time \(\tau _\mathrm{f}\). At higher energy, \(\tau _\mathrm{cross}\) is expected to be always lower than \(\tau _\mathrm{f}\) [438] except maybe for backwardrapidity ranges. This means that the \({c\overline{c}}\) is nearly always in a preresonant state when traversing the nuclear matter and the nuclear breakup should be the same for the \(\psi \text {(2S)}\) and \(\mathrm {J}/\psi \).
The PHENIX experiment has measured \(R_{\mathrm {dAu}} =0.54\pm 0.11\mathrm { (stat)}^{+0.19}_{0.16}\mathrm { (syst)}\) for the \(\psi \text {(2S)}\) and \(R_{\mathrm {dAu}} =0.77\pm 0.41\mathrm { (stat)}\pm 0.18\mathrm { (syst)}\) for the \(\chi _c\) for \(y<0.35\) [320]. While the large uncertainty prevents any conclusion for the \(\chi _c\), the relative modification factor of the \(\psi \text {(2S)}\) to inclusive \(\mathrm {J}/\psi \) in d–Au collisions, \([ \psi \text {(2S)}/\mathrm {J}/\psi ]_\mathrm{dAu} / [ \psi \text {(2S)}/\mathrm {J}/\psi ]_\mathrm{pp}\) equivalent to \(R_{\mathrm {dAu}} ^{\psi \text {(2S)}}/R_{\mathrm {dAu}} ^{\mathrm {J}/\psi }\), has been found to be \(0.68\pm 0.14\mathrm { (stat)}^{+0.21}_{0.18}\mathrm { (syst)}\), i.e. 1.3 \(\sigma \) lower than 1. The relative modification factor as a function of \(\mathrm {N_{coll}}\) is shown in the left panel of Fig. 44. In the most central collisions, the \(\psi \text {(2S)}\) is more suppressed than the \(\mathrm {J}/\psi \) by about \(2 \,\sigma \).
ALICE has also measured in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 5.02 TeV the \(\psi \text {(2S)}\) to \(\mathrm {J}/\psi \) relative modification factor and has found \(0.52\pm 0.09\mathrm { (stat)}\pm 0.08\mathrm { (syst)}\) for \(4.46<y <2.96\) and \(0.69\pm 0.09\mathrm { (stat)}\pm 0.10\mathrm { (syst)}\) for \(2.03<y <3.53\) [326], respectively \(4 \,\sigma \) and \(2 \,\sigma \) lower than unity. In the right panel of Fig. 44, the relative modification factor is shown as a function of rapidity. This double ratio has also been measured as a function of \(p_{\mathrm {T}}\) [326] and does not exhibit a significant \(p_{\mathrm {T}}\) dependence. In addition, preliminary results [439] show that the nuclear modification factor of the \(\psi \text {(2S)}\) follows a similar trend as the \(\mathrm {J}/\psi \) as a function of event activity at forward rapidity but is significantly more suppressed at backward rapidity towards central events.
Models based on initialstate effects [363, 438] or coherent energy loss [397] do not predict a relative suppression of the \(\psi \text {(2S)}\) production with respect to the \(\mathrm {J}/\psi \) one. These measurements could indicate that the \(\psi \text {(2S)}\) production is sensitive to finalstate effects in p–A collisions. A recent theoretical work uses EPS09 LO nPDF and includes the interactions of the quarkonium states with a comoving medium [436] (COMOV). The COMOV calculations are shown in Fig. 44. They describe fairly well the PHENIX and ALICE results. Hot nuclear matter effects were also proposed as a possible explanation for the \(\psi \text {(2S)}\) relative suppression in central p–Pb collisions at the LHC [440].
The data are compared to models based on nPDFs (CEM EPS09 NLO, EXT EPS09 LO), coherent energy loss (COH.ELOSS) and gluon saturation (SAT). Given the limited statistics, the data cannot constrain the models in most of the phase space and is in good agreement with the theory calculations. Only at midrapidity at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 200\(~\text {GeV}\), the observed suppression is challenging for all the models, where no suppression is expected. In the nPDF based model, the rapidity range where \(R_{\mathrm {pA}}\) is higher than unity corresponds to the antishadowing region. Clearly the data is not precise enough to conclude on the strength of gluon antishadowing.
As in the \(\mathrm {J}/\psi \) case, the ratio \(R_{\mathrm {FB}}\) of the nuclear modification factors for a rapidity range symmetric with respect to \(y \sim 0\) has also been extracted for the \(\Upsilon \text {(1S)}\) at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 5.02 \(\text {TeV}\) [328, 331].
Comparison of \(\Upsilon \text {(1S)}\) \(R_{\mathrm {pPb}}\) to the one from open beauty from Fig. 37 can give a hint on finalstate effects on \(\Upsilon \text {(1S)}\). A similar level of suppression is observed for the \(\Upsilon \text {(1S)}\) and the \(\mathrm {J}/\psi \) from B mesons. Larger statistics data however would be needed to rule out any finalstate effect on \(\Upsilon \text {(1S)}\) production in p–Pb.
The study of excited bottomonium states in p–Pb collisions may indicate the presence of finalstate effects in bottomonium production. The \(\Upsilon \text {(3S)}\) has the smallest binding energy (0.2\(~\text {GeV}\)), followed by the \(\Upsilon \text {(2S)}\) (0.54 \(~\text {GeV}\)) and the \(\Upsilon \text {(1S)}\) (1.10\(~\text {GeV}\)) [437]. Since the bottomonium formation time is expected to be larger than the nuclear size, the suppression in p–Pb is expected to be the same for all \(\Upsilon \) states.
To better quantify the modification between \(\mathrm pp\) and p–Pb and cancel out some of the systematic uncertainties from the detector setup, the double ratio \([ \Upsilon \text {(nS)}/\Upsilon \text {(1S)} ]_\mathrm{pPb} / [ \Upsilon \text {(nS)}/\Upsilon \text {(1S)} ]_\mathrm{pp}\) has also been evaluated by CMS at midrapidity using \(\mathrm pp\) collisions at 2.76 \(\text {TeV}\) [268] and is displayed in the right panel of Fig. 46. The double ratio in p–Pb is lower than one by \(2.4\,\sigma \) for \(\Upsilon \text {(2S)}\) and \(\Upsilon \text {(3S)}\). The double ratios signal the presence of different or stronger finalstate effect acting on the excited states compared to the ground state from \(\mathrm pp\) to p–Pb collisions.
As for the charmonium production, the excited states are not expected to be differentially suppressed by any of the models that include initialstate effects nor from the coherent energyloss effect. A possible explanation may come from a suppression associated to the comoving medium. Precise measurements in a larger rapidity range, which covers different comoving medium density, would help to confirm this hypothesis.
CMS has also performed measurements as a function of the event activity at forward (\(4<\eta <5.2\) for the transverse energy \(E_\mathrm{T}\)) and midrapidity (\(\eta <2.4\) for the chargedtrack multiplicity \(N_\mathrm{tracklets}\)) [268]. Figure 47 shows the \(\Upsilon \) selfnormalised cross section ratios \(\Upsilon \text {(1S)}\)/\(\langle \Upsilon \text {(1S)} \rangle \) where \(\langle \Upsilon \text {(1S)} \rangle \) is the eventactivity integrated value for \(\mathrm pp\), p–Pb and Pb–Pb collisions. The selfnormalised cross section ratios are found to rise with the event activity as measured by these two estimators and similar results are obtained for \(\Upsilon \text {(2S)}\) and \(\Upsilon \text {(3S)}\). When Pb ions are involved, the increase can be related to the increase in the number of nucleon–nucleon collisions. A possible interpretation of the positive correlation between the \(\Upsilon \) production yield and the underlying activity of the \(\mathrm pp\) event is related to MultipleParton Interactions (MPI) occurring in a single \(\mathrm pp\) collisions. Linear fits performed separately for the three collision systems show that the selfnormalised ratios have a slope consistent with unity in the case of forward event activity. Hence, no significant difference between \(\mathrm pp\), p–Pb and Pb–Pb is observed when correlating \(\Upsilon \) production yields with forward event activity. On the contrary in the case of midrapidity event activity, different slopes are found for the three collisions systems. These observations are also related to the single cross section ratios \(\Upsilon \text {(nS)}/\Upsilon \text {(1S)} \) as shown in Fig. 19 and discussed in detail in Sect. 2.4.1.
3.4 Extrapolation of CNM effects from p–A to AA collisions
It is an important question to know whether cold nuclear matter effects can be simply extrapolated from p–A to AA collisions. Some of the CNM effects discussed in Sect. 3.2 can in principle be extrapolated to AA collisions. This is the case of nPDF and coherent energyloss effects, discussed below. Some other approaches, on the contrary, are affected by interference effects between the two nuclei involved in the collision, making delicate an extrapolation to AA collisions.
In principle, this factorisation hypothesis can also be applied to open heavy flavour.
Multiple scattering and energy loss Let us first discuss how predictions can be extrapolated in the coherent energyloss model. In a generic A–B collision both incoming partons, respectively from the ‘projectile’ nucleus A and the ‘target’ nucleus B, might suffer multiple scattering in the nucleus B and A, respectively. Consequently, gluon radiation off both partons can interfere with that of the finalstate particle (here, the compact colour octet \({Q\overline{Q}} \) pair), making a priori difficult the calculation of the mediuminduced gluon spectrum in the collision of two heavy ions.
However, it was shown in [404] that the gluon radiation induced by rescattering in nuclei A and B occurs in distinct regions of phase space (see Fig. 49). As a consequence, the energy loss induced by the presence of each nucleus can be combined in a probabilistic manner, making a rather straightforward extrapolation of the model predictions from p–A to AA collisions. Remarkably, it is possible to show that the quarkonium suppression in AA collisions follows the factorisation hypothesis (see Eq. (26)). However, since the energyloss effects do not scale with the momentum fraction \(x_2\), the datadriven extrapolation of p–Pb data at \(\sqrt{s_{\mathrm{NN}}} =5\) TeV to Pb–Pb data at \(\sqrt{s_{\mathrm{NN}}} =2.76\) TeV, discussed below, and which assumes nPDF effects only is not expected to hold [404].
The model by Sharma and Vitev can also easily be generalised to AA reactions where both incoming and outgoing partons undergo elastic, inelastic and coherent soft interactions in the large nuclei. In contrast, the Kopeliovich, Potashnikova and Schmidt approach for charmonium production cannot be simply extrapolated from p–A to AA collisions, because nucleus–nucleus collisions include new effects of double colour filtering and a boosted saturation scale, as explained in detail in [385].
Datadriven extrapolation At RHIC, the d–Au collisions are performed with symmetrical beam energies, so that \(y_\mathrm{CM/lab} = 0\), and at the same nucleon–nucleon centreofmass energy than for heavyion collisions. The direct comparison of d–Au data to heavyion data is then easier. In this context, the PHENIX experiment has evaluated the \(\mathrm {J}/\psi \) breakup (i.e. absorption) cross section by fitting \(R_{\mathrm {dAu}}\) as a function of the rapidity, and also as a function of the average number of binary collisions (\(\mathrm {N_{coll}}\)), and by assuming different shadowing scenarios (EKS and NDSG) [317]. The two shadowing scenarios with their resulting breakup cross section were applied to \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\), both for Cu–Cu and Au–Au collisions. Moreover, an alternative datadriven method [443] was applied to PHENIX data [317]. This method assumes that all cold nuclear matter effects are parametrised with a modification factor consisting of a function of the radial position in the nucleus. Note that the use of d–Au data in [443] may not be appropriate for peripheral collisions where the size of the deuteron causes significant averaging over impact parameter; on the contrary it should be adequated in central collisions for which the averaging is not so important. An attempt to solve this issue has been proposed in [64] where an estimate of \(R_{\mathrm {pAu}}\) was derived from \(R_{\mathrm {dAu}} \) using a Glauber model including EKS98 nuclear PDF.
A more recent investigation of RHIC data by Ferreiro et al. [374] showed how the use of \(2 \rightarrow 2\) partonic process instead of the usual \(2 \rightarrow 1\) can imply a different value of the absorption cross section [444], since the antishadowing peak is systematically shifted towards larger rapidities in d–Au. The other noticeable consequence is that \(R_{\mathrm {dAu}}\) versus y is not symmetric anymore around \(y \approx 0\). This implies that the CNM effects in \(R_{\mathrm {AA}}\) at RHIC will also show a rapidity dependence, with less suppression from CNM effects at midrapidity than at forward rapidity, in the same direction as the one exhibited by the Au–Au and Cu–Cu data from PHENIX (see extensive comparisons in [374]). This is quite important since this shape of \(R_{\mathrm {AA}}\) at RHIC was also considered as a possible hint for hot inmedium recombination effects, while it might come from CNM effects only.
At LHC, the p–Pb results cannot easily be compared to Pb–Pb collisions. Indeed, the nucleon–nucleon centreofmass energies are not the same (5.02 versus 2.76 TeV) and, moreover, the p and the Pb beam energies per nucleon are different, leading to a rapidity shift of the centreofmass frame with respect to the lab frame. But assuming factorisation and Eq. (26), a datadriven extrapolation of p–A data to AA can be performed.
This datadriven extrapolation of p–A collisions to AA collisions applied by the ALICE Collaboration to \(\mathrm {J}/\psi \) production lead to [325]: \([R_{\mathrm {pPb}} (2<y<3.5) \cdot R_{\mathrm {pPb}} (4.5<y<3)]^{\mathrm {J}/\psi } = 0.75 \pm 0.10 \pm 0.12\), the first uncertainty being the quadratic combination of statistical and uncorrelated systematic uncertainties and the second one the linear combination of correlated uncertainties. The application of this result to the interpretation of Pb–Pb data is discussed in Sect. 5.3.
In summary, according to the theoretical and datadriven extrapolation approaches, one can conclude that there are nonnegligible CNM effects on AA results at the LHC (up to 50 % at low \(p_{\mathrm {T}}\)). A \(p_{\mathrm {T}}\) dependence of \(\mathrm {J}/\psi \) \(R_{\mathrm {pPb}}\) factorisation will be presented in Sect. 5.1.2.
3.5 Summary and outlook

The nuclear modification factor of open heavyflavour decay leptons in d–Au collisions at RHIC shows a dependence on centrality and on rapidity, with values smaller than unity at forward rapidity and larger than unity at mid and backward rapidity in the most central collisions.

In p–Pb collisions at the LHC, the Dmeson nuclear modification factor at midrapidity and \(1<p_{\mathrm {T}} <16\) \(~\text {GeV}/c\) is consistent with unity within uncertainties of about 20 %.

The \(R_{\mathrm {pA}}\) of \(\mathrm {J}/\psi \) from B mesons at the LHC shows a modest suppression at forward rapidity and is consistent with unity at backward rapidity.

A rapidity dependence of \(\mathrm {J}/\psi \) suppression has been measured at RHIC and LHC. At both energies the suppression is more pronounced at forward than at midrapidity. At backward rapidity, \(\mathrm {J}/\psi \) production is slightly suppressed at RHIC but is compatible with no suppression at the LHC.

There is no evidence of \(\mathrm {J}/\psi \) suppression at large \(p_{\mathrm {T}} \) in the full rapidity range at RHIC and LHC.

At RHIC, open heavy flavour from lepton decay and \(\mathrm {J}/\psi \) suppression for \(p_{\mathrm {T}} > 1~\text {GeV}/c \) are of the same order at forward rapidity but not at backward and midrapidity: suppression mechanisms from finalstate effects may be at play on \(\mathrm {J}/\psi \) production at backward and midrapidity.

\(\Upsilon \text {(1S)}\) \(R_{\mathrm {pA}}\) measurements are compatible with unity except at midrapidity at RHIC and forward rapidity at the LHC. Similar level of suppression is observed for the \(\Upsilon \text {(1S)}\) and the \(\mathrm {J}/\psi \) from Bmesons at the LHC. However, the \(\Upsilon \text {(1S)}\) \(R_{\mathrm {pA}}\) measurements have large statistical uncertainties.

Excited states are more suppressed than 1S states at RHIC and LHC suggesting the presence of finalstate CNM effects.

Open heavyflavour current data do not allow one to favour specific models based on nuclear PDF, parton saturation, or initialparton energy loss.
 Regarding \(\mathrm {J}/\psi \) production, one can conclude the following:

The nuclear PDFs describe well the \(R_{\mathrm {pA}}\) despite large theoretical uncertainties at forward rapidity where the data would require strong shadowing effects. At backward rapidity, while the nPDF models describe correctly the LHC data, they do not describe the RHIC data without considering additional effects such as nuclear absorption.

The early CGC prediction of \(\mathrm {J}/\psi \) \(R_{\mathrm {pA}}\) by Fuji and Watanabe is ruled out by the present LHC data at forward rapidity. The calculations do not describe either the \(p_{\mathrm {T}}\) dependence of the RHIC data at forward rapidity. Refinements of the model have now been proposed, leading to less disagreement with data.

The predictions of the coherent energyloss model describes well the rapidity dependence of \(\mathrm {J}/\psi \) \(R_{\mathrm {pA}}\) both at RHIC and at LHC. Regarding the \(p_{\mathrm {T}} \) dependence, the shape of the data is also rather well captured, although the dependence is slightly more abrupt in the model than in the data, especially at forward rapidity. The predicted \(\mathrm {J}/\psi \) suppression expected in the dipole propagation model by Kopeliovich, Potashnikova and Schmidt seems much larger than seen in data, suggesting the need for additional effects to compensate the suppression. Finally, the approach based on energy loss and power corrections by Sharma and Vitev predicts a moderate and flat \(\mathrm {J}/\psi \) and \(\Upsilon \) suppression as a function of \(p_{\mathrm {T}}\), above \(p_{\mathrm {T}} =4\) GeV/c, somehow in contradiction with data.


The suppression of excited states relative to 1S state is described so far only by considering the effect from a comoving medium.
Regarding the experimental uncertainties, in the case of rare probes like B mesons, \(\psi \text {(2S)}\) and \(\Upsilon \), but also high \(p_{\mathrm {T}}\) yields, the experimental data suffer from limited statistics. For more abundant probes, like heavyflavour decay leptons. D mesons, B from \(\mathrm {J}/\psi \) and prompt \(\mathrm {J}/\psi \), the size of the systematic uncertainties is the main limitation.
To address part of these issues, a reference \(\mathrm pp\) period at \(\sqrt{s} =5.02\) \(\text {TeV}\) and a higherstatistics p–Pb period at \(\sqrt{s} =5.02\) \(\text {TeV}\), which will allow a better control of the systematics, during the LHC Run 2 would be very helpful to improve the precision of the current measurements. However, for the probes which are using the full LHC luminosity and have already a \(\mathrm pp\) reference at 8 \(\text {TeV}\) from the 2012 Run 1 data taking period, it would be more interesting to get a new p–Pb run at \(\sqrt{s_{\mathrm{NN}}} =8\) \(\text {TeV}\) in order to study the CNM effects at higher energy. These aspects have to be balanced in order to choose the energy for the p–Pb run in Run 2.
New observables could help to disentangle the various CNM effects. First studies of the heavyflavour azimuthal correlations at RHIC and LHC were carried out and (at RHIC) suggest a modification of charm production kinematics in d–Au. A comparison of open to hidden heavy flavour production from \(p_{\mathrm {T}} =0\) would allow one to separate initial from finalstate effects on quarkonia. Another open question is related to quarkonium polarisation: can the CNM effects modify the polarisation of quarkonia? In addition, the RHIC capability to collide a polarisedproton beam with nuclei can be used to explore new observables.
Finally, it is not clear whether the CNM effects can be extrapolated from p–A to AA collisions. At present, only phenomenological works based on nPDF and coherent energyloss effects have shown that this extrapolation is possible, although with some caveats. On the one hand, in the nPDFbased models, the main parameter is the probed momentum fraction x. Since there is a rapidity shift of the centreofmass in p–Pb collisions at LHC, the optimal strategy would be to choose the LHC beam energy according to Eq. (27). On the other hand, in the coherent energyloss model, the relevant parameter is the \(\sqrt{s_{\mathrm{NN}}}\) value and p–A collisions can be directly related to AA collisions only if taken at the same energy.
4 Open heavy flavour in nucleus–nucleus collisions
Heavyflavour hadrons are effective probes of the conditions of the highenergydensity QGP medium formed in ultrarelativistic nucleus–nucleus collisions.
Heavy quarks, are produced in primary hard QCD scattering processes in the early stage of the nucleus–nucleus collision and the time scale of their production (or coherence time) is, generally, shorter than the formation time of the QGP, \(\tau _0\sim 0.1\)–1 fm / c. More in detail, the coherence time of the heavy quarkantiquark pair is of the order of the inverse of the virtuality Q of the hard scattering, \(\Delta \tau \sim 1/Q\). The minimum virtuality \(2\,m_{c,b}\) in the production of a \({c\overline{c}}\) or \({b\overline{b}}\) pair implies a spacetime scale of \(\sim 1/3~\text {GeV} ^{1}\sim 0.07~\mathrm{fm}\) and \(\sim 1/10~\text {GeV} ^{1}\sim 0.02~\mathrm{fm}\) for charm and for beauty, respectively. One exception to this picture are configurations where the quark and antiquark are produced with a small relative opening angle in the socalled gluon splitting processes \(g\rightarrow {q\overline{q}} \). In this case, the coherence time is increased by a boost factor \(E_g/(2\,m_{c,b})\sim E_{c,b}/m_{c,b}\) and becomes \(\Delta t \sim E_{c,b}/(2\,m_{c,b}^2)\). This results, for example, in a coherence time of about 1 fm/c (0.1 fm/c) for charm (beauty) quarks with energy of 15\(~\text {GeV}\), and of about 1 fm/c for beauty quark jets with energy of about 150\(~\text {GeV}\). The fraction of heavy quarks produced in gluon splitting processes has been estimated using perturbative calculations and Monte Carlo event generators, resulting in moderate values of the order of 10–20 % for charm [445, 446] and large values of the order of 50 % for beauty [447]. Given that the gluon splitting fraction is moderate for charm and the coherence time is small for beauty from gluon splitting when \(p_{\mathrm {T}} \) is smaller than about 50\(~\text {GeV}/c\), it is reasonable to conclude that heavyflavour hadrons in this \(p_{\mathrm {T}}\) range probe the heavy quark inmedium interactions.
During their propagation through the medium, heavy quarks interact with its constituents and lose a part of their energy, thus being sensitive to the medium properties. Various approaches have been developed to describe the interaction of the heavy quarks with the surrounding plasma. In a perturbative treatment, QCD energy loss is expected to occur via both inelastic (radiative energy loss, via mediuminduced gluon radiation) [448, 449] and elastic (collisional energy loss) [450, 451, 452] processes. However, this distinction is no longer meaningful in strongly coupled approaches relying for instance on the AdS/CFT conjecture [453, 454]. In QCD, quarks have a smaller colour coupling factor with respect to gluons, so that the energy loss for quarks is expected to be smaller than for gluons. In addition, the “deadcone effect” should reduce smallangle gluon radiation for heavy quarks with moderate energyovermass values, thus further attenuating the effect of the medium. This idea was first introduced in [455]. Further theoretical studies confirmed the reduction of the total induced gluon radiation [456, 457, 458, 459], although they did not support the expectation of a “dead cone”. Other mechanisms such as inmedium hadron formation and dissociation [422, 460], would determine a stronger suppression effect on heavyflavour hadrons than lightflavour hadrons, because of their smaller formationtimes.
In contrast to light quarks and gluons, which can be produced or annihilated during the entire evolution of the medium, heavy quarks are produced in initial hardscattering processes and their annihilation rate is small [461]. Secondary “thermal” charm production from processes like \(gg\rightarrow {c\overline{c}} \) in the QGP is expected to be negligible, unless initial QGP temperatures much larger than that accessible at RHIC and LHC are assumed [462]. Therefore, heavy quarks preserve their flavour and mass identity while traversing the medium and can be tagged throughout all momentum ranges, from low to high \(p_{\mathrm {T}}\), through the measurement of heavyflavour hadrons in the final state of thecollision.
Open heavy flavour published measurements in Au–Au and Cu–Cu collisions at RHIC. The nucleon–nucleon energy in the centreofmass system (\(\sqrt{s_{\mathrm{NN}}}\)), the covered kinematic ranges and the observables are indicated
Probe  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (\(\text {TeV}\))  \(y_\mathrm{cms}\) (or \(\eta _\mathrm{cms}\))  \(p_{\mathrm {T}}\) (\(\text {GeV}/c\))  Observables  References 

PHENIX  
\(\mathrm{HF} \rightarrow e^{\pm }\)  Au–Au  62.4  \(y<0.35\)  1–5  Yields (\(p_{\mathrm {T}}\),centrality)  [467] 
\(R_{\mathrm {CP}} \)(\(p_{\mathrm {T}}\))  
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
\(R_{\mathrm {AA}} \)(\(\mathrm {N_{coll}}\),\(p_{\mathrm {T}}\))  
1.3–3.5  \(v_{2} \)(\(p_{\mathrm {T}}\),centrality)  
1.3–2.5  \(v_{2} \)(\(\sqrt{s_{\mathrm{NN}}}\),centrality)  
130  \(y<0.35\)  0.4–3  Yields (\(p_{\mathrm {T}}\),centrality)  [468]  
200  \(\eta <0.35\)  0.3–9  Yields (\(p_{\mathrm {T}}\),centrality)  
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
\(R_{\mathrm {AA}} \)(\(\mathrm {N_{part}}\),\(p_{\mathrm {T}}\))  
\(>0.4\)  \(\frac{\mathrm {d}\sigma _{NN}}{\mathrm {d}y}\)(\(\mathrm {N_{coll}}\))  
\(>0\)  \(\frac{\mathrm {d}\sigma _{NN}}{\mathrm {d}y}\)(centrality)  
\(\sigma _{NN}^{{c\overline{c}}}\)(centrality)  
0.3–5  \(v_{2} \)(\(p_{\mathrm {T}}\),centrality)  
200  \(y<0.35\)  2–4  \(\frac{1}{N_{\mathrm {trig}}^{e_{HF}}}\frac{\mathrm {d}N_{\mathrm {assoc.}}^{\mathrm {h}}}{\mathrm {d}p_{\mathrm {T}}}(p_{\mathrm {T}} ^{\mathrm {h}},\Delta \phi )\)  [113]  
\(I_{\mathrm {AA}} ^{e_{HF}h}(p_{\mathrm {T}} ^{\mathrm {h}},\Delta \phi )\)  
2–3  \(R_{\mathrm {HS}}(p_{\mathrm {T}} ^{\mathrm {h}})\)  
Cu–Cu  200  \(y<0.35\)  0.5–7  Yields (\(p_{\mathrm {T}}\),centrality)  [473]  
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
\(R_{\mathrm {AA}} \)(\(\mathrm {N_{coll}}\),\(p_{\mathrm {T}}\))  
\(R_{\mathrm {AA}} \)(\(\mathrm {N_{part}}\),\(p_{\mathrm {T}}\))  
0.5–6  \(R_{\mathrm {CP}} \)(\(p_{\mathrm {T}}\))  
\(\mathrm{HF} \rightarrow \mu ^{\pm }\)  Cu–Cu  200  \(1.4<y<1.9\)  1–4  Yields (\(p_{\mathrm {T}}\),centrality)  [474] 
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
\(R_{\mathrm {AA}} \)(\(\mathrm {N_{part}}\))  
STAR  
\(\mathrm {D}^{0}\)  Au–Au  200  \(y<1\)  0–6  Yields (\(p_{\mathrm {T}}\),centrality)  [475] 
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
0–8  \(R_{\mathrm {AA}} \)(\(\langle \mathrm {N_{part}} \rangle \),\(p_{\mathrm {T}}\))  
\(\mathrm{HF} \rightarrow e^{\pm }\)  Au–Au  200  \(0<\eta <0.7\)  1.2–8.4  Yields (\(p_{\mathrm {T}}\),centrality)  [420] 
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\),centrality)  
39, 62.4, 200  \(\eta <0.7\)  0–7  \(v_{2} \)(\(p_{\mathrm {T}}\))  [476] 
These questions can be addressed both with the study of the \(R_{\mathrm {AA}}\) at low and intermediate \(p_{\mathrm {T}}\) (smaller than about five times the heavyquark mass) and with azimuthal anisotropy measurements of heavyflavour hadron production with respect to the reaction plane, defined by the beam axis and the impact parameter of the collision. For noncentral collisions, the two nuclei overlap in an approximately lenticular region, the short axis of which lies in the reaction plane. Hard partons are produced at an early stage, when the geometrical anisotropy is not yet reduced by the system expansion. Therefore, partons emitted in the direction of the reaction plane (inplane) have, on average, a shorter inmedium path length than partons emitted orthogonally (outofplane), leading a priori to a stronger high\(p_{\mathrm {T}}\) suppression in the latter case. In the lowmomentum region, the inmedium interactions can also modify the parton emission directions, thus translating the initial spatial anisotropy into a momentum anisotropy of the finalstate particles. Both effects cause a momentum anisotropy that can be characterised with the coefficients \(v_n\) and the symmetry planes \(\Psi _n\) of the Fourier expansion of the \(p_{\mathrm {T}}\)dependent particle distribution \(\mathrm {d}^2N/\mathrm {d}p_{\mathrm {T}} \mathrm {d}\phi \) in azimuthal angle \(\phi \). The elliptic flow is the second Fourier coefficient\(v_{2}\).
The final ambitious goal of the heavyflavour experimental programmes in nucleus–nucleus collisions is the characterisation of the properties of the produced QCD matter, in particular getting access to the transport coefficients of the QGP. Theoretical calculations encoding the interaction of the heavy quarks with the plasma into a few transport coefficients (see e.g. [466]) provide the tools to achieve this goal: through a comparison of the experimental data with the numerical outcomes obtained with different choices of the transport coefficients it should be possible, in principle, to put tight constraints on the values of the latter. This would be the analogous of the way of extracting information on the QGP viscosity through the comparison of softparticle spectra with predictions from fluid dynamic models. An even more intriguing challenge would be to derive the heavyflavour transport coefficients through a first principle QCD calculation and confront them with experimental data, via model implementations that describe the medium evolution. This chapter reviews the present status of this quest, from the experimental and theoreticalviewpoints.
Open heavy flavour published measurements in Pb–Pb collisions at LHC. The nucleon–nucleon energy in the centreofmass system (\(\sqrt{s_{\mathrm{NN}}}\)), the covered kinematic ranges and the observables are indicated
Probe  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (\(\text {TeV}\))  \(y_\mathrm{cms}\) (or \(\eta _\mathrm{cms}\))  \(p_{\mathrm {T}}\) (\(\text {GeV}/c\))  Observables  References 

ALICE  
\(\mathrm {D}^{0}\), \(\mathrm {D}^{+}\), \(\mathrm {D}^{*+}\)  Pb–Pb  2.76  \(y<0.5\)  2–16  Yields (\(p_{\mathrm {T}}\))  [477] 
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\))  
2–12  \(R_{\mathrm {AA}} (\mathrm{centrality})\)  
6–12  \(R_{\mathrm {AA}} (\mathrm{centrality})\)  
\(y<0.8\)  2–16  \(v_{2} \)(\(p_{\mathrm {T}}\))  
\(v_{2} \)(centrality,\(p_{\mathrm {T}}\))  
\(R_{\mathrm {AA}} ^{\text {in/out plane}}\)(\(p_{\mathrm {T}}\))  
\(\mathrm{HF} \rightarrow \mu ^{\pm }\)  Pb–Pb  2.76  \(2.5<y<4\)  4–10  \(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\))  [120] 
6–10  \(R_{\mathrm {AA}} (\mathrm{centrality})\)  
Nonprompt \(\mathrm {J}/\psi \)  Pb–Pb  2.76  \(y<0.8\)  1.5–10  \(R_{\mathrm {AA}}\) (\(p_{\mathrm {T}}\))  [480] 
CMS  
bJets  Pb–Pb  2.76  \(\eta <2\)  80–250  Yields (\(p_{\mathrm {T}}\))  [481] 
\(R_{\mathrm {AA}} \)(\(p_{\mathrm {T}}\))  
80–110  \(R_{\mathrm {AA}} (\mathrm{centrality})\)  
Nonprompt \(\mathrm {J}/\psi \)  Pb–Pb  2.76  \(y<2.4\)  6.5–30  Yields (centrality)  [482] 
\(R_{\mathrm {AA}} \)(centrality) 
4.1 Experimental overview: production and nuclear modification factor measurements
4.1.1 Inclusive measurements with leptons
Heavyflavour production can be measured inclusively via the semileptonic decay channels. The key points of the measurement are the lepton identification and background subtraction.
In PHENIX, muons are measured with two muon spectrometers that provide pion rejection at the level of \(2.5 \times 10^{4}\) in the pseudorapidity range \(2.2 < \eta < 1.2\) and \(1.2 < \eta < 2.4\) over the full azimuth.
Muons are detected in ALICE with the forward muon spectrometer in the pseudorapidity range \(4 < \eta < 2.5\). The extraction of the heavyflavour contribution to the single muon spectra requires the subtraction of muons from the decay in flight of pions and kaons, estimated through the extrapolations of the measurements at midrapidity.
In ATLAS, muons are reconstructed in the pseudorapidity range \(\eta <1.05\) by matching the tracks in the Inner silicon Detector (ID) with the ones in the Muon Spectrometer (MS), surrounding the electromagnetic and hadronic calorimeters. The background muons arise from pion and kaon decays, muons produced in showers in the calorimeters and misassociation of MS and ID tracks. The signal component is extracted through a MC template fit of a discriminant variable that depends on the difference between the ID and MS measurements of the muon momentum, after accounting for energy loss in the calorimeters, and the deflections in the trajectory resulting from decay in flight [483].
At the LHC, heavyflavour production was measured in the leptonic decay channels in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 2.76\) \(\text {TeV}\). Figure 52 shows the nuclear modification factors of muons from heavyflavour decays in \(2.5<y<4\) measured by ALICE as a function of \(p_{\mathrm {T}}\) in the 10 % most central collisions (left panel) and as a function of centrality in \(6 < p_{\mathrm {T}} < 10\) \(~\text {GeV}/c\) (right panel) [120]. The observed suppression increases from peripheral to central collisions, up to a factor of three in central collisions. The result is consistent with a preliminary measurement of the \(R_{\mathrm {AA}}\) of heavyflavour decay electrons at midrapidity (with \(4<p_{\mathrm {T}} <18\) \(~\text {GeV}/c\)) [485]. Moreover, it is also in qualitative agreement with a preliminary measurement of the heavyflavour decay muon centraltoperipheral nuclear modification factor \(R_{\mathrm {CP}}\) at midrapidity (with \(4<p_{\mathrm {T}} <14\) \(~\text {GeV}/c\)), carried out by the ATLAS Collaboration [486], which shows a suppression of a factor about two, independent of \(p_{\mathrm {T}}\), for the centrality ratio 0–10 %/60–80 %. The comparison of the results at forward and midrapidity indicate a weak dependence on this variable in the rapidity region \(y<4\).
4.1.2 D meson measurements
The differential charm production cross section is determined from measurements of open charm mesons (STAR and ALICE). D mesons are reconstructed via the hadronic decays \(\mathrm {D}^{0} \rightarrow \mathrm {K} ^{} \pi ^{+}\), \(\mathrm {D}^{+} \rightarrow \mathrm {K} ^{} \pi ^{+} \pi ^{+}\), and \(\mathrm {D}^{*+} (2010) \rightarrow \mathrm {D}^{0} \pi ^{+}\) with \(\mathrm {D}^{0} \rightarrow \mathrm {K} ^{} \pi ^{+}\), and their charge conjugates. The mean proper decay lengths of \(\mathrm {D}^{0} \) and \(\mathrm {D}^{+} \) are of about 120 and 300 \(\upmu \mathrm{m}\), respectively, while the \(\mathrm {D}^{*+} \) decays strongly with no significant separation from the interaction vertex. In the STAR and ALICE experiments, charmed hadrons are measured with an invariant mass analysis of the fully reconstructed decay topologies in the hadronic decay channels. In both experiments, the kaon and pion identification is performed by combining the information of the Time Of Flight and of the specific ionisation energy loss in the TPC [103, 104, 125]. The spatial resolution of the ALICE silicon tracker allows, in addition, to reconstruct the decay vertex and apply a topological selection on its separation from the interaction vertex [104].
The left panel of Fig. 53 shows the transverse momentum dependence of the nuclear modification factor \(R_{\mathrm {AA}}\) for \(\mathrm {D}^{0}\) mesons in the most central Au–Au collisions at \(\sqrt{s_{\mathrm{NN}}} = 200\) \(~\text {GeV}\) from the STAR experiment [475]. The \(R_{\mathrm {AA}}\) is enhanced at around 1.5\(~\text {GeV}/c\) and shows a strong suppression at \(p_{\mathrm {T}} > 3\) \(~\text {GeV}/c\). STAR also measured \(\mathrm {D}^{0}\) mesons in U–U collisions at \(\sqrt{s_{\mathrm{NN}}} = 193\) \(~\text {GeV}\) and observed a similar trend for the \(R_{\mathrm {AA}}\) as seen in Au–Au collisions [487].
The ALICE experiment measured the production of prompt \(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \) and \(\mathrm {D}^{*+} \) mesons in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 2.76\) \(\text {TeV}\) [477]. The average \(R_{\mathrm {AA}}\) of D mesons for two centrality classes is shown in the right panel of Fig. 53. The high\(p_{\mathrm {T}}\) Dmeson yield for the most central events is strongly suppressed (by factor of about four at 10\(~\text {GeV}/c\)). The analysis of the Pb–Pb data collected in 2011 allowed to extend the measurement to higher transverse momenta: a similar suppression pattern is observed up to \(p_{\mathrm {T}} =30\) \(~\text {GeV}/c\) in the 7.5 % most central collisions [488]. In addition, the \(\mathrm {D}^{+}_{s}\) meson, consisting of a charm and an antistrange quark, was measured for the first time in Pb–Pb collisions [489]. The \(\mathrm {D}^{+}_{s}\) meson is expected to be sensitive to the possible hadronisation of charm quarks via recombination with light quarks from the medium: the expected abundance of strange quarks in the QGP may lead to an increase of the ratio of strange over nonstrange D mesons with respect to \(\mathrm pp\) collisions in the momentum range where recombination can be relevant [490, 491]. The observed central value of the \(\mathrm {D}^{+}_{s}\) \(R_{\mathrm {AA}}\) is larger than that of \(\mathrm {D}^{0}\), \(\mathrm {D}^{+}\) and \(\mathrm {D}^{*+}\) mesons at low \(p_{\mathrm {T}}\), although the large statistical and systematic uncertainties prevent from drawing any conclusion.
4.1.3 Beauty production measurements
The detection and identification of beauty hadrons usually exploit their long life times, with \(c\tau \) values of about 500 \(\upmu \)m. Precise charged particle tracking and vertexing are of crucial importance, with the required resolution of the track impact parameter in the transverse plane being of the order of 100 \(\upmu \)m. Most decay channels proceed as a \(b \rightarrow c\) hadron cascade, giving rise to a topology that contains both a secondary and a tertiary decay vertex.
Lepton identification is often exploited in beauty measurements, as the semileptonic branching ratio is about 20 %, taking into account both decay vertices. The beauty contribution can be extracted from the semielectronic decays of heavy flavours through a fit of the impactparameter distribution. This approach was applied by the ALICE Collaboration in \(\mathrm pp\) collisions at the LHC [107, 123] (see Sect. 2.2.3) and recently also in Pb–Pb collisions [492], where preliminary results indicate \(R_{\mathrm {AA}}\) values below unity for electron \(p_{\mathrm {T}}\) larger than about 5\(~\text {GeV}/c\). The charm and beauty contribution can be disentangled also by studying the correlations between electrons and associated charged hadrons, exploiting the larger width of the nearside peak for B hadron decays [107, 115, 493]. The main limitation of the beauty measurements via single electrons (or muons) is the very broad correlation between the momentum of the measured electron and the momentum of the parent B meson.
A more direct measurement is achieved using the inclusive \(\mathrm{B}\rightarrow \mathrm {J}/\psi + X\) decay mode. Such decays can be measured inclusively by decomposing the \(\mathrm {J}/\psi \) yield into its prompt and nonprompt components, using a fit to the lifetime distribution. The first measurement with this technique in heavyion collisions was performed by the CMS Collaboration, using data from the 2010 Pb–Pb run. The \(R_{\mathrm {AA}}\) of nonprompt \(\mathrm {J}/\psi \) in \(6.5 < p_{\mathrm {T}} < 30\) \(~\text {GeV}/c\) and \(y < 2.4\) was measured to be \(0.37 \pm 0.08 \mathrm{(stat.)} \pm 0.02 \mathrm{(syst.)}\) in the 20 % most central collisions (see left panel of Fig. 54) [482]. Preliminary measurements from the larger 2011 dataset explore the \(p_{\mathrm {T}}\) dependence of \(R_{\mathrm {AA}}\) [494]. A recent measurement from the ALICE Collaboration [480] (left panel of Fig. 54) shows a similar value of \(R_{\mathrm {AA}}\) in a close kinematic range (\(4.5<p_{\mathrm {T}} <10\) \(~\text {GeV}/c\) and \(y<0.8\)).
Further insights into the parton energy loss can be provided through measurements of reconstructed jets and comparison with theory [495], which is complementary to the studies on B hadrons as the reconstructed jet energy is closely related to that of the \(b\) quark. Assuming that the quark hadronises outside the medium, to first approximation the jet energy represents the sum of the parton energy after its interaction with the medium, as well as any transferred energy that remains inside the jet cone. CMS has performed a measurement of \(b\) jets in Pb–Pb collisions by direct reconstruction of displaced vertices associated to the jets [482]. Despite the large underlying Pb–Pb event, a light jet rejection factor of about 100 can still be achieved in central Pb–Pb events. The \(R_{\mathrm {AA}}\) of \(b\) jets as a function of centrality is shown in Fig. 54 (right), for two ranges of jet \(p_{\mathrm {T}}\). The observed suppression, which reaches a value of about 2.5 in central collisions, does not show any significant difference compared to a similar measurement of the inclusive jet \(R_{\mathrm {AA}}\) [496] within the sizeable systematic uncertainties. While quark mass effects may not play a role at such large values of \(p_{\mathrm {T}}\), the difference in energy loss between quarks and gluons should manifest itself as a difference in \(R_{\mathrm {AA}}\) for \(b\) jets and inclusive jets, as the latter are dominated by gluon jets up to very large \(p_{\mathrm {T}}\). It should be noted, however, that the \(b\) jets do not always originate from a primary \(b\) quark. As discussed in the introduction to this Section, at LHC energies, a significant component of \(b\) quarks are produced by splitting of gluons into \({b\overline{b}}\) pairs [447]. For \(b\) jets with very large \(p_{\mathrm {T}}\) a significant part of the inmedium path length is likely to be traversed by the parent gluon, as opposed to the \(b\) quarks (for example, about 1–2 fm for \(b\) quarks with \(p_{\mathrm {T}}\) of 100–200\(~\text {GeV}/c\)). The gluon splitting contribution can be minimised by selecting hard fragments, although this is complicated by the fact that the \(b\)–hadron kinematics are not fully measured. An alternative is to select backtoback \(b\)tagged jets, a configuration in which the gluon splitting contribution is negligible. The dijet asymmetry of \(b\) jets can then be compared to that of inclusive jets, a measurement that should be feasible with the luminosity expected from the upcoming LHC Run 2.
4.1.4 Comparison of \(R_{\mathrm {AA}}\) for charm, beauty and lightflavour hadrons
Preliminary measurements based on higherstatistics data from the 2011 Pb–Pb run at LHC provide a first indication that the nuclear modification factor of B mesons is larger than that of D at transverse momentum of about 10\(~\text {GeV}/c\). The measurements were carried out, as a function of collision centrality, for D mesons with \(8<p_{\mathrm {T}} <16\) \(~\text {GeV}/c\) and \(y<0.5\) by the ALICE Collaboration [497] and for nonprompt \(\mathrm {J}/\psi \) mesons with \(6.5<p_{\mathrm {T}} <30\) \(~\text {GeV}/c\) and \(y<1.2\) by the CMS Collaboration [494]. With these \(p_{\mathrm {T}}\) intervals, the average \(p_{\mathrm {T}}\) values of the probed D and B mesons are both about 10–11\(~\text {GeV}/c\). In central collisions (centrality classes 0–10 and 10–20 %) the \(R_{\mathrm {AA}}\) values are of about 0.2 and 0.4 for D and nonprompt \(\mathrm {J}/\psi \) mesons, respectively, and they are not compatible within experimental uncertainties. This experimental observation alone does not allow one to draw conclusions on the comparison of energy loss of charm and beauty quarks, because several kinematic effects contribute to the \(R_{\mathrm {AA}}\) resulting from a given partonic energy loss. In particular, the shape of the quark \(p_{\mathrm {T}}\) distribution (which is steeper for charm than for beauty quarks) and the shape of the fragmentation function (which is harder for \(b\rightarrow \mathrm B\) than for \(c\rightarrow \mathrm D\)) have to be taken into account. Model calculations of heavyquark production, inmedium propagation and fragmentation provide a tool to consistently consider these effects in the comparison of charm and beauty measurements. In Sect. 4.5 we will show that model calculations including a massdependent energy loss result in \(R_{\mathrm {AA}}\) values significantly larger for \(\mathrm {J}/\psi \) from B decays than for D mesons, consistently with the preliminary results from the ALICE and CMS experiments.
4.2 Experimental overview: azimuthal anisotropy measurements
As mentioned in the introduction to this chapter, the azimuthal anisotropy of particle production in heavyion collisions is measured using the Fourier expansion of the azimuthal angle (\(\phi \)) and the \(p_{\mathrm {T}}\)dependent particle distribution \(\mathrm {d}^2 N / \mathrm {d}p_{\mathrm {T}} \mathrm {d}\phi \). The second coefficient, \(v_{2}\) or elliptic flow, which is the dominant component of the anisotropy in noncentral nucleus–nucleus collisions, is measured using these three methods: event plane (EP) [498], scalar product (SP) [499] and multiparticle cumulants [500]. In the following, an overview of the elliptic flow measurements for heavyflavour particles is presented: the published measurements at RHIC use heavyflavour decay electrons (Sect. 4.2.1); the published measurements at LHC use D mesons (Sect. 4.2.2).
4.2.1 Inclusive measurements with electrons
The measurement of the production of heavyflavour decay electrons has been presented in Sect. 4.1.1. In order to determine the heavyflavour decayelectron \(v_{2}\), the starting point is the measurement of \(v_{2}\) for inclusive electrons. Inclusive electrons include, mainly, the socalled photonic (or background) electrons (from photon conversion in the detector material and internal conversions in the Dalitz decays of light mesons), a possible contamination from hadrons, and heavyflavour decay electrons. Exploiting the additivity of \(v_{2}\), the heavyflavour decayelectron \(v_{2}\) is obtained by subtracting from the inclusive electron \(v_{2}\) the \(v_{2}\) of photonic electrons and hadrons, weighted by the corresponding contributions to the inclusive yield.
decreasing trend, although the statistical uncertainties prevent a firm conclusion. The study of the centrality dependence of \(v_{2}\) (not shown) indicates a maximum effect in the two semiperipheral centrality classes (20–40 and 40–60 %), for which the initial spatial anisotropy is largest [469]. The central value of the heavyflavour electron \(v_{2}\) in Au–Au collisions at \(\sqrt{s_{\mathrm{NN}}} = 62.4\) \(~\text {GeV}\) [467] is significantly lower than at 200\(~\text {GeV}\) [see Fig. 56 (right)]. However, the statistical and systematic uncertainties are sizeable and do not allow one to conclude firmly on the energy dependence of \(v_{2}\). In Fig. 56 (right) the measurements for heavyflavour decay electrons with \(1.3<p_{\mathrm {T}} <2.5\) \(~\text {GeV}/c\) are compared with those for neutral pions with the same \(p_{\mathrm {T}}\): the pions exhibit a larger \(v_{2}\) than the electrons; however, this comparison should be taken with care, because the \(p_{\mathrm {T}}\) of the heavyflavour mesons is significantly larger than that of their decay electrons.
The STAR Collaboration measured the heavyflavour decayelectron \(v_{2}\) in Au–Au collisions at \(\sqrt{s_{\mathrm{NN}}} = 39, ~62.4\) and 200\(~\text {GeV}\) [476]. The twoparticle cumulant method was used to measure the elliptic flow for the two lower collision energies. The event plane, and both two and fourparticle cumulant methods were used at \(\sqrt{s_{\mathrm{NN}}} = 200\) \(~\text {GeV}\). Figure 57 shows the \(v_{2}\) measured with twoparticle cumulants at the three centreofmass energies. At \(\sqrt{s_{\mathrm{NN}}} = 200\) \(~\text {GeV}\) the measurement shows a \(v_{2}\) larger than zero for \(p_{\mathrm {T}} >0.3\) \(~\text {GeV}/c\), compatible with the measurement by the PHENIX Collaboration in the same centrality class (see comparison in [476]). At \(\sqrt{s_{\mathrm{NN}}} = 39\) and 62.4\(~\text {GeV}\), the \(v_{2} \{2\}\) values are consistent with zero within uncertainties.
Preliminary results by the ALICE Collaboration on the elliptic flow of heavyflavour decay electrons at central rapidity (\(y<0.6\)) and of heavyflavour decay muons at forward rapidity (\(2.5<y<4\)) in Pb–Pb collisions at the LHC show a \(v_{2}\) significantly larger than zero in both rapidity regions and with central values similar to those measured at top RHIC energy [501].
4.2.2 Dmeson measurements
The ALICE Collaboration measured the \(v_{2}\) of prompt D mesons in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =2.76\) \(\text {TeV}\) [478, 479]. The D mesons (\(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \) and \(\mathrm {D}^{*+} \)) were measured in \(y < 0.8\) and \(2 < p_{\mathrm {T}} < 16\) \(~\text {GeV}/c\) using their hadronic decay channels, and exploiting the separation of a few hundred \(\upmu \)m of the decay vertex from the interaction vertex to reduce the combinatorial background. The measurement of Dmeson \(v_{2}\) was carried out using the event plane, the scalar product and the twoparticle cumulant methods.
Figure 58 (left) shows the average of the \(v_{2}\) measurements for \(\mathrm {D}^{0} \), \(\mathrm {D}^{+} \) and \(\mathrm {D}^{*+} \) in the centrality class 30–50 % as a function of \(p_{\mathrm {T}}\) [479]. The measurement shows a \(v_{2}\) larger than zero in the interval \(2 < p_{\mathrm {T}} < 6\) \(~\text {GeV}/c\) with a \(5.7\sigma \) significance. In the same figure, the \(v_{2}\) of charged particles for the same centrality class is reported for comparison: the magnitude of \(v_{2}\) is similar for charmed and lightflavour hadrons. Figure 58 (right) shows the dependence on collision centrality of the \(\mathrm {D}^{0} \) meson \(v_{2}\) for three \(p_{\mathrm {T}}\) intervals. An increasing trend of \(v_{2}\) towards more peripheral collisions is observed, as expected due to the increasing initial spatial anisotropy.
As discussed at the beginning of this Section, the azimuthal dependence of the nuclear modification factor \(R_{\mathrm {AA}}\) can provide insight into the path length dependence of heavyquark energy loss. The nuclear modification factor of \(\mathrm {D}^{0} \) mesons in Pb–Pb collisions (30–50 % centrality class) was measured by the ALICE Collaboration in the direction of the event plane (inplane) and in the direction orthogonal to the event plane (outofplane) [478]. The results, shown in Fig. 59, exhibit a larger high\(p_{\mathrm {T}}\) suppression in the outofplane direction, where the average path length in the medium is expected to be larger. It is worth noting that the difference between the values of \(R_{\mathrm {AA}}\) inplane and \(R_{\mathrm {AA}}\) outofplane is equivalent to the observation of \(v_{2} > 0\), because the three observables are directly correlated.
4.3 Theoretical overview: heavy flavour interactions in the medium
All the approaches include a description of the interactions that occur between the heavy quarks and the partonic constituents of the QGP. For ultrarelativistic heavy quarks (\(p_Q\gg m_Q\), say \(>10\,m_Q\)), the dominant source of the energy loss is commonly considered to be the radiation of gluons resulting from the scattering of the heavy quark on the medium constituents. These are \(2\rightarrow 3\) processes \(q(g)Q\rightarrow q(g)Qg\), where q(g) is a medium light quark (or gluon). As this mechanism proceeds through long formation times, several scatterings contribute coherently and quantities like the total energy loss \(\Delta E(L)=p_Q^\mathrm{in}p_Q^\mathrm{fin}\) can only be evaluated at the end of the inmedium path length L. This feature is shared by all schemes that have been developed to evaluate radiative energy loss of ultrarelativistic partons [456, 457, 458]. For merely relativistic heavy quarks (say \(p_Q<10\,m_Q\)), elastic (collisional) processes are believed to have an important role as well. These are \(2\rightarrow 2\) process \(q(g)Q\rightarrow q(g)Q\). The inmedium interactions are gauged by the following, closely related, variables: the mean free path \(\lambda =1/(\sigma \rho )\) is related to the medium density \(\rho \) and to the cross section \(\sigma \) of the partonmedium interaction (for \(2\rightarrow 2\) or \(2\rightarrow 3\) processes); the Debye mass \(m_D\) is the inverse of the screening length of the colour electric fields in the medium and it is proportional to the temperature T of the medium; the transport coefficients encode the momentum transfers with the medium (more details are given in the next paragraph).
In the relativistic regime, the gluon formation time for radiative processes becomes small enough that the energyloss probability \(\mathcal{P}(\Delta E)\) can be evaluated as a result of some local transport equation– like the Boltzmann equation, relying on local cross sections – evolving from initial to final time. This simplification can be applied also to collisional processes. When the average momentum transfer is small with respect to the quark mass,^{19} the Boltzmann equation can be reduced to the Fokker–Planck equation, which is often further simplified to the Langevin stochastic equation (see [466] for a recent review). These linear partialdifferential equations describe the timeevolution of the momentum distribution \(f_Q\) of heavy quarks. The medium properties are encoded in three transport coefficients: (a) the drift coefficient – also called friction or drag coefficient – which represents the fractional momentum loss per unit of time in the absence of fluctuations and admits various equivalent symbolic representations (\(\eta _D,\,A_Q,\,\ldots \)) and (b) the longitudinal and transverse momentum diffusion coefficients \(B_\mathrm{L}\) and \(B_\mathrm{T}\) (or \(B_1\) and \(B_0\), \(\kappa _\mathrm{L}\) and \(\kappa _\mathrm{T}\),..., depending on the authors), which represent the increase of the variance of \(f_Q\) per unit of time. For small momentum, the drift and diffusion coefficients are linked through the Einstein relation \(B=m_Q\,\eta _D\,T\) and also uniquely related to the spatial diffusion coefficient \(D_s\), which describes the spread of the distribution in physical space with time. Although the Fokker–Planck approach has some drawbacks,^{20} it can also be deduced from more general considerations [503], so that it may still be considered as a valid approach for describing heavyquark transport even when the Boltzmann equation does not apply, as for instance in the strongcoupling limit.
Some of the approaches consider only partonic interactions and define the \(\mathcal{P}_{Q\rightarrow H}\) probability as a convolution of \(\mathcal{P}_{Q\rightarrow Q'}(p_Q^\mathrm{in},p_Q^\mathrm{fin})\) – the probability for the heavy quark to lose \(p_Q^\mathrm{in}p_Q^\mathrm{fin}\) in the medium – with the unmodified fragmentation function. A number of approaches also include, for lowintermediate momentum heavy quarks, a contribution of hadronisation via recombination (also indicated as coalescence). Finally, some approaches consider latestage interactions of the heavyflavour hadrons with the partonic or hadronic medium.

Sections 4.3.1 and 4.3.2 are devoted to pQCD and pQCDinspired calculations of radiative and collisional energy loss, as developed by Gossiaux et al. (MC\(@_s\)HQ), Beraudo et al. (POWLANG), Djordjevic et al., Vitev et al. and Uphoff et al. (BAMPS); examples of the relative energy loss (\(\Delta E/E\)) and the momentum loss per unit length (\(\mathrm {d}P/\mathrm {d}t\)) for \(c\) and \(b\) quarks are shown.

Section 4.3.3 focuses on the calculation by Vitev et al. of inmedium formation and dissociation of heavyflavour hadrons; this proposed mechanism is expected to effectively induce an additional momentum loss with respect to radiative and collisional heavyquark inmedium interactions alone.

Sections 4.3.4 and 4.3.5 describe the calculation of transport coefficients through Tmatrix approach supplemented with a nonperturbative potential extracted from lattice QCD (Rapp et al., TAMU) or through direct ab initio latticeQCD calculations (Beraudo et al., POWLANG); the transport coefficients that are discussed are the spatial diffusion coefficient (or friction coefficient), for which examples are shown, and the momentum diffusion coefficient.

Section 4.3.6 presents the AdS/CFT approach for the calculation of the transport coefficients, developed by Horowitz et al..
The implementation of these various approaches in full models that allow one to compute the final heavyflavour hadron kinematic distributions will be described in Sect. 4.4, with particular emphasis on the modelling of the QGP and its evolution.
Given the focus of this review, we have chosen not to discuss the theoretical approaches that were not yet applied to LHC energies at the time of writing the document. For example, the modelling of heavy quark energy loss within the Dynamical QuasiParticle Model (DQPM) approach in [504, 505], recently integrated in the PHSD transport theory [506], appears to be quite promising.
4.3.1 pQCD energy loss in a dynamical QCD medium
The radiative processes, which are neglected in the model described above, are taken into account in other approaches. Djordjevic et al. developed a stateoftheart dynamical energyloss formalism, which (i) is applicable for both light and heavy partons, (ii) computes both radiative [511, 512] and collisional [513] energy loss in the same theoretical framework, (iii) takes into account recoil of the medium constituents, i.e. the fact that medium partons are moving (i.e. dynamical) particles, (iv) includes realistic finite size effects, i.e. the fact that the partons are produced inside the medium and that the medium has finite size. Recently, the formalism was also extended to include (v) finite magnetic mass effects [514] and (vi) running coupling (momentum dependence of \(\alpha _s\)) [515].
Note that this dynamical energy loss presents an extension of the wellknown static DGLV [457, 518] energyloss formalism to the dynamical QCD medium. The connection between dynamical and static energy losses was discussed in Refs. [511, 512]. That is, static energy loss can be obtained from the above dynamical energyloss expression by replacing the dynamical mean free path and effective potential by equivalent expressions for a static QCD medium: \(v_\mathrm {dyn}(\vec {q})\rightarrow v_\mathrm {stat}(\vec {q})=\frac{\mu _E^2}{(\vec {q}^2{+}\mu _E^2)^2}\) and \(\lambda _\mathrm {dyn}^{1}\rightarrow \lambda _\mathrm {stat}^{1}= 6 \frac{1.202}{\pi ^2} \frac{1{+}\frac{n_f}{4}}{1{+}\frac{n_f}{6}} \lambda _\mathrm {dyn}^{1}\). Note that the static DGLV formalism was also used in the WHDG model [459, 519], as well as for the quark energyloss calculation by Vitev et al. (see Sect. 4.3.3).
The dynamical energyloss formalism was further extended to the case of finite magnetic mass, since various nonperturbative approaches suggest a nonzero magnetic mass at RHIC and LHC collision energies (see e.g. Refs. [520, 521, 522, 523, 524]). The finite magnetic mass is introduced through generalised sumrules [514]. The main effect of the inclusion of finite magnetic mass turns out to be the modification of effective cross section \(v_\mathrm {dyn}(\vec {q})\) in Eq. (33) to \(v(\vec {q})=\frac{\mu _E^2\mu _M^2}{(\vec {q}^2+\mu _M^2) (\vec {q}^2{+}\mu _E^2)}\), where \(\mu _M\) is the magnetic mass. In Fig. 61, the fractional energy loss \(\frac{\Delta E}{E}\) corresponding to the full model described above is shown, for a path length \(L=5~\mathrm{fm}\) and an effective constant temperature of \(T=304~\text {MeV} \). For charm quarks, radiative energy loss starts to dominate for \(p_{\mathrm {T}} > 10\) \(~\text {GeV}/c\), while this transition happens for \(p_{\mathrm {T}} >25\) \(~\text {GeV}/c\) for beauty quark. The comparison of radiative energy loss for the two quark species clearly illustrates the dead cone effect, as well as its disappearance when \(p_{\mathrm {T}} \gg m_Q\).
The same group recently published predictions for photontagged and Bmesontagged bjet production at LHC [527].
4.3.2 A pQCDinspired running \(\alpha _s\) energyloss model in MC\(@_s\)HQ and BAMPS
In the Monte Carlo at Heavy Quark approach [528, 529, 530] (MC\(@_s\)HQ), heavy quarks lose and gain energy by interacting with light partons from the medium (assumed to be in thermal equilibrium) according to rates which include both collisional and radiative types of processes.
A similar model is implemented in BAMPS [537, 538, 539, 540], although with some variations. In BAMPS the Debye mass \(m_D^2\) is calculated dynamically from the nonequilibrium distribution functions f of gluons and light quarks via [541] \(m_D^2 = \pi \alpha _s \nu _g \int \frac{\mathrm {d}^3p}{(2\pi )^3} \frac{1}{p} ( N_c f_g + n_f f_q) \), where \(N_c=3\) denotes the number of colours and \(\nu _g = 16\) is the gluon degeneracy. While MC@\(_s\)HQ applies the equilibrium Debye mass with quantum statistics for temperatures extracted from the fluid dynamic background, BAMPS treats all particles as Boltzmann particles, due to the nonequilibrium nature of the cascade. Moreover, in BAMPS the scale of the running coupling in the Debye mass is evaluated at the momentum transfer of the process, e.g. \(\alpha _s(t)\). The differences in the treatment lead to a larger energy loss of about a factor of two in MC@\(_s\)HQ compared to BAMPS.
Figure 62 illustrates two properties of this energyloss model as implemented in MC@sHQ model. Both the pure elastic case and a combination of the elastic and LPMradiative energy loss are considered. In both cases, the model is calibrated by applying a multiplicative Kfactor to the interaction cross sections, in order to describe the \(R_{\mathrm {AA}}\) of D mesons for intermediate \(p_{\mathrm {T}}\) range in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =2.76\) \(\text {TeV}\) in the 0–20 % centrality class. This leads to \(K_\mathrm{el}=1.5\) and \(K_\mathrm{el+LPMrad}=0.8\), while one obtains \(K_\mathrm{el}=1.8\) and \(K_\mathrm{el+LPMrad}=1\) following a similar procedure at RHIC. For the spatial diffusion coefficient \(D_s\), one sees that both combinations are compatible with the lQCD calculations of Refs. [550, 551] and thus provide some systematic “error band” of the approach. The corresponding average momentum loss per unit of time (or length), shown on the right panel of Fig. 62, illustrates the masshierarchy, found to be stronger for the radiative component (black lines in the figure).
4.3.3 Collisional dissociation of heavy mesons and quarkonia in the QGP
Heavy flavour dynamics in dense QCD matter critically depends on the time scales involved in the underlying reaction. Two of these time scales, the formation time of the QGP \(\tau _0\) and its transverse size \(L_{QGP}\), can be related to the nuclear geometry, the QGP expansion, and the bulk particle properties. The formation time \(\tau _\mathrm{form}\) of heavy mesons and quarkonia, on the other hand, can be evaluated from the virtuality of the heavy quark Q decay into D, B mesons [422, 460] or the time for the \({Q\overline{Q}}\) pair to expand to the size of the \(\mathrm {J}/\psi \) or \(\Upsilon \) wave function [83]. For a \(\pi ^0\) with an energy of 10\(~\text {GeV}\), \(\tau _\mathrm{form} \sim 25\) fm \(\gg L_{QGP}\) affords a relatively simple interpretation of light hadron quenching in terms of radiative and collisional partonlevel energy loss [552]. On the other hand, for D, B, \(\mathrm {J}/\psi \) and \(\Upsilon (1\mathrm{S})\), one obtains \( \tau _\mathrm{form} \sim \) 1.6, 0.4, 3.3 and 1.4 fm \(\ll L_{QGP}\). Such short formation times necessitate understanding of heavy meson and quarkonium propagation and dissociation in strongly interactingmatter.
4.3.4 Tmatrix approach to heavyquark interactions in the QGP
The thermodynamic Tmatrix approach is a firstprinciples framework to selfconsistently compute one and twobody correlations in hot and dense matter. It has been widely applied to, e.g., electromagnetic plasmas [553] and the nuclear manybody problem [554, 555]. Its main assumption is that the basic twobody interaction can be cast into the form of a potential, V(t), with the fourmomentum transfer approximated by \(t=q^2=q_0^2\vec q^{\,2}\simeq \vec q^{\,2}\). This relation is satisfied for charm and beauty quarks (\(Q = c,b\)) in a QGP up to temperatures of 2–3 \(T_\mathrm{c}\), since their large masses imply \(q_0^2\simeq (\vec q^{\,2}/2m_Q)^2\ll \vec q^{\,2}\) with typical momentum transfers of \(\vec q^{\, 2} \sim T^2\). Therefore, the Tmatrix formalism is a promising framework to treat the nonperturbative physics of heavyquark (HQ) interactions in the QGP [556, 557, 558]. It can be applied to both hidden and open heavyflavour states, and it provides a comprehensive treatment of bound and scattering states [558]. It can be systematically constrained by lattice data [559], and implemented to calculate heavyflavour observables in heavyion collisions [560, 561].
4.3.5 LatticeQCD
4.3.6 Heavyflavour interaction with medium in AdS/CFT
4.4 Theoretical overview: medium modelling and mediuminduced modification of heavyflavour production
Besides modelling the energy loss as described in Sect. 4.3, each model aiming at explaining open heavy flavour observables in AA collisions needs to include several key ingredients. These are: the “initial” production of heavy flavour (see Sect. 2.1.1) possibly affected by cold nuclear matter effects (see Sect. 3), a spacetime description of the QGP evolution up to the freezeout, mechanisms for hadronisation (including specific processes like the socalled “coalescence”) and, ultimately, Dmeson and Bmeson interactions in the ensuing hadronic matter. For a given energyloss model, it has been shown that various choices of these auxiliary ingredients could generate a factor of 2 in the observables [575]. In this section, the solutions adopted in the various models are described in order to better understand their predictions for the modification of heavyflavour production in AA.
4.4.1 pQCD energy loss in a static fireball (Djordjevic et al.)
The dynamical energyloss formalism discussed in Sect. 4.3.1 (Djordjevic et al.) was incorporated by the same authors into a numerical procedure in order to calculate mediummodified heavyflavour hadron momentum distributions. This procedure includes (i) production of light and heavyflavour partons based on the nonzeromass variable flavour number scheme VFNS [391], NLO [422] and FONLL calculations [44], respectively, (ii) multigluon [576] and pathlength [519, 577] fluctuations, (iii) light [578] and heavy [579, 580, 581] flavour fragmentation functions, and (iv) decay of heavyflavour mesons to single electrons/muons and \(\mathrm {J}/\psi \) [44]. This model does not include hadronisation via recombination. Inmedium path length is sampled from a distribution corresponding to a static fireball at fixed effective temperature. The \(R_{\mathrm {AA}}\) predictions are provided for both RHIC and LHC energies, various light and heavyflavour probes and different collision centralities. This model does not include any free parameter. All implementation details are provided in Ref. [582]. A representative set of these predictions will be presented in Sect. 4.5, while other predictions and detailed comparison with experimental data are provided in Ref. [515, 582, 583]. In summary, this formalism provides a robust agreement with experimental data, across diverse probes, experiments and experimental conditions.
4.4.2 pQCD embedded in viscous hydro (POWLANG and Duke)
The starting point of the POWLANG setup [507] is the generation of the \({Q\overline{Q}}\) pairs in elementary nucleon–nucleon collisions. For this purpose the POWHEGBOX package [48, 152] is employed: the latter deals with the initial production in the hard pQCD event (evaluated at NLO accuracy), interfacing it to PYTHIA 6.4 [151] to simulate the associated initial and finalstate radiation and other effects, like e.g. the intrinsic\(k_{\mathrm {T}}\) broadening. In the AA case, EPS09 nuclear corrections [364] are applied to the PDFs and the \({Q\overline{Q}}\) pairs are distributed in the transverse plane according to the local density of binary collisions given by the geometric Glauber model. Furthermore, a further \(k_{\mathrm {T}}\) broadening depending on the crossed nuclear matter thickness is introduced, as described in Ref. [508]. Both in the \(\mathrm pp\) benchmark and in the AA case (at the decoupling from the fireball) hadronisation is modelled through independent fragmentation of the heavy quarks, with invacuum fragmentation functions tuned by the FONLL authors [431]. Concerning the modelling of the fireball evolution, the latter is taken from the output of the \((2+1)\)d viscous fluid dynamics code of Ref. [584]. At each time step, the update of the heavyquark momentum according to the Langevin equation is performed in the local rest frame of the fluid, boosting then back the results into the laboratory frame. In setting the transport coefficients entering into the Langevin equation, the approach adopted in Ref. [507] was to derive the momentum broadening \(\kappa _\mathrm{T/L}(p)\) and to fix the friction coefficient \(\eta _D(p)\) so to satisfy the Einstein relation. For the former, two different sets of values were explored: the ones from a weakcoupling calculations described in Sect. 4.3.1 and the ones provided by the lattice QCD calculations described in Sect. 4.3.5. The local character of these energyloss models indeed allows for their implementation with fluid dynamics as a background.
In first phenomenological studies performed with the POWLANG setup [507, 508] hadronisation was modelled as occurring in the vacuum, neglecting the possibility of recombination of the heavy quarks with light thermal partons from the medium. Hence no modification of the heavy flavour spectra or hadrochemistry at hadronisation was considered, charm and beauty going into hadrons with the same fragmentation fractions as in the vacuum. A mediummodified hadronisation scheme has been recently developed in Refs. [585, 586]. Note that the recombination with light thermal quarks would make the final charmed hadrons inherit part of the flow of the medium, moving present POWLANG results closer to the experimental data. First numerical results [585, 586] show that this is actually the case, in particular for what concerns the elliptic flow of D mesons at LHC and their \(R_{\mathrm {AA}}\) at low \(p_{\mathrm {T}}\) at RHIC.
The spacetime evolution of the temperature and collective flow profiles of the thermalised medium are described with a \((2+1)\)d viscous fluid dynamics [588, 589, 590]. At the end of the QGP phase, the hadronisation of heavy quarks is modelled with a hybrid fragmentation plus recombination scenario. Fragmentation processes are simulated by PYTHIA 6.4 [151] while the heavy quark coalescence with light quarks is treated with the “sudden recombination” approach developed in [591].
4.4.3 pQCDinspired energy loss with running \(\alpha _s\) in a fluiddynamical medium and in Boltzmann transport
The implementation of the microscopic models based on a running coupling constant – described in Sect. 4.3.2 – in the MC@\(_s\)HQ and BAMPS frameworks is presented here. In its latest version, MC@\(_s\)HQ couples a Boltzmann transport of heavy quarks to the (\(3+1\))d idealfluiddynamical evolution from EPOS2 initial conditions. In its integral version, which includes the hadronic finalstate interactions, the EPOS2 model describes very well a large variety of observables measured in the lightflavour sector in nucleus–nucleus, proton–nucleus and proton–proton collisions at RHIC and LHC [264, 592, 593, 594]. Therefore, it provides as a reliable description of the medium from which the thermal scattering partners of the heavy quarks are sampled. Due to the fluctuating initial conditions of the fluid dynamics, the heavyquark evolution can be treated in an eventbyevent setup. Initially, the heavy quarks are produced at the spatial scattering points of the incoming nucleons with a momentum distribution from either FONLL [41, 44, 595] or MC@NLO [47, 153]. The latter combines nexttoleading order pQCD cross sections with a parton shower evolution, which provides more realistic distributions for the initial correlations of heavy quark–antiquark pairs than the backtoback initialisation applied to single inclusive spectra obtained with FONLL. In recent implementations [596], a convolution of the initial \(p_{\mathrm {T}}\) spectrum was applied in order to include (anti)shadowing at (high) low \(p_{\mathrm {T}}\) in central collisions at the LHC according to the EPS09 nuclear shadowing effect [364]. After propagation in the deconfined medium heavy quarks hadronise at a transition temperature of \(T=155\) \(~\text {MeV}\), which is well in the range of transition temperatures given by lattice QCD calculations [597]. As described in [529], hadronisation of heavy quarks into D and B mesons can proceed through coalescence (predominant at low \(p_{\mathrm {T}}\)) or fragmentation (predominant at large \(p_{\mathrm {T}}\)). Recently, MC@\(_s\)HQ\(+\)EPOS2 has also been used to study heavyflavour correlation observables [530] (see Sect. 4.6) and higherorder flow coefficients [598].
In the BAMPS model [541, 599], the initial heavy quark distribution is obtained from MC@NLO [47, 153] for \(\mathrm pp\) collisions through scaling with the number of binary collisions to heavyion collisions without taking cold nuclear matter effects into account. After the QGP evolution (that is, after the local energy density has fallen below \(\epsilon = 0.6\) \(~\text {GeV}\)/fm\(^3\)) heavy quarks are fragmented via Peterson fragmentation [600] to D and B mesons. Recombination processes are not considered for the hadronisation.
RHIC heavyflavour decayelectron data can be reproduced with only collisional interactions if their cross section is increased by a Kfactor of 3.5 [601]. With this parameter fixed, BAMPS predictions [601] for \(v_{2}\) at LHC for various heavyflavour particles can describe the data, but the \(R_{\mathrm {AA}}\) is slightly underestimated. However, the need of the phenomenological Kfactor is rather unsatisfying from the theory perspective, especially if K is found to deviate vastly from unity. Therefore, radiative processes were recently included in BAMPS [540] and the Kfactor mocking higherorder effects abandoned.^{21} The ensuing predictions are in satisfactory agreement with the data, which seems to favour this recent development of the BAMPS model.
4.4.4 Nonperturbative Tmatrix approach in a fluiddynamic model (TAMU) and in UrQMD transport
The Tmatrix approach for heavyflavour diffusion through QGP, hadronisation and hadronic matter [558, 565] described in Sect. 4.3.4 has been implemented into a fluiddynamic background medium [561]. The latter is based on the (\(2+1\))dimensional idealfluid dynamics code of Kolb and Heinz [602], but several amendments have been implemented to allow for an improved description of bulkhadron observables at RHIC and LHC [603]. First, the quasiparticle QGP equation of state (EoS) with firstorder transition has been replaced by a latticeQCD EoS which allows for a nearsmooth matching into the hadronresonance gas. Second, the initialisation at the thermalisation time has been augmented to account for a nontrivial flow field, in particular a significant radial flow [604]. Third, the initial energydensity profile has been chosen in more compact form, close to a collision profile that turns out to resemble initial states from saturation models. All three amendments generate a more violent transverse expansion of the medium, which, e.g., have been identified as important ingredients to solve the discrepancy between the fluid dynamics predictions and the measured HBT radii at RHIC (the socalled HBT puzzle [605]). These features furthermore lead to an “early” saturation of the bulkmedium \(v_{2}\) [603], close to the phasetransition region. Consequently, multistrange hadrons (\(\phi \), \(\Xi \) and \(\Omega ^{}\)) need to freeze out at this point to properly describe their \(p_{\mathrm {T}}\) spectra and \(v_{2}\). This provides a natural explanation of the phenomenologically wellestablished universal kineticenergy scaling of hadron \(v_{2}\) at RHIC. For the medium evolution at LHC, an initial radial flow is phenomenologically less compelling, and has not been included in the tune of fluid dynamics for Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} =2.76\) \(\text {TeV}\), while the thermalisation time (\(\tau _0=0.4\) fm/c) is assumed to be shorter than at RHIC (0.6 fm/c). Representative bulkhadron observables at LHC are reasonably well described as a function of both \(p_{\mathrm {T}}\) and centrality [606].
Heavyflavour diffusion is implemented into the fluiddynamical medium employing relativistic Langevin simulations of the Fokker–Planck equation. The pertinent nonperturbative transport coefficients from the heavylight Tmatrix in the QGP and effective hadronic theory for D mesons in the hadronic phase are utilised in the local rest frame of the expanding medium. The initial heavyquark momentum distributions are taken from FONLL pQCD calculations [595], which describe \(\mathrm pp\) spectra with suitable fragmentation functions. After diffusion through the QGP the HQ distributions are converted into D/D\(^*\) mesons using the resonance recombination model (RRM) [607] with \(p_{\mathrm {T}}\)dependent formation probabilities from the heavylight Tmatrices in the coloursinglet channels. The hadronisation is carried out on the hypersurface corresponding to \(T_\mathrm{pc}=170\) \(~\text {MeV}\). The heavy quarks that do not recombine are hadronised via the same fragmentation function as used in \(\mathrm pp\) collisions. The resulting Dmeson distributions are further evolved through the hadronic phase until kinetic freezeout of the fluiddynamical medium. However, the distributions of \(\mathrm {D}^{+}_{s} =(c{\overline{s}})\) mesons, which do not contain any light quarks, are frozen out right after hadronisation, in line with the early freezeout of multistrange mesons.
In recent years, another model [608, 609, 610] implementing the nonperturbative Tmatrix approach described in Sect. 4.3.4 has been put forward. It was motivated by the necessity of a realistic description for the bulk evolution of the fireball created in ultrarelativistic heavyion collisions. For this purpose, a transport fluiddynamics hybrid model of the bulk has been developed [611]. It combines the Ultrarelativistic Quantum Molecular Dynamics (UrQMD) to describe the initial and final stages and idealfluid dynamics for the intermediate stage of the evolution. In this model, the initial collision of the two nuclei is simulated with the UrQMD cascade model [612, 613]. After a time \(t_{\text {start}}=2 R/\sqrt{\gamma _{\text {cm}}^21}\), (R: radius of the colliding nuclei), when the Lorentzcontracted nuclei have passed through each other (\(\gamma _{\text {cm}}\): the Lorentzcontraction factor in the centreofmass frame) the evolution is switched to a relativistic idealfluid simulation using the full (\(3+1\))dimensional SHASTA algorithm [614, 615, 616] by mapping the energy, baryon number, and momenta of all particles within UrQMD onto a spatial grid. Thermal freezeout is assumed to occur approximately on equal propertime hypersurfaces and performed in terms of the usual Cooper–Frye prescription [617].
The diffusion of heavy quarks is described during the fluiddynamics stage of the simulation using a Fokker–Planck description [531, 560, 618, 619, 620, 621, 622, 623] (“Brownian motion”) employing a MonteCarlo implementation of the relativistic Langevin approach, with quarkQ (lightquark–heavyquark) drag and diffusion coefficients calculated as explained in Sect. 4.3.4.^{22} The elastic gluon–Q interaction is computed using a leadingorder pQCD cross section [369] with a Debye screening mass of \(m_{Dg}=g T\) in the gluon propagators, which regularises the tchannel singularities in the matrix elements. The strong coupling constant is set to the constant value \(\alpha _s=g^2/(4 \pi )=0.4\).
Heavyquark production is evaluated perturbatively on the timedependent background by UrQMD/hybrid. A first UrQMD run is used to determine the collision coordinates of the nucleons within the nuclei according to a Glauber initialstate geometry. The corresponding spacetime coordinates are saved and used in a second full UrQMD run as possible production coordinates for the heavy quarks. The initial \(p_{\mathrm {T}}\) distributions of heavy quarks at \(\sqrt{s} =200\) \(~\text {GeV}\) is an ad hoc parametrisation, such that the decayelectron \(p_{\mathrm {T}}\) distribution from the calculation describes the distribution measured in pp collisions at RHIC [560, 622]. For LHC energy, heavyquark \(p_{\mathrm {T}}\) distributions obtained from the PYTHIA event generator are used. Finally, at freezeout temperature the heavy quarks decouple and are hadronised either via Peterson [600] fragmentation or coalescence.
4.4.5 LatticeQCD embedded in viscous fluid dynamics (POWLANG)
In the POWLANG framework (see Sect. 4.4.3) a set of diffusion coefficients \(\kappa \) provided by the lattice QCD calculations and described in Sect. 4.3.5 was also implemented.
The main limitation of the lattice QCD approach, providing in principle a nonperturbative result, is the absence of any information on the momentum dependence of \(\kappa \). The authors of POWLANG make the choice of keeping \(\kappa \) constant. On the contrary, in the weakcoupling pQCD calculation the longitudinal momentum broadening coefficient \(\kappa _\mathrm{L}(p)\), although starting from a much lower value than the lQCD one, displays a steep rise with the heavyquark momentum, which for high enough energy makes it overshoot the latticeQCD result, taken as constant. Experimental data on the \(R_{\mathrm {AA}}\) of D mesons and heavyflavour decay electrons seem to favour an intermediate scenario.
4.4.6 AdS/CFT calculations in a static fireball
In the model of Refs. [624, 625], FONLL [44] provides both the heavyflavour production and the fragmentation to D and B mesons. The medium is described with a static fireball with a transverse profile \(T(\vec {x},\,\tau )\propto \rho _{part}(\vec {x})\tau ^{1/3}\) based on the Glauber model. The energy loss of a heavy quark propagating through the plasma is then given by the AdS/CFT drag derivation, Eq. (51), starting at an early thermalisation time \(\tau _0 = 0.6\) fm and continuing until \(T = T_{\mathrm{hadronisation}} = 160\) \(~\text {MeV}\). Path lengths are sampled through a participant transverse density distribution taking into account the nuclear diffuseness.
It is nontrivial to connect the parameters of QCD to those of the SYM theory in which the AdS/CFT derivations were performed. Two common prescriptions [573] for determining the parameters in the SYM theory are to take: (i) \(\alpha _{SYM} = \alpha _s\) and \(T_{SYM} = T_{QCD}\) or (ii) \(\lambda _{SYM} = 5.5\) and \(e_{SYM} = e_{QCD}\) (and hence \(T_{SYM} = T_{QCD} / 3^{1/4}\)). In the first prescription, the SYM coupling is taken equal to the QCD coupling and the temperatures are equated. In the second prescription, the energy densities of QCD and SYM are equated and the coupling is fitted by comparing the static quark–antiquark force from AdS/CFT to lattice results.
Comparative overview of the models for heavyquark energy loss or transport in the medium described in the previous sections
Model  Heavyquark production  Medium modelling  Quark–medium interactions  Heavyquark hadronisation  Tuning of medium coupling (or density) parameter(s) 

FONLL no PDF shadowing  Glauber model nuclear overlap no fl. dyn. evolution  Rad. + coll. energy loss finite magnetic mass  Fragmentation  Medium temperature fixed separately at RHIC and LHC  
FONLL no PDF shadowing  Glauber model nuclear overlap no fl. dyn. evolution  Rad. + coll. energy loss  Fragmentation  RHIC (then scaled with \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \))  
Nonzeromass VFNS no PDF shadowing  Glauber model nuclear overlap ideal fl. dyn. \(1+1\)d Bjorken expansion  Radiative energy loss inmedium meson dissociation  Fragmentation  RHIC (then scaled with \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \))  
FONLL no PDF shadowing  Glauber model nuclear overlap no fl. dyn. evolution  AdS/CFT drag  Fragmentation  RHIC (then scaled with \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \))  
POWHEG (NLO) EPS09 (NLO) PDF shadowing  \(2+1\)d expansion with viscous fl. dyn. evolution  Transport with Langevin eq. collisional energy loss  Fragmentation recombination  Assume pQCD (or lQCD U potential)  
FONLL EPS09 (LO) PDF shadowing  \(3+1\)d expansion (EPOS model)  Transport with Boltzmann eq. rad. \(+\) coll. energy loss  Fragmentation recombination  QGP transport coefficient fixed at LHC, slightly adapted for RHIC  
MC@NLO no PDF shadowing  \(3+1\)d expansion parton cascade  Transport with Boltzmann eq. rad. + coll. energy loss  Fragmentation  RHIC (then scaled with \(\mathrm{d}N_\mathrm{ch}/\mathrm{d}\eta \))  
FONLL EPS09 (NLO) PDF shadowing  2\(+\)1d expansion ideal fl. dyn.  Transport with Langevin eq. collisional energy loss diffusion in hadronic phase  Fragmentation recombination  Assume lQCD U potential  
PYTHIA no PDF shadowing  3\(+\)1d expansion ideal fl. dyn.  Transport with Langevin eq. collisional energy loss  fragmentation recombination  Assume lQCD U potential  
PYTHIA EPS09 (LO) PDF shadowing  2\(+\)1d expansion viscous fl. dyn.  Transport with Langevin eq. rad. + coll. energy loss  Fragmentation recombination  QGP transport coefficient fixed at RHIC and LHC (same value) 
The kinematic range of applicability of the model can be estimated through the \(p_{\mathrm {T}}\) scale at which including momentum fluctuations becomes important. By comparing the momentum lost to drag to the potential momentum gain of the fluctuations, one expects that momentum fluctuations become important at a scale \(\gamma _{\mathrm{crit}}^{\Delta p^2} = m_Q^2 / 4 \, T^2\). One can see from above that the speed limit at which the entire calculational framework breaks down, \(\gamma _{\mathrm{crit}}^{SL}\), is parametrically in \(\lambda \) smaller than \(\gamma _{\mathrm{crit}}^{\Delta p^2}\); however, numerically for the finite values of \(\lambda \) phenomenologically relevant at RHIC and LHC \(\gamma _{\mathrm{crit}}^{\Delta p^2} < \gamma _{\mathrm{crit}}^{SL}\). In particular, one expects nontrivial corrections to the drag results for \(e^\pm \) and D mesons from open heavy flavour for \(p_{\mathrm {T}} <4\)–5\(~\text {GeV}/c\). Other calculations [621, 627] have attempted to include the effect of fluctuations; however, their diffusion coefficients were set by the Einstein relations and not those derived from AdS/CFT (recall that the derived diffusion coefficients are qualitatively different from those based on the fluctuation–dissipation theorem except in the limit of \(v = 0\)).
4.5 Comparative overview of model features and comparison with data
The theoretical models described in the previous sections are compared in Table 11 in terms of their “ingredients” for heavyquark production, medium modelling, quark–medium interactions, and heavyquark hadronisation.
Figure 66 shows the comparison for D mesons in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}} = 2.76~\text {TeV} \), measured by the ALICE Collaboration [477, 479]. The left panels show \(R_{\mathrm {AA}}\) in the centrality class 0–20 %, the right panels show \(v_{2}\) in the centrality class 30–50 %. The models that include only collisional energy loss are shown in the upper panels. These models provide in general a good description of \(v_{2}\). The original version of the POWLANG model does not exhibit a clear maximum in \(v_{2}\) like the other models, which could be due to the fact that it does not include, in such a version, hadronisation via recombination. The latter has been recently introduced in the POWLANG model and the additional flow inherited by the D mesons from the light quarks moves the calculations to higher values of \(v_{2}\). In the TAMU model the decrease of \(v_{2}\) towards high \(p_{\mathrm {T}}\) is faster than in the other models, which reflects a moderate coupling with the medium, also seen in the rise of \(R_{\mathrm {AA}}\) of D mesons at large \(p_{\mathrm {T}}\). In this range, some of the other models oversuppress the \(R_{\mathrm {AA}}\) and one observes large discrepancies between them, which mostly originate from the medium description as well as from the transport coefficients adopted in each model. At low \(p_{\mathrm {T}}\) the models (UrQMD, BAMPS) that do not include PDF shadowing give a \(R_{\mathrm {AA}}\) value larger than observed in the data. The models that include both radiative and collisional energy loss are shown in the central panels. All these models provide a good description of \(R_{\mathrm {AA}}\), but most of them underestimate the maximum of \(v_{2}\) observed in data. This could be due to the fact that the inclusion of radiative process reduces the weight of collisional process, which are more effective in building up the azimuthal anisotropy. In addition, some of these models (Djordjevic et al., WHDG, Vitev et al.) do not include a fluiddynamical medium (for this reason, the Djordjevic et al. and Vitev et al. models do not provide a calculation for \(v_{2}\)), and none of them implements the detailed balance reaction which is mandatory to reach thermalisation and then undergo the full drift from the medium. The POWLANG model with lQCDbased transport coefficient and the AdS/CFT predictions are plotted in the lower panels. POWLANG provides a good description of \(R_{\mathrm {AA}}\), while, for what concerns \(v_{2}\), the results depend crucially on the way hadronisation is described, recombination scenarios leading to a larger elliptic flow (although still smaller than the experimental data in the accessible \(p_{\mathrm {T}}\) range). AdS/CFT, on the other hand, overpredicts the suppression in the full \(p_{\mathrm {T}}\) range explored.
Figure 67 shows the comparison for the \(\mathrm {D}^{0}\) meson \(R_{\mathrm {AA}}\) in Au–Au collisions at \(\sqrt{s_{\mathrm{NN}}} =200\) \(~\text {GeV}\), measured by the STAR Collaboration [475]. The models that include collisional interactions in an expanding fluiddynamical medium (TAMU, BAMPS, Duke, MC@\(_s\)HQ, POWLANG) describe qualitatively the shape of \(R_{\mathrm {AA}}\) in the interval 0–3\(~\text {GeV}/c\), with a rise, a maximum at 1.5\(~\text {GeV}/c\) with \(R_{\mathrm {AA}} >1\), and a decrease. In these models, this shape is the effect of radial flow on light and charm quarks. The TAMU model also includes flow in the hadronic phase. It can be noted that these models predict a similar bump also at LHC energy (left panels of Fig. 66): the bump reaches \(R_{\mathrm {AA}} >1\) for the models that do not include PDF shadowing, while it stays below \(R_{\mathrm {AA}} =0.8\) for the models that include it. The present ALICE data for \(p_{\mathrm {T}} >2\) \(~\text {GeV}/c\) do not allow one to draw a strong conclusion. However, the preliminary ALICE data reaching down to 1\(~\text {GeV}/c\) in the centrality class 0–7.5 % [488] do not favour models that predict a bump with \(R_{\mathrm {AA}} >1\).
The comparisons with measurements of heavyflavour decay leptons at RHIC and LHC are shown in Figs. 68 and 69, respectively. The \(R_{\mathrm {AA}}\) and \(v_{2}\) of heavyflavour decay electrons in Au–Au collisions at top RHIC energy, measured by PHENIX [469] and STAR [420], are well described by all model calculations. Note that in some of the models the quark–medium coupling (medium density or temperature or interaction cross section) is tuned to describe the \(R_{\mathrm {AA}}\) of pions (Djordjevic et al., WHDG, Vitev et al.) or electrons (BAMPS) at RHIC. The \(R_{\mathrm {AA}}\) of heavyflavour decay muons at forward rapidity (\(2.5<y<4\)), measured by ALICE in central Pb–Pb collisions at LHC [120], is well described by most of the models. The BAMPS model tends to oversuppress this \(R_{\mathrm {AA}}\), as observed also for the high\(p_{\mathrm {T}}\) \(R_{\mathrm {AA}}\) of D mesons at RHIC and LHC. The MC@\(_s\)HQ model describes the data better when radiative energy loss is not included. In general, it can be noted that the differences between the various model predictions are less pronounced in the case of heavyflavour decay lepton observables than in the case of D mesons. This is due to the fact that the former include a \(p_{\mathrm {T}}\)dependent contribution of charm and beauty decays. In addition, the decay kinematics shifts the lepton spectra towards low momentum, reducing the impact on \(R_{\mathrm {AA}}\) of effects like PDF shadowing, radial flow and recombination.

the Dmeson \(v_{2}\) measurements at LHC are best described by the models that include collisional interactions within a fluiddynamical expanding medium, as well as hadronisation via recombination;

however, theoretical predictions of the \(R_{\mathrm {AA}}\) of D mesons from these models are scattered, both at RHIC and LHC, which leaves room for theoretical improvement in the future before reliable conclusions can be drawn;

on the contrary, the models that include radiative and collisional energy loss provide a good description of the Dmeson \(R_{\mathrm {AA}}\), but they underestimate the value of \(v_{2}\) at LHC;

the models that include collisional energy loss in a fluiddynamical expanding medium, hence radial flow, exhibit a bump in the low\(p_{\mathrm {T}}\) Dmeson \(R_{\mathrm {AA}}\), which is qualitatively consistent with the RHIC data;

these models predict a bump also at LHC energy, the size of which depends strongly on the nuclear modification of the PDFs (shadowing); the current data at LHC are not precise enough to be conclusive in this respect;

most of the models can describe within uncertainties the measurements of \(R_{\mathrm {AA}}\) and \(v_{2}\) for heavyflavour decay electrons at RHIC (in some models, the quark–medium coupling is tuned to describe these data) and of \(R_{\mathrm {AA}}\) for heavyflavour decay muons at LHC;

all models predict that the \(R_{\mathrm {AA}}\) of nonprompt \(\mathrm {J}/\psi \) from B decays is larger than that of D mesons by about 0.2–0.3 units for the \(p_{\mathrm {T}}\) region (\(\sim 10\) \(~\text {GeV}/c\)) for which preliminary data from the LHC experiments exist.
4.6 Heavyflavour correlations in heavyion collisions: status and prospects
Recent works [530, 539, 586, 610, 636, 637] have shown that the azimuthal distributions of heavy quarkantiquark pairs are sensitive to the different interaction mechanisms, collisional and radiative. The relative angular broadening of the \({Q\overline{Q}}\) pair does not only depend on the drag coefficient discussed above (see Sect. 4.3) but also on the momentum broadening in the direction perpendicular to the initial quark momentum, \(\langle p_\perp ^2\rangle \), which is not probed directly in the traditional \(R_{\mathrm {AA}}\) and \(v_{2}\) observables. This is one of the motivation for measuring azimuthal correlations of heavyflavour.
The experimental challenges in measurements like D–\( \overline{\mathrm{D}}\) correlations in heavyion collisions come from the reconstruction of both the hadronic decays of the backtoback D mesons, which require large statistics to cope with low branching ratios and low signaltobackground in nucleus–nucleus collisions. As an alternative, correlations of D mesons with charged hadrons (D–h), correlations of electrons/muons from decays of heavyflavour particles with charged hadrons (\(e\)–h) and correlations of D–\(e\), \(e^{+}e^{}\), \(\mu ^{+}\mu ^{}\) and \(e\)–\(\mu \) pairs (where electrons and muons come from heavyflavour decays) can be studied. Such observables might, however, hide decorrelation effects intrinsic to the decay of heavy mesons. In addition, in the case of correlations triggered by electrons or muons from heavyflavour decays, the interpretation of the results is complicated by the fact that the lepton carries only a fraction of the momentum of the parent meson. This makes the understanding of \(\mathrm pp\) collisions as baseline a very crucial aspect of these analyses (see Sect. 2.4.2).
Measurements of heavyflavour correlations in nucleus–nucleus collisions were carried out at both RHIC and LHC with \(e\)–h correlations [113, 639, 640] (where electrons come from heavyflavour decays), but the current statistics prevents us from drawing quantitative conclusions. Such measurements are expected to provide more information as regards how the charm and beauty quarks propagate through the hot and dense medium and how this affects and modifies the correlation structures. In particular, PHENIX reported a decrease of the ratio of yields in the awayside region (\(2.51<\Delta \phi <\pi \)) to those in the shoulder region (\(1.25<\Delta \phi <2.51\)) from \(\mathrm pp\) to Au–Au collisions (left panel of Fig. 73).
Further measurements of heavyflavour triggered azimuthal correlations will be promising in future data takings at both RHIC (with the new silicon tracker detectors) and LHC (with the machine and detector upgrades). As reported in Fig. 73, right, the relative uncertainty on the awayside yield in D–h correlations in central Pb–Pb collisions with the ALICE and LHC upgrades will be \(\approx \) 15 % for low\(p_{\mathrm {T}}\) D mesons and only a few percent for intermediate/high \(p_{\mathrm {T}}\).
Several theoretical works have recently addressed angular correlations of heavyflavour particles in nucleus–nucleus collisions [530, 539, 586, 610, 636, 637]. However, none of these approaches presently includes the interactions of D and B mesons in the hadronic phase present in the late stages of the system evolution. These interactions could add a further smearing on top of QGPinduced modification of the heavyquark angular correlations. For the traditional \(R_{\mathrm {AA}}\) and \(v_{2}\) observables, a first step in this direction was made in Refs. [491, 641, 642], with effects found of the order of 20 % at most. We now focus on a particular model in order to illustrate the sensitivity of heavyflavour angular correlations to the type of interaction mechanism [530]. In Fig. 74 (left), the transverse momentum broadening per unit of time is shown as a function of the initial momentum \(p_{}^\mathrm{ini}\) of charm quarks for the purely collisional and collisional plus radiative (LPM) interactions as applied within the MC@\(_s\)HQ model (see Sect. 4.3.2). For all initial momenta, \(\langle p_\perp ^2\rangle \) is larger in a purely collisional interaction mechanism. \(\langle p_\perp ^2\rangle \) has similar numerical values for charm and for beauty quarks. A larger \(\langle p_\perp ^2\rangle \) leads to a more significant change of the initial relative pair azimuthal angle \(\Delta \phi \) during the evolution in the medium. This means that for a purely collisional interaction mechanism one expects a stronger broadening of the initial correlation at \(\Delta \phi =\pi \), as seen in the central and right panels of Fig. 74. In the central panel, the \(\Delta \phi \) distribution of all initially correlated pairs is shown after hadronisation into \(\mathrm {D\overline{D}}\) pairs. Since no cut in \(p_{\mathrm {T}}\) is applied, these distributions are dominated by lowmomentum pairs, while in the right panel a cut of \(p_{\mathrm {T}} >3\) \(~\text {GeV}/c\) is applied. The lowmomentum pairs show the influence of the radial flow of the underlying QGP medium, which tends to align the directions of the quark and the antiquarks toward smaller opening angles. It again happens more efficiently for larger \(\langle p_\perp ^2\rangle \) of the underlying interaction mechanism. This effect, which was called “partonic wind” [643], is thus only seen for the purely collisional interaction mechanism. A \(p_{\mathrm {T}}\) threshold reveals clearly the residual correlation around \(\Delta \phi \sim \pi \). Here in the purely collisional scenario one sees a larger background of pairs that decorrelated during the evolution in the QGP than for the collisional plus radiative (LPM) scenario.
For these calculations an initial backtoback correlation has been assumed. Nexttoleading order processes, however, destroy this strict initial correlation already in proton–proton collisions. Unfortunately the theoretical uncertainties on these initial distributions are very large, especially for charm quarks. Here, a thorough experimental study of heavyflavour correlations in proton–proton and proton–nucleus collisions is very important for validating different initial models. Also enhanced theoretical effort in these reference systems is necessary.
4.7 Summary and outlook
The LHC Run 1 has provided a wealth of measurements of heavyflavour production in heavyion collisions, which have extended and complemented the results from the RHIC programme. The main observations and their present interpretation are summarised in the following.

The \(R_{\mathrm {AA}} \) measurements show a strong suppression with respect to binary scaling in central nucleus–nucleus collisions for D mesons, heavyflavour decay leptons and J\(/\psi \) from B decays. The suppression of D mesons and heavyflavour decay leptons is similar, within uncertainties, at RHIC and LHC energies. Given that a suppression is not observed in proton(deuteron)–nucleus collisions, the effect in nucleus–nucleus collisions can be attributed to inmedium energy loss.

The suppression of D mesons with average \(p_{\mathrm {T}} \) of about 10 GeV/c is stronger than that of J/\(\psi \) decaying from B mesons with similar average \(p_{\mathrm {T}} \). This observation, still based on preliminary results, is consistent with the expectation of lower energy loss for heavier quarks and it is described by model calculations that implement radiative and collisional energy loss with this feature.

The suppression of D mesons and pions is consistent within uncertainties at both RHIC and LHC. While there is no experimental evidence of the colourcharge dependence of the energy loss, model calculations indicate that similar \(R_{\mathrm {AA}} \) values can result from the combined effect of colourcharge dependent energy loss and the softer \(p_{\mathrm {T}} \) distribution and fragmentation function of gluons with respect to c quarks.

At very high\(p_{\mathrm {T}} \) (above 100 GeV/c), a similar \(R_{\mathrm {AA}} \) is observed for btagged jets and inclusive jets. This observation is consistent with a negligible effect of the heavy quark mass at these scales.

The measurements of electrons (in particular) and D mesons at RHIC show that the total production of charm quarks is consistent with binary scaling within uncertainties of about 30–40 %. The available measurements at LHC do not extend to sufficiently small \(p_{\mathrm {T}} \) to provide an estimate of the total yields.

The Dmeson \(R_{\mathrm {AA}} \) at RHIC energy shows a pronounced maximum at \(p_{\mathrm {T}} \) of about 1–2 GeV/c (where \(R_{\mathrm {AA}} \) becomes larger than unity). This feature is not observed in the measurements at LHC energy. Model calculations including collisional (elastic) interaction processes in an expanding medium and a contribution of hadronisation via inmedium quark recombination, as well as initialstate gluon shadowing, describe qualitatively the behaviour observed at both energies. In these models the bump at RHIC is due to radial flow and the effect on \(R_{\mathrm {AA}} \) at LHC is strongly reduced because of the harder \(p_{\mathrm {T}} \) distributions and of the effect of gluon shadowing.

A positive elliptic flow \(v_2\) is measured in noncentral collisions for D mesons at LHC and heavyflavour decay leptons at RHIC and LHC. The Dmeson \(v_2\) at LHC is comparable to that of lightflavour hadrons (within uncertainties of about 30 %). These measurements indicate that the interaction with the medium constituents transfers information as regards the azimuthal anisotropy of the system to charm quarks. The \(v_2\) measurements are best described by the models that include collisional interactions within a fluiddynamical expanding medium, as well as hadronisation via recombination.

Does the total charm and beauty production follow binary scaling or is there a significant gluon shadowing effect? This requires a precise measurement of charm and beauty production down to zero \(p_{\mathrm {T}} \), in proton–proton, proton–nucleus and nucleus–nucleus collisions.

Can there be an experimental evidence of the colourcharge dependence of the energy loss? This requires a precise comparison of D mesons and pions in the intermediate \(p_{\mathrm {T}} \) region, at both RHIC and LHC.

Is the difference in the nuclear modification factor of charm and beauty hadrons consistent with the quark mass dependent mechanisms of the energy loss? Can it provide further insight on these mechanisms (for example, the gluon radiation angular distribution)? This requires a precise measurement of D and B meson (or J/\(\psi \) from B) \(R_{\mathrm {AA}} \) over a wide \(p_{\mathrm {T}} \) range and as a function of collision centrality. This will also be mandatory in order to extract the precise pathlength dependence of the energy loss, which cannot be extracted from the actual data.

Does the positive elliptic flow observed for D mesons and heavyflavour decay leptons result from the charm quark interactions in the expanding medium? Are charm quarks thermalised in the medium? Is there a contribution (dominant?) inherited from light quarks via the recombination process? What is the contribution from the path length dependence of the energy loss? This requires precise measurements of the elliptic flow and of the higherorder flow coefficients of charm and beauty hadrons over a wide \(p_{\mathrm {T}} \) interval, and their comparison with lightflavour hadrons.

What is the role of inmedium hadronisation and of radial flow for heavy quarks? This requires measurements of \(R_{\mathrm {AA}} \) and \(v_2\) of heavy flavour hadrons with different quark composition and different masses, namely D, \(\mathrm{D}_s\), B, \(\mathrm{B}_s\), \(\Lambda _c\), \(\Xi _c\), \(\Lambda _b\).

What is the relevance of radiative and collisional processes in heavy quark energy loss? What is the path length dependence of the two types of processes? This requires precise simultaneous measurements of the \(R_{\mathrm {AA}} \) and \(v_2\) and their comparison with model calculations. Heavyquark correlations are also regarded as a promising tool in this context.
From the theoretical point of view, a wide range of models, also with somewhat different “ingredients”, can describe most of the available data, at least qualitatively. The main challenges in the theory sector is thus to connect the data with the fundamental properties of the QGP and of the theory of the strong interaction. For this purpose, it is important to identify the features of the quark–medium interaction that are needed for an optimal description of all aspects of the data and to reach a uniform treatment of the “external inputs” in the models (e.g. using stateoftheart pQCD baseline, fragmentation functions and fluiddynamical medium description, and fixing transport coefficients on those that will be ultimately obtained from lattice calculations for finite \(p_{\mathrm {T}}\)).
5 Quarkonia in nucleus–nucleus collisions
Quarkonia are considered important probes of the QGP formed in heavyion collisions. In a hot and deconfined medium quarkonium production is expected to be significantly suppressed with respect to the proton–proton yield, scaled by the number of binary nucleon–nucleon collisions, as long as the total charm cross section remains unmodified.^{23} The origin of such a suppression, taking place in the QGP, is thought to be the colour screening of the force that binds the \({c\overline{c}}\) (\({b\overline{b}}\)) state [644]. In this scenario, quarkonium suppression should occur sequentially, according to the binding energy of each meson: strongly bound states, such as the \(\Upsilon \text {(1S)}\) or the \(\mathrm {J}/\psi \), should melt at higher temperatures with respect to the more loosely bound ones, such as the \(\chi _b\), \(\Upsilon \text {(2S)}\), or \(\Upsilon \text {(3S)}\) for the bottomonium family or the \(\psi \text {(2S)}\) and the \(\chi _c\) for the charmonium one. As a consequence, the inmedium dissociation probability of these states should provide an estimate of the initial temperature reached in the collisions [645]. However, the prediction of a sequential suppression pattern is complicated by several factors. Feeddown decays of highermass resonances, and of bhadrons in the case of charmonium, contribute to the observed yield of quarkonium states. Furthermore, other hot and cold matter effects can play a role, competing with the suppression mechanism.
On the one hand, the production of c and \(\overline{c}\) quarks increases with increasing centreofmass energy. Therefore, at high energies, as at the LHC, the abundance of c and \(\overline{c}\) quarks might lead to a new charmonium production source: the (re)combination of these quarks throughout the collision evolution [646] or at the hadronisation stage [647, 648]. This additional charmonium production mechanism, taking place in a deconfined medium, enhances the \(\mathrm {J}/\psi \) yield and might counterbalance the expected \(\mathrm {J}/\psi \) suppression. Also the \({b\overline{b}}\) cross section increases with energy, but, given the smaller number of \({b\overline{b}}\) pairs, with respect to \({c\overline{c}}\), this contribution is less important for bottomonia even in high\(\sqrt{s_{\mathrm{NN}}}\) collisions.
On the other hand, quarkonium production is also affected by several effects related to cold matter (the socalled cold nuclear matter effects, CNM) discussed in Sect. 3. For example, the production cross section of the \({Q\overline{Q}}\) pair is influenced by the kinematic parton distributions in nuclei, which are different from those in free protons and neutrons (the socalled nuclear PDF effects). In a similar way, approaches based on the ColourGlass Condensate (CGC) effective theory assume that a gluon saturation effect sets in at high energies. This effect influences the quarkonium production occurring through fusion of gluons carrying small values of the Bjorkenx in nuclei. Furthermore, parton energy loss in the nucleus may decrease the pair momentum, causing a reduction of the quarkonium production at large longitudinal momenta. Finally, while the \({Q\overline{Q}}\) pair evolves towards the fully formed quarkonium state, it may also interact with partons of the crossing nuclei and eventually breakup. This effect is expected to play a dominant role only for low\(\sqrt{s_{\mathrm{NN}}}\) collisions, where the crossing time of the (pre)resonant state in the nuclear environment is rather large. On the contrary, this contribution should be negligible at high\(\sqrt{s_{\mathrm{NN}}}\), where, due to the decreased crossing time, resonances are expected to form outside the nuclei. Cold nuclear matter effects are investigated in proton–nucleus collisions. Since these effects are present also in nucleus–nucleus interactions, a precise knowledge of their role is crucial in order to correctly quantify the effects related to the formation of hot QCD matter.
The information from \(R_{\mathrm {AA}}\) can be complemented by the study of the quarkonium azimuthal distribution with respect to the reaction plane, defined by the beam axis and the impactparameter vector of the colliding nuclei. The second coefficient of the Fourier expansion of the azimuthal distribution, \(v_{2}\), is called elliptic flow, as explained in Sect. 4. Being sensitive to the dynamics of the partonic stages of heavyion collisions, \(v_{2}\) can provide details on the quarkonium production mechanisms: in particular, \(\mathrm {J}/\psi \) produced through a recombination mechanism, should inherit the elliptic flow of the charm quarks in the QGP, acquiring a positive \(v_{2}\).
Studies performed for thirty years, first at the SPS (\(\sqrt{s_{\mathrm{NN}}}\) \(=\) 17\(~\text {GeV}\)) and then at RHIC (\(\sqrt{s_{\mathrm{NN}}}\) \(=\) 39–200\(~\text {GeV}\)),^{24} have indeed shown a reduction of the \(\mathrm {J}/\psi \) yield beyond the expectations from cold nuclear matter effects (such as nuclear shadowing and \({c\overline{c}}\) breakup). Even if the centreofmass energies differ by a factor of ten, the amount of suppression, with respect to \(\mathrm pp\) collisions, observed by SPS and RHIC experiments at midrapidity is rather similar. This observation suggests the existence of an additional contribution to \(\mathrm {J}/\psi \) production, the previously mentioned (re)combination process, which sets in already at RHIC energies and can compensate for some of the quarkonium suppression due to screening in the QGP. Furthermore, \(\mathrm {J}/\psi \) suppression at RHIC is, unexpectedly, smaller at midrapidity than at forward rapidity (y), in spite of the higher energy density which is reached close to \(y \sim 0\). The stronger \(\mathrm {J}/\psi \) suppression at forwardy might be considered a further indication of the role played by (re)combination processes. Note, however, that the rapidity dependence of the (re)combination contribution is expected to be rather small [654, 655]. On the other hand, at RHIC energies, cold nuclear matter effects can also explain the observed difference [64], at least partially.
Quarkonium results obtained in AA at SPS. The nucleon–nucleon energy in the centreofmass frame (\(\sqrt{s_{\mathrm{NN}}}\)), the covered kinematic range, the probes and observables are reported
Probe  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (\(~\text {GeV}\))  y  \(p_{\mathrm {T}}\) (\(\text {GeV}/c\))  Observables  References 

NA38  
\(\mathrm {J}/\psi \)  S–U  17.2  \(0<y<1\)  \(p_{\mathrm {T}} >0\)  \(\sigma _{\mathrm {J}/\psi }\), \(\sigma _{\mathrm {J}/\psi }/\sigma _{\text {Drell}{}\text {Yan}}(\text {cent.})\)  [656] 
\(\psi \text {(2S)}\)  \(\sigma _{\psi \text {(2S)}}\), \(\sigma _{\psi \text {(2S)}}/\sigma _{\text {Drell}{}\text {Yan}}(\text {cent.})\)  
NA50  
\(\mathrm {J}/\psi \)  Pb–Pb  17.2  \(0<y<1\)  \(p_{\mathrm {T}} >0\)  Yield(\(p_{\mathrm {T}}\)), \(\sigma _{\mathrm {J}/\psi }\) and \(\sigma _{\mathrm {J}/\psi }/\sigma _{\text {Drell}{}\text {Yan}}(\text {cent.})\)  
\(\psi \text {(2S)}\)  Yield(\(p_{\mathrm {T}}\)), \(\sigma _{\psi \text {(2S)}}/\sigma _{\text {Drell}{}\text {Yan}}\) and \(\sigma _{\psi \text {(2S)}}/\sigma _{\mathrm {J}/\psi }(\text {cent.})\)  
NA60  
\(\mathrm {J}/\psi \)  In–In  17.2  \(0<y<1\)  \(p_{\mathrm {T}} >0\)  \(\sigma _{\mathrm {J}/\psi }/\sigma _{\text {Drell}{}\text {Yan}}(\text {cent.})\)  
Polarisation  [236] 
Quarkonium results obtained in AA from RHIC experiments. The experiment, the probes, the collision energy (\(\sqrt{s_{\mathrm{NN}}}\)), the covered kinematic range and the observables are indicated
Probe  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (\(~\text {GeV}\))  y  \(p_{\mathrm {T}}\) (\(\text {GeV}/c\))  Observables  References 

PHENIX  
\(\mathrm {J}/\psi \)  Au–Au  200  \(1.2<y<2.2\)  \(p_{\mathrm {T}} >0\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  
\(y<0.35\)  
\(0<p_{\mathrm {T}} <5\)  \(v_{2} (p_{\mathrm {T}},\,y)\)  [669]  
\(1.2<y<2.2\)  [670]  
Cu–Cu  \(1.2<y<2.2\)  \(p_{\mathrm {T}} >0\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  [671]  
\(y<0.35\)  
Cu–Au  \(1.2<y<2.2\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,y)\)  [672]  
U–U  193  \(1.2<y<2.2\)  \(p_{\mathrm {T}} >0\)  \(R_{\mathrm {AA}} (\text {cent.})\)  [673]  
Au–Au  62.4  Yield\((\text {cent.},\,p_{\mathrm {T}})\), \(R_{\mathrm {AA}} (\text {cent.})\)  [674]  
39  
\(\Upsilon \text {(1S+2S+3S)}\)  200  \(y<0.35\)  Yield, \(R_{\mathrm {AA}} \text {(cent.)}\)  [675]  
STAR  
\(\mathrm {J}/\psi \)  Au–Au  200  \(y<1\)  \(p_{\mathrm {T}} >0\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}})\)  
\(v_{2} (\text {cent.},\,p_{\mathrm {T}})\)  [677]  
Cu–Cu  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}})\)  
U–U  193  \(R_{\mathrm {AA}} (p_{\mathrm {T}})\)  [678]  
Au–Au  62.4  Yield, \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}})\)  
39  
\(\Upsilon \text {(1S)}\)  200  \(\sigma \) and \(R_{\mathrm {AA}} (\text {cent.})\)  [323]  
\(\Upsilon \text {(1S+2S+3S)}\)  \(R_{\mathrm {AA}} (\text {cent.})\)  
U–U  193  [678] 
The four large LHC experiments (ALICE, ATLAS, CMS, and LHCb) have carried out studies on quarkonium production either in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 2.76 \(\text {TeV}\) ^{25} or in p–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 5.02 \(\text {TeV}\). Quarkonium production has also been investigated in \(\mathrm pp\) interactions at \(\sqrt{s}\) \(=\) 2.76, 7 and 8 \(\text {TeV}\). The four experiments are characterised by different kinematic coverages, allowing one to investigate quarkonium production in \(y<4\), down to zero transverse momentum.
ATLAS and CMS are designed to measure quarkonium production by reconstructing the various states in their dimuon decay channel. They both cover the midrapidity region: depending on the quarkonium state under study and on the \(p_{\mathrm {T}}\) range investigated, the CMS rapidity coverage can reach up to \(y<2.4\), and a similar y range is also covered by ATLAS. ALICE measures quarkonium in two rapidity regions: at midrapidity (\(y<0.9\)) in the dielectron decay channel and at forward rapidity (\(2.5<y<4\)) in the dimuon decay channel, in both cases down to zero transverse momentum. LHCb has taken part only in the \(\mathrm pp\) and p–A LHC programmes during Run 1 and their results on quarkonium production, reconstructed through the dimuon decay channel, are provided at forward rapidity (\(2<y<4.5\)), down to zero \(p_{\mathrm {T}}\). As an example, the \(p_{\mathrm {T}}\)y acceptance coverages of the ALICE and CMS experiments are sketched in Fig. 75 for \(\mathrm {J}/\psi \) (left) and \(\Upsilon \) (right).
Quarkonium results obtained in AA from LHC experiments. The experiment, the probes, the collision energy (\(\sqrt{s_{\mathrm{NN}}}\)), the covered kinematic range and the observables are indicated
Probe  Colliding system  \(\sqrt{s_{\mathrm{NN}}}\) (\(\text {TeV}\))  y  \(p_{\mathrm {T}}\) (\(\text {GeV}/c\))  Observables  References 

ALICE  
\(\mathrm {J}/\psi \)  Pb–Pb  2.76  \(y<0.9\)  \(p_{\mathrm {T}} >0\)  \(R_{\mathrm {AA}} (\text {cent.,}\,p_{\mathrm {T}})\)  
\(2.5<y<4\)  \(p_{\mathrm {T}} >0\)  \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  
\(0<p_{\mathrm {T}} <10\)  \(v_{2} (\text {cent.},\,p_{\mathrm {T}})\)  [681]  
\(\psi \text {(2S)}\)  \(p_{\mathrm {T}} <3\)  \(\frac{(N_{\psi \text {(2S)}}/N_{\mathrm {J}/\psi })_{\mathrm {Pb}{}\mathrm {Pb}}}{(N_{\psi \text {(2S)}}/N_{\mathrm {J}/\psi })_{\mathrm {pp}}}(\text {cent.})\)  [682]  
\(3<p_{\mathrm {T}} <8\)  
\(\Upsilon \text {(1S)}\)  \(p_{\mathrm {T}} >0\)  \(R_{\mathrm {AA}} (\text {cent.},\,y)\)  [683]  
ATLAS  
\(\mathrm {J}/\psi \)  Pb–Pb  2.76  \(\eta <2.5\)  \(p_{\mathrm {T}} \gtrsim 6.5\)  \(R_{\mathrm {CP}} (\text {cent.})\)  [684] 
CMS  
\(\mathrm {J}/\psi \) (prompt)  Pb–Pb  2.76  \(y<2.4\)  \(6.5<p_{\mathrm {T}} <30\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  [482] 
\(v_{2} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  [685]  
\(1.6<y<2.4\)  \(3<p_{\mathrm {T}} <30\)  
\(y<1.2\)  \(6.5<p_{\mathrm {T}} <30\)  Yield and \(R_{\mathrm {AA}}\)  [482]  
\(1.2<y<1.6\)  \(5.5<p_{\mathrm {T}} <30\)  
\(1.6<y<2.4\)  \(3<p_{\mathrm {T}} <30\)  
\(\psi \text {(2S)}\) (prompt)  \(1.6<y<2.4\)  \(3<p_{\mathrm {T}} <30\)  \(R_{\mathrm {AA}}\), \(\frac{(N_{\psi \text {(2S)}}/N_{\mathrm {J}/\psi })_{\mathrm {Pb}{}\mathrm {Pb}}}{(N_{\psi \text {(2S)}}/N_{\mathrm {J}/\psi })_{\mathrm {pp}}}(\text {cent.})\)  [686]  
\(y<1.6\)  \(6.5<p_{\mathrm {T}} <30\)  
\(\Upsilon \text {(1S)}\)  \(y<2.4\)  \(p_{\mathrm {T}} >0\)  Yield and \(R_{\mathrm {AA}} (\text {cent.},\,p_{\mathrm {T}},\,y)\)  [482]  
\(\Upsilon \text {(nS)}\)  \(y<2.4\)  \(p_{\mathrm {T}} >0\)  \(R_{\mathrm {AA}} (\text {cent.})\)  
\(\frac{(N_{\Upsilon \text {(2S)}}/N_{\Upsilon \text {(1S)}})_{\mathrm {Pb}{}\mathrm {Pb}}}{(N_{\Upsilon \text {(2S)}}/N_{\Upsilon \text {(1S)}})_{\mathrm {pp}}}(\text {cent.})\)  [268] 
This section is organised as follows. In the first part, a theoretical overview is presented, in which the sequential suppression pattern of quarkonia and the lattice calculations are introduced. Other effects, such as modifications of the parton distribution functions inside nuclei and their influence on nucleus–nucleus collisions are discussed. Along with the suppression, the enhancement of quarkonia is also considered through two different approaches to (re)generation: the statistical hadronisation model and transport models. In the context of bottomonium studies, nonequilibrium effects on quarkonium suppression in the anisotropic hydrodynamic framework are also discussed. Finally, the collisional dissociation model and the comover interaction model are briefly introduced.
In the second part, experimental quarkonium results are reviewed. The recent LHC results, starting with a brief discussion on the quarkonium production cross sections in \(\mathrm pp\) collisions as necessary references to build the nuclear modification factors, are presented. The description of the experimental \(R_{\mathrm {AA}}\) results for \(\mathrm {J}/\psi \) production, both at low and high \(p_{\mathrm {T}}\) is then addressed. The LHC results are compared to those at RHIC energies and to theoretical models. A similar discussion is also introduced for the \(\mathrm {J}/\psi \) azimuthal anisotropy. Results obtained at RHIC from variations of the beam energy and collision system are also addressed. The charmonium section is concluded with a discussion of \(\psi \text {(2S)}\) production. Next, the bottomonium results on ground and excited states at RHIC and LHC energies are discussed.
Finally, other possible references for the quarkonium behaviour in nucleus–nucleus collisions, namely proton–nucleus collisions and open heavy flavour, production are discussed.
5.1 Theory overview
5.1.1 Sequential suppression and lattice QCD
Mass, binding energy and radius for charmonia and bottomonia [437]
State  \(J/\psi \)  \(\chi _c \text {(1P)}\)  \(\psi \text {(2S)} \)  \(\Upsilon \text {(1S)} \)  \(\chi _b \text {(1P)}\)  \(\Upsilon \text {(2S)}\)  \(\chi _b \text {(2P)}\)  \(\Upsilon \text {(3S)} \) 

Mass (GeV\(/c^2\))  3.07  3.53  3.68  9.46  9.99  10.02  10.26  10.36 
\(\mathrm{Binding}\) (GeV)  0.64  0.20  0.05  1.10  0.67  0.54  0.31  0.20 
Radius (fm)  0.25  0.36  0.45  0.14  0.22  0.28  0.34  0.39 
The QGP consists of deconfined colour charges, so that the binding of a \({Q\overline{Q}}\) pair is subject to the effect of colour screening which limits the range of strong interactions. Intuitively, the fate of heavy quark bound states in a QGP depends on the size of the colour screening radius \(r_D\) (which is inversely proportional to the temperature, so that it decreases with increasing temperature) in comparison to the quarkonium binding radius \(r_Q\): if \(r_D \gg r_Q\), the medium does not really affect the heavy quark binding. Once \(r_D \ll r_Q\), however, the two heavy quarks cannot “see” each other any more and hence the bound state will melt. It is therefore expected that quarkonia will survive in a QGP through some range of temperatures above \(T_c\), and then dissociate once T becomes large enough. Recent studies have shown that the Debyescreened potential develops an imaginary part, implying a class of thermal effects that generate a finite width for the quarkonium peak in the spectral function. These results can be used to study quarkonium in a weakly coupled Quark Gluon Plasma within an Effective Field Theories (EFT) framework [689]. On the other hand latticeQCD enables ab initio study of quarkonium correlation functions in the strongly coupled regime. The sequential dissociation scenario is confirmed by all these approaches [64].
In vacuum, progress in lattice calculations and effective field theories have turned quarkonium physics into a powerful tool to determine the heavyquark masses and the strength of the QCD coupling, with an accuracy comparable to other techniques. The measurements of quarkonia in heavyion collisions provide quantitative inputs for the study of QCD at high density and temperature, providing an experimental basis for analytical and lattice studies to extract the inmedium properties of heavyflavor particles and the implications for the QCD medium [64, 690, 691, 692].
Finitetemperature lattice studies on quarkonium mostly consist of calculations of spectral functions for temperatures in the range explored by the experiments. The spectral function \(\rho (\omega )\) is the basic quantity encoding the equilibrium properties of a quarkonium state. It characterises the spectral distribution of binding strength as a function of energy \(\omega \). Bound or resonance states manifest themselves as peaks with welldefined mass and spectral width. The inmedium spectral properties of quarkonia are related to phenomenology, since the masses determine the equilibrium abundances, their inelastic widths determine formation and destruction rates (or chemical equilibration times) and their elastic widths affect momentum spectra (and determine the kinetic equilibration times).
Spectral functions play an important role in understanding how elementary excitations are modified in a thermal medium. They are the power spectrum of autocorrelation functions in real time, hence provide a direct information on large time propagation. In the lattice approach such realtime evolution is not directly accessible: the theory is formulated in a fourdimensional box – three dimensions are spatial dimensions, the fourth is the imaginary (Euclidean) time \(\tau \). The lattice temperature \(T_L\) is realised through (anti)periodic boundary conditions in the Euclidean time direction – \(T_L = 1/N_\tau \), where \(N_\tau \) is the extent of the time direction, and can be converted to physical units once the lattice spacing is known. The spectral functions appear now in the decomposition of a (zeromomentum) Euclidean propagator \(G(\tau )\): \( G(\tau ) = \int _{0}^\infty \rho (\omega ) \frac{\mathrm {d}\omega }{2\pi }\, K(\tau ,\omega )\), with \(K(\tau ,\omega ) = \frac{(e^{\omega \tau } + e^{\omega (1/T  \tau )})}{1  e^{\omega /T}}\). The \(\tau \) dependence of the kernel K reflects the periodicity of the relativistic propagator in imaginary time, as well as its T symmetry. The Bose–Einstein distribution, intuitively, describes the wrapping around the periodic box, which becomes increasingly important at higher temperatures.
The procedure is, then, based on the generation of an appropriate ensemble of lattice gauge fields at a temperature of choice, on the computation on such an ensemble of the Euclidean propagators \(G(\tau )\), and on the extraction of the spectral functions. All such quarkonium studies yield qualitatively the same result: a given quarkonium state melts at a temperature above, or possibly at, the phasetransition temperature. There is, however, disagreement between different calculations in the precise temperatures for the following reasons. First, experience with lattice calculations has demonstrated that it is extremely important to have results in the continuum limit, and with the proper matter content. This means that the masses of the dynamical quark fields which are used in the generation of the gauge ensembles must be as close as possible to the physical ones, and the lattice spacing should be fine enough to allow for making contact with continuum physics. These systematic effects, which have been studied in detail for bulk thermodynamics, are still under scrutiny for the spectral functions. Second, the calculation of spectral functions using Euclidean propagators as an input is a difficult, possibly illdefined, problem. It has been mostly tackled by using the Maximum Entropy Method (MEM) [693], which has proven successful in a variety of applications. Recently, an alternative Bayesian reconstruction of the spectral functions has been proposed in Refs. [694, 695] and applied to the analysis of configurations from the HotQCD Collaboration [696].
Most calculations of charmonium spectral functions have been performed in the quenched approximation – neglecting quark loops –, although recently the spectral functions of the charmonium states have been studied as a function of both temperature and momentum, using as input relativistic propagators with two light quarks [697, 698] and, more recently, including the strange quark, for temperatures ranging between 0.76 and \(1.9\,T_c\). The sequential dissolution of the peaks corresponding to the S and Pwave states is clearly seen. The results are consistent with the expectation that charmonium melts at high temperature, however, as of today they lack quantitative precision and control over systematic errors.
5.1.2 Effect of nuclear PDFs on quarkonium production in nucleus–nucleus collisions
Figure 77 shows the results for the dependence of shadowing on rapidity, transverse momentum, and centrality are shown for \(\mathrm {J}/\psi \) and \(\Upsilon \) production in Pb–Pb collisions at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 2.76 \(\text {TeV}\), neglecting absorption. Results obtained in the colour evaporation model (CEM) at nexttoleading order (NLO) in the total cross section (leading order in \(p_{\mathrm {T}}\)) are discussed first, followed by results from a leadingorder colour singlet model (CSM) calculation.
The upper lefthand panel of Fig. 77 shows the uncertainty in the shadowing effect on \(\mathrm {J}/\psi \) due to the variations in the 30 EPS09 NLO sets [364] (red). The uncertainty band calculated in the CEM at LO with the EPS09 LO sets is shown for comparison (blue). It is clear that the LO results exhibit a larger shadowing effect. This difference between the LO results, also shown in Ref. [363], and the NLO calculations arises because the EPS09 LO and NLO gluon shadowing parametrisations differ significantly at low x [364].
In principle, the shadowing results should be the same for LO and NLO. Unfortunately, however, the gluon modifications, particularly at low x and moderate \(Q^2\), are not yet sufficiently constrained. The lower left panel shows the same calculation for \(\Upsilon \) production. Here, the difference between the LO and NLO calculations is reduced because the mass scale, as well as the range of x values probed, is larger. Differences in LO results relative to, e.g., the colour singlet model arise less from the production mechanism than from the different mass and scale values assumed, as we discuss below.
It should be noted that the convolution of the two nuclear parton densities results in a \(\sim \)20 % suppression at NLO for \(y\le 2.5\) with a mild decrease in suppression at more forward rapidities. The gluon antishadowing peak at \(y \sim 4\) for \(\mathrm {J}/\psi \) and \(y \sim 2\) for \(\Upsilon \) with large x in the nucleus is mitigated by the shadowing at low x in the opposite nucleus with the NLO parametrisation. The overall effect due to NLO nPDFs in both nuclei is a result with moderate rapidity dependence and \(R_\mathrm{AA}^{\mathrm {J}/\psi } \sim 0.7\) for \(y\le 5\) and \(R_\mathrm{AA}^{\Upsilon } \sim 0.84\) for \(y \le 3\). The nPDF effect gives more suppression at central rapidity than at forward rapidity, albeit less so for the LHC energies than for RHIC where the antishadowing peak at \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 200\(~\text {GeV}\) is at \(y \sim 2\). The difference between the central value of \(R_\mathrm{AA}\) at LO and NLO is \(\sim \)30 % for the \(\mathrm {J}/\psi \) and \(\sim \)10 % for the \(\Upsilon \). If a different nPDF set with LO and NLO parametrisations, such as nDSg [367], is used, the difference between LO and NLO is reduced to a few percent since the difference between the underlying LO and NLO proton parton densities at low x is much smaller for nDSg than for EPS09 [707].
The right panels of Fig. 77 show the \(p_{\mathrm {T}}\) dependence of the effect at forward rapidity for \(\mathrm {J}/\psi \) (upper) and \(\Upsilon \) (lower). The effect is rather mild and increases slowly with \(p_{\mathrm {T}}\). There is little difference between the \(\mathrm {J}/\psi \) and \(\Upsilon \) results for \(R_{\text {{PbPb}}}(p_{\mathrm {T}})\) because, for \(p_{\mathrm {T}}\) above a few GeV, the \(p_{\mathrm {T}}\) scale is dominant. There is no LO comparison here because the \(p_{\mathrm {T}}\) dependence cannot be calculated in the LO CEM.
However, the leadingorder colour single model calculation (LO CSM) of \(\mathrm {J}/\psi \) production, shown to be compatible with the magnitude of the of the \(p_{\mathrm {T}}\)integrated cross sections, is a \(2 \rightarrow 2\) process, \(g + g \rightarrow \mathrm {J}/\psi + g\), which has a calculable \(p_{\mathrm {T}}\) dependence at LO, as in the socalled extrinsic scheme [432].
In this approach, one can use the partonic differential cross section computed from any \(2 \rightarrow 2\) theoretical model that satisfactorily describes the data down to low \(p_{\mathrm {T}}\). Here, a generic \(2\rightarrow 2\) matrix element which matches the \(p_{\mathrm {T}}\) dependence of the data has been used and the parametrisations EKS98 LO [401] and nDSg LO [367] have been employed. The former coincides with the mid value of EPS09 LO [364]. The error bands for the EKS98 and nDSg models shown in Fig. 78 correspond to the variation of the factorisation scale (\(0.5 m_\mathrm{T} < \mu _F < 2 m_\mathrm{T}\)).
The spatial dependence of the nPDF has been included in this approach through a probabilistic Glauber MonteCarlo framework, JIN [373], assuming an inhomogeneous shadowing proportional to the local density [370, 371]. Results are shown in Fig. 79.
5.1.3 Statistical (re)generation models
Over the past 20 years thorough evidence has been gathered that production of hadrons with u, d, svalence quarks in heavyion collisions can be described using a statistical model reflecting a hadrochemical equilibrium approach [648, 708]. Hadron yields from top AGS energy (\(\sim \)10\(~\text {GeV}\)) up to the LHC are reproduced over many orders of magnitude employing a statistical operator that incorporates a complete hadronresonance gas. In a grand canonical treatment, the only thermal parameters are the chemical freezeout temperature T and the baryochemical potential \(\mu _b\) (and the fireball volume V, in case yields rather than ratios of yields are fitted). These parameters are fitted to data for every collision system as a function of collision energy. The temperature initially rises with \(\sqrt{s_{\mathrm{NN}}}\) and flattens at a value of \((159\pm 2)~\text {MeV} \) close to top SPS energy, while the baryochemical potential drops smoothly and reaches a value compatible with zero at LHC energies. In the energy range where T saturates, it has been found to coincide with the (quasi)critical temperature found in lattice QCD.
Deconfinement of quarks is expected in a QGP and for heavy quarks, in particular, this has been formulated via modification of the heavy quark potential in a process analogous to Debye screening in QED [644] (see Sect. 5.1.1). Heavy quarks are not expected to be produced thermally but rather in initial hardscattering processes. Even at top LHC energy thermal production is only a correction at maximally the 10 % level [709]. Therefore a scenario was proposed, in which charm quarks, formed in a high energy nuclear collision in initial hard scattering, find themselves colourscreened, therefore deconfined in a QGP, and hadronise with light quarks and gluons at the phase boundary [647, 710, 711]. At hadronisation open charm hadrons as well as charmonia are formed according to their statistical weights and the mass spectrum of charmed hadrons.
Since for each beam energy the values of T and \(\mu _b\) are already fixed by the measured light hadron yields, the only additional input needed is the initial charm production cross section per unit rapidity in the appropriate rapidity interval. The conservation of the number of charm quarks is introduced in the statistical model via a fugacity \(g_c\), where all open charm hadron yields scale proportional to \(g_c\), while charmonia scale with \(g_c^2\) since they are formed from a charm and an anticharm quark. A logical consequence of this is that at energies below LHC energy, where the charm yield is small, charmonium production is suppressed in comparison to scaled pp collisions, while for LHC energies, the charm yield is larger and the charmonium yield is enhanced [647, 710, 711].
Already a comparison to first data on \(\mathrm {J}/\psi \) production from PHENIX at RHIC using a charm cross section from perturbative QCD proved successful [465]. When more data became available it was found that in particular the rapidity and centrality dependence of \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) from RHIC and the \(\psi \text {(2S)}\) to \(\mathrm {J}/\psi \) ratio from NA50 at the SPS were well reproduced by this approach [712, 713]. In order to treat properly the centrality dependence, also production in the dilute corona using the pp production cross section of \(\mathrm {J}/\psi \) is considered [712, 713]. While it was clear that for LHC energies larger values for \(R_{\mathrm {AA}}\) of \(\mathrm {J}/\psi \) are expected than at RHIC, \(R_{\mathrm {AA}}\) depends linearly on the unknown \({c\overline{c}}\) cross section. Predictions for an expected range were given in [714].
The comparison of the statistical hadronisation predictions with the LHC data require the knowledge of the \({c\overline{c}}\) cross section. This quantity has been measured in \(\mathrm pp\) collisions at \(\sqrt{s}\) \(=\) 7 \(\text {TeV}\) and is then extrapolated to the lower Pb–Pb beam energy, i.e. \(\sqrt{s_{\mathrm{NN}}}\) \(=\) 2.76 \(\text {TeV}\). Since the current data are for half the LHC design energy, the open charm cross section is at the lower end of the range considered in Ref. [714]. The uncertainty on this model prediction comes from the uncertainty on the \({c\overline{c}}\) cross section and it stems from the measurement of the \({c\overline{c}}\) cross section itself, \(\sqrt{s}\), and shadowing extrapolations.
As it will be discussed in Sect. 5.2.2, the statistical model reproduces the significant increase observed, for central collisions, in the \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) from RHIC to the LHC (see Fig. 85).
Another characteristic feature of the statistical hadronisation model is an excited state population driven by Boltzmann factors at the hadronisation temperature. So far the only successful test of this prediction is the \(\psi \text {(2S)}\)/\(\mathrm {J}/\psi \) ratio at the SPS. Data for \(\psi \text {(2S)}\) and \(\chi _c\) at LHC and RHIC will be crucial tests of this model and will allow one to differentiate between transport model predictions (see Sect. 5.1.4) and statistical hadronisation at the phase boundary, if measured with sufficient precision (10–20 %).
5.1.4 Transport approach for inmedium quarkonia
The reaction rate can be calculated from inelastic scattering amplitudes of quarkonia on the constituents of the medium (light quarks and gluons, or light hadrons). The relevant processes depend on the (inmedium) properties of the bound state [717]. In the QGP, for a tightly bound state (binding energy \(E_B\ge T\)), an efficient process is gluodissociation [718], \(g+\mathcal{Q}\rightarrow Q+\overline{Q}\), where all of the incoming gluon energy is available for breakup. However, for loosely bound states (\(E_B <T\) for excited and partially screened states), the phase space for gluodissociation rapidly shuts off, rendering “quasifree” dissociation, \(p+\mathcal{Q}\rightarrow Q+\overline{Q}+p\) (\(p=q,\overline{q},g\)), the dominant process [717], cf. Fig. 80 (left). Gluodissociation and inelastic parton scatteringdissociation of quarkonia have also been studied within an EFT approach [719].
The equilibrium number densities are simply those of Q quarks (with spincolour and particleantiparticle degeneracy \(6 \times 2\)) and quarkonium states (summed over including their spin degeneracies \(d_\mathcal{Q}\)).
The quarkonium equilibrium limit is thus coupled to the open HF spectrum in medium; e.g., a smaller cquark mass increases the cquark density, which decreases \(\gamma _c\) and therefore reduces \(N_{J/\psi }^\mathrm{eq}\), by up to an order of magnitude for \(m_c=1.8 \rightarrow 1.5\) GeV/\(c^2\), cf. Fig. 80 (right).
In practice, further corrections to \(N_\mathcal{Q}^\mathrm{eq}\) are needed for more realistic applications in heavyion collisions. First, heavy quarks cannot be expected to be thermalised throughout the course of a heavyion collision; harder heavyquark (HQ) momentum distributions imply reduced phasespace overlap for quarkonium production, thus suppressing the gain term. In the rate equation approach this has been implemented through a relaxation factor \(\mathcal{R} = 1\exp (\int \mathrm {d}\tau /\tau _Q^\mathrm{therm})\) multiplying \(N_\mathcal{Q}^\mathrm{eq}\), where \(\tau _Q^\mathrm{therm}\) represents the kinetic relaxation time of the HQ distributions [715, 720]. This approximation has been quantitatively verified in Ref. [721]. Second, since HQ pairs are produced in essentially pointlike hard collisions, they do not necessarily explore the full volume in the fireball. This has been accounted for by introducing a correlation volume in the argument of the Bessel functions, in analogy to strangeness production at lower energies [722].
An important aspect of this transport approach is a controlled implementation of inmedium properties of the quarkonia [715, 723]. Colourscreening of the QCD potential reduces the quarkonium binding energies, which, together with the inmedium HQ mass, \(m_Q^*\), determines the boundstate mass, \(m_\mathcal{Q} = 2m_Q^* E_B\). As discussed above, the interplay of \(m_\mathcal{Q}\) and \(m_Q^*\) determines the equilibrium limit, \(N_\mathcal{Q}^\mathrm{eq}\), while \(E_B\) also affects the inelastic reaction rate, \(\Gamma _\mathcal{Q}(T)\). To constrain these properties, pertinent spectral functions have been used to compute Euclidean correlators for charmonia, and required to approximately agree with results from lattice QCD [723]. Two basic scenarios have been put forward for tests against charmonium data at the SPS and RHIC: a strongbinding scenario (SBS), where the J\(/\psi \) survives up to temperatures of about 2 \(T_c\), and a weakbinding scenario (WBS) with \(T_\mathrm{diss}\simeq 1.2\,T_c\), cf. Fig. 81. These scenarios are motivated by microscopic Tmatrix calculations [558] where the HQ internal (\(U_{{Q\overline{Q}}}\)) or free energy (\(F_{{Q\overline{Q}}}\)) have been used as potential, respectively. A more rigorous definition of the HQ potential, and a more direct implementation of the quarkonium properties from the Tmatrix approach is warranted for future work. The effects of the hadronic phase are generally small for \(\mathrm {J}/\psi \) and bottomonia, but important for the \(\psi \text {(2S)}\), especially, if its direct decay channel \(\psi \text {(2S)} \rightarrow \overline{\mathrm{D}}\mathrm{D}\) is opened (due to reduced masses and/or finite widths of the D mesons) [715, 720].
The rate equation approach has been extended to compute \(p_{\mathrm {T}}\) spectra of charmonia in heavyion collisions [654]. Toward this end, the loss term was solved with a threemomentum dependent dissociation rate and a spatial dependence of the charmonium distribution function, while for the gain term blastwave distributions at the phase transition were assumed (this should be improved in the future by an explicit evaluation of the gain term from the Boltzmann equation using realistic timeevolving HQ distributions; see Ref. [724] for initial studies) [725]. In addition, formation time effects are included, which affect quarkonium suppression at high \(p_{\mathrm {T}}\) [726].
To close the quarkonium rate equations, several input quantities are required which are generally taken from experimental data in \(\mathrm pp\) and p–A collisions, e.g., quarkonia and HQ production cross sections (with shadowing corrections), and primordial nuclear absorption effects encoded in phenomenological absorption cross sections. Feeddown effects from excited quarkonia (and bhadron decays into charmonium) are accounted for. The spacetime evolution of the medium is constructed using an isotropically expanding fireball model reproducing the measured hadron yields and their \(p_{\mathrm {T}}\) spectra. The fireball resembles the basic features of hydrodynamic models [727], but an explicit use of the latter is desirable for future purposes.
Two main model parameters have been utilised to calibrate the rate equation approach for charmonia using the centrality dependence of inclusive \(\mathrm {J}/\psi \) production in Pb–Pb collisions at the SPS (\(\sqrt{s_{\mathrm{NN}}}\) \(=\) 17 GeV) and in Au–Au collisions at RHIC (\(\sqrt{s_{\mathrm{NN}}}\) \(=\) 200\(~\text {GeV}\)): the strongcoupling constant \(\alpha _s\), controlling the inelastic reaction rate, and the cquark relaxation time affecting the gain term through the amended charmonium equilibrium limit. With \(\alpha _s\simeq 0.3\) and \(\tau _c^{\text {therm}}\simeq \) 4–6 (1.5–2) fm/c for the SBS (WBS), the inclusive \(\mathrm {J}/\psi \) data at SPS and RHIC can be reasonably well reproduced, albeit with different decompositions into primordial and regenerated yields (the former are larger in the SBS than in the WBS). The \(\tau _c^{\text {therm}}\) obtained in the SBS is in the range of values calculated microscopically from the Tmatrix approach using the Upotential [558], while for the WBS it is much smaller than calculated from the Tmatrix using the Fpotential. Thus, from a theoretical point of view, the SBS is the preferred scenario.
With this setup, namely the TAMU transport model, quantitative predictions for Pb–Pb collisions at the LHC (\(\sqrt{s_{\mathrm{NN}}}\) \(=\) 2.76 TeV) have been carried out for the centrality dependence and \(p_{\mathrm {T}}\) spectra of \(\mathrm {J}/\psi \) [728], as well as for \(\Upsilon \text {(1S)}\), \(\chi _b\), and \(\Upsilon \text {(2S)}\) production [729].
Similar results are obtained in the transport approach THU developed by the Tsinghua group [730, 731], which differs in details of the implementation, but overall asserts the robustness of the conclusions. In the THU model, the quarkonium distribution is also governed by the Boltzmanntype transport equation. The cold nuclear matter effects change the initial quarkonium distribution and heavy quark distribution at \(\tau _0\). The interaction between the quarkonia and the medium is reflected in the loss and gain terms and depends on the local temperature \(T(\vec {r},\tau )\) and velocity \(u_\mu (\vec {r},\tau )\), which are controlled by the energymomentum and charge conservations of the medium, \(\partial _\mu T^{\mu \nu }=0\) and \(\partial _\mu n^\mu =0\).
Within this approach, the centrality dependence of the nuclear modification factor \(R_{\mathrm {AA}}\) can be obtained and compared to experimental results at low \(p_{\mathrm {T}}\). In contrast to collisions at SPS and RHIC energies, at LHC energies the large abundance of c and \(\overline{c}\) quarks increases their combining probability to form charmonia. Hence this regeneration mechanism becomes the dominant source of charmonium production for semicentral and central collisions at the LHC. The competition between dissociation and regeneration leads to a flat structure of the \(\mathrm {J}/\psi \) yield as a function of centrality. This flat behaviour should disappear at higher energies or, regeneration being a \(p_{\mathrm {T}}\)dependent mechanism, with increasing \(p_{\mathrm {T}}\).
The charmonium transverse momentum distribution contains more dynamic information on the hot medium and can be calculated within the transport approach. The regeneration occurs in the fireball, and therefore the thermally produced charmonia are mainly distributed at low \(p_{\mathrm {T}}\), their contribution increasing with centrality. On the other hand, those charmonia from the initial hard processes carry high momenta and dominate the high \(p_{\mathrm {T}}\) region at all centralities. This different \(p_{\mathrm {T}}\) behaviour of the initially produced and regenerated charmonia can even lead to a minimum located at intermediate \(p_{\mathrm {T}}\). Moreover, this particular \(p_{\mathrm {T}}\) behaviour will lead to an evolution of the mean transverse momentum, \(\langle p_{\mathrm {T}} \rangle \), with centrality that would be higher for SPS than for LHC nuclear collisions, once normalised to the corresponding proton–proton \(\langle p_{\mathrm {T}} \rangle \) [732, 733]. At the SPS, almost all the measured \(\mathrm {J}/\psi \) are produced through initial hard processes and carry high momentum. At RHIC, the regeneration starts to play a role and even becomes equally important as the initial production in central collisions. At the LHC, regeneration becomes dominant, and results in a decreasing of \(\langle p_{\mathrm {T}} \rangle \) with increasing centrality.
5.1.5 Nonequilibrium effects on quarkonium suppression
Since heavy quarkonium states have a short formation time in their rest frame (\(< 1~\text {fm}/c\)), they are sensitive to the earlytime dynamics of the QGP. As a consequence, it is necessary to have dynamical models that can accurately describe the bulk dynamics of the QGP during the first \(\text {fm}/c\) of its lifetime. This is complicated by the fact that, at the earliest times after the initial nuclear impact, the QGP is momentumspace anisotropic in the local rest frame. The existence of large QGP momentumspace anisotropies is found in both the weak and the strong coupling limits (see e.g. Ref. [734, 735, 736, 737]). In both limits, one finds that the longitudinal pressure, \(\mathcal{P}_L = T^{zz}\), is much less than the transverse pressure, \(\mathcal{P}_T = (T^{xx}+T^{yy})/2\), at times smaller than \(1~\text {fm}/c\). During the QGP evolution this momentumspace anisotropy relaxes to zero, but it does so only on a time scale of several \(\text {fm}/c\). In addition, the momentumspace anisotropy grows larger as one approaches the transverse edge of the QGP, where the system is colder. The existence of such momentumspace anisotropies is consistent with first and secondorder viscous hydrodynamics; however, since these approaches rely on linearisation around an isotropic background, it is not clear that these methods can be applied in a farfromequilibrium situation. In order to address this issue, a nonperturbative framework, called anisotropic hydrodynamics (aHYDRO), has been developed. This framework allows the system to be arbitrarily anisotropic [738, 739, 740, 741].
The timeevolution provided by aHYDRO has to be folded together with the nonequilibrium (anisotropic) quarkonium rates. These were first considered in Refs. [742, 743, 744, 745, 746] where the effect of momentumspace anisotropy was included for both the real and imaginary parts of potential. In this context, the imaginary part of the potential plays the most important role, as it sets the inmedium decay rate of heavy quarkonium states. The calculations of the resulting decay rates in Ref. [746] demonstrated that these inmedium decay rates were large with the corresponding lifetime of the states being on the order of \(\text {fm}/c\). In practice, one integrates the decay rate over the lifetime of the state in the plasma as a function of its threedimensional position in the system and its transverse momentum. The result of this is a prediction for the \(R_{\mathrm {AA}}\) that depends on the assumed shearviscosity to entropy density ratio (\(\eta /s\)) of the QGP since this ratio determines the degree to which the system remains isotropic. The results obtained for the inclusive \(\Upsilon \text {(1S)}\) and \(\Upsilon \text {(2S)}\) suppression [747, 748, 749] have a significant dependence on the assumed value of \(\eta /s\), in particular for the inclusive \(\Upsilon \text {(1S)}\). This ratio can be determined from independent collective flow measurements and at the energies probed by the LHC one finds that \(4\pi \,\eta /s \sim 1\text {}3\) [748, 750]. The upper limit of this range seems to be compatible with the CMS data (the comparison will be shown in Sect. 5.2.7); however, since the model used did not include any regeneration effects, it is possible that the final \(\eta /s\) could be a bit lower than three times the lower bound. Furthermore, it should be pointed out that feeddown fractions based on CDF measurements with \(p_{\mathrm {T}} >8~\text {GeV}/c \) are used [209, 441], which with \(\approx \)50 % is larger than the fraction one would obtain when using the recent \(\chi _b \text {nP}\rightarrow \Upsilon \text {(1S)} \) measurements by LHCb that extend to slightly lower \(p_{\mathrm {T}}\) [203]. In the later case the total \(\Upsilon \text {(1S)}\) feeddown contribution is \(\approx \)30 % for \(p_{\mathrm {T}} >6~\text {GeV}/c \).
5.1.6 Collisional dissociation of quarkonia from finalstate interactions
The model described in Sect. 4.3.3 can also be modified to describe the dissociation dynamics of quarkonia in the QGP. The model includes both initialstate cold nuclear matter energyloss and finalstate effects, such as radiative energy loss for the colouroctet state and collisional dissociation for quarkonia, as they traverse the created hot medium. The main differences with respect to the formalism discussed in Sect. 4.3.3 are (a) that once a high\(p_{\mathrm {T}}\) quarkonium is dissociated, it is unlikely that it will fragment again to form a new quarkonium, (b) the formation time is given not by fragmentation dynamics but by binding energies. A selfconsistent description of the formation of a quarkonium in a thermal QGP is a challenging problem [83] and assumes that the formation time lies between \(1/(2E_b)\) and \(1/(E_b)\), and that the wave function does not show significant thermal effects in this short time.
When compared to the \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) results, obtained by the CMS experiment, the model is consistent with the observations for the peripheral events, but underestimates the suppression for the most central events, suggesting that thermalisation effects on the wave functions may be substantial.
5.1.7 Comover models
5.1.8 Summary of theoretical models for experimental comparison
Different theoretical models are available for comparison. Among them, the statistical hadronisation model, the transport model, the collisional dissociation model, and the comover model will be compared to charmonium experimental results in the next section. Their principal characteristics can be summarised as follows.
In the statistical hadronisation model, the charm (beauty) quarks and antiquarks, produced in initial hard collisions, thermalise in a QGP and form hadrons at chemical freezeout. It is assumed that no quarkonium state survives in the deconfined state (full suppression) and, as a consequence, also CNM effects are not included in this model. An important aspect in this scenario is the canonical suppression of open charm or beauty hadrons, which determines the centrality dependence of production yields in this model. The overall magnitude is determined by the input charm (beauty) production cross section.
Kinetic (re)combination of heavy quarks and antiquarks in a QGP provides an alternative quarkonium production mechanism. In transport models, there is continuous dissociation and (re)generation of quarkonia over the entire lifetime of the deconfined state. A hydrodynamicallike expansion of the fireball of deconfined matter, constrained by data, is part of such models, alongside an implementation of the screening mechanism with inputs from lattice QCD. Other important ingredients are partonlevel cross sections. Cold nuclear matter effects are incorporated by means of an overall effective absorption cross section that accounts for (anti)shadowing, nuclear absorption, and Cronin effects.
The collisional dissociation model considers, in addition to modifications of the binding potential by the QGP and cold nuclear matter effects, radiative energy loss of the colour octet quarkonium precursor and collisional dissociation processes inside the QGP.
Similarly, the comover interaction model includes dissociation of quarkonia by interactions with the comoving medium of hadronic and partonic origin. Regeneration reactions are also included. Their magnitude is determined by the production cross section of \({c\overline{c}}\) pairs and quarkonium states. Cold nuclear matter effects are taken into account by means of (anti)shadowing models.

Statistical hadronisation assumes full suppression of primordial quarkonia and regeneration at the phase boundary.

Transport models include cold nuclear absorption, direct suppression, and regeneration.

Collisional dissociation models include initialstate cold nuclear matter effects and finalstate effects based on radiative energy loss and collisional dissociation.

Comover models include shadowing, interaction with comoving medium, and regeneration.
In order to compare with experimental data on bottomonium, also the hydrodynamical formalism assuming finite local momentumspace anisotropy due to finite shear viscosity will be considered. The main ingredients are: a screened potential, an hydrodynamicallike evolution of the QGP, and feeddown from highermass states. Neither cold nuclear matter effects nor recombination are included.
5.2 Experimental overview of quarkonium results at RHIC and LHC
5.2.1 Proton–proton collisions as a reference for \(R_{\mathrm {AA}}\) at the LHC
The medium effects on quarkonia are usually quantified via the nuclear modification factor \(R_{\mathrm {AA}}\), basically comparing the quarkonium yields in AA to the \(\mathrm pp\) ones. A crucial ingredient for the \(R_{\mathrm {AA}}\) evaluation is, therefore, \(\sigma _\mathrm{pp}\), the quarkonium production cross section in \(\mathrm pp\) collisions measured at the same energy as the AA data.
During LHC Run 1, \(\mathrm pp\) data at \(\sqrt{s}\) \(=\) 2.76 \(\text {TeV}\) were collected in two short data taking periods in 2011 and 2013. When the collected data sample was large enough, the quarkonium \(\sigma _\mathrm{pp}\) was experimentally measured, otherwise an interpolation of results obtained at other energies was made.
More in detail, the \(\mathrm {J}/\psi \) cross section (\(\sigma _\mathrm{pp}^{\mathrm {J}/\psi }\)) adopted by ALICE for the forward rapidity \(R_{\mathrm {AA}}\) results is based on the 2011 \(\mathrm pp\) data taking. The \(\mathcal {L} _{\text {int}} = 19.9\text {~nb}^{1} \) integrated luminosity, corresponding to \(1364\pm 53\) \(\mathrm {J}/\psi \) reconstructed in the dimuon decay channel, allows for the extraction of both the integrated as well as the \(p_{\mathrm {T}}\) and y differential cross sections [412]. The statistical uncertainty is 4 % for the integrated result, while it ranges between 6 and 20 % for the differential measurement. Systematic uncertainties are \(\sim \)8 %. The collected data (\(\mathcal {L} _{\text {int}} = 1.1\text {~nb}^{1} \)) allow for the evaluation of \(\sigma _\mathrm{pp}^{\mathrm {J}/\psi }\) also in the ALICE midrapidity region, where \(\mathrm {J}/\psi \) are reconstructed through their dielectron decay. The measurement is, in this case, affected by larger statistical and systematic uncertainties, of about 23 and 18 %, respectively. Therefore, the \(\sigma _\mathrm{pp}^{\mathrm {J}/\psi }\) reference for the \(R_{\mathrm {AA}}\) result at midrapidity was obtained performing an interpolation based on midrapidity results from PHENIX at \(\sqrt{s}\) \(=\) 0.2 \(\text {TeV}\) [411], CDF at \(\sqrt{s}\) \(=\) 1.96 \(\text {TeV}\) [414], and ALICE at \(\sqrt{s}\) \(=\) 2.76 [412] and 7 \(\text {TeV}\) [413]. The interpolation is done by fitting the data points with several functions assuming a linear, an exponential, a power law, or a polynomial \(\sqrt{s}\)dependence. The resulting systematic uncertainty is, in this case, 10 %, i.e. smaller than the one obtained directly from the data at \(\sqrt{s}\) \(=\) 2.76 \(\text {TeV}\).
The limited size of the \(\mathrm pp\) data sample at \(\sqrt{s}\) = 2.76 \(\text {TeV}\) has not allowed ALICE to measure the \(\Upsilon \) cross section. The reference adopted by ALICE for the \(R_{\mathrm {AA}}\) studies [683] is, in this case, based on the \(\mathrm pp\) measurement by LHCb [419]. However, since the LHCb result is obtained in a rapidity range (\(2<y<4.5\)) not exactly matching the ALICE one (\(2.5<y<4\)), the measurement is corrected through a rapidity interpolation based on a Gaussian shape.
For the \(\Upsilon \) \(R_{\mathrm {AA}}\), CMS results are based on the \(\mathrm pp\) reference cross section extracted from \(\mathrm pp\) data at \(\sqrt{s}\) = 2.76 \(\text {TeV}\) [482]. The number of \(\Upsilon \text {(1S)}\) with \(y<2.4\) and \(0<p_{\mathrm {T}} <20~\text {GeV}/c \) is \(101\pm 12\), with a systematic uncertainty on the signal extraction of \(\sim \)10 %.
Overview of the \(\mathrm pp\) datasets and approaches adopted for the evaluation of the \(\sigma _\mathrm{pp}\) production cross section for the quarkonium states under study
ALICE  CMS  

\(\mathrm {J}/\psi \)  Forwardy: \(\sigma ^{\mathrm {J}/\psi }_\mathrm{pp}\) from \(\mathrm pp\) data at \(\sqrt{s}\) = 2.76 \(\text {TeV}\)  \(\sigma ^{\mathrm {J}/\psi }_\mathrm{pp}\) from \(\mathrm pp\) data at \(\sqrt{s}\) = 2.76 \(\text {TeV}\) 
Midy: \(\sigma ^{\mathrm {J}/\psi }_\mathrm{pp}\) from interpolation of ALICE, CDF and PHENIX data  
\(\Upsilon \)  \(\sigma ^{\Upsilon }_\mathrm{pp}\) from LHCb \(\mathrm pp\) data at \(\sqrt{s}\) = 2.76 \(\text {TeV}\) + yinterpolation  \(\sigma ^{\Upsilon }_\mathrm{pp}\) from \(\mathrm pp\) data at \(\sqrt{s}\) = 2.76 \(\text {TeV}\) 
5.2.2 \(\mathrm J/\psi \) \(R_{\mathrm {AA}}\) results at low \(p_{\mathrm {T}}\)
The experiments ALICE at the LHC and PHENIX and STAR at RHIC measure the inclusive \(\mathrm {J}/\psi \) production (prompt \(\mathrm {J}/\psi \) plus those coming from bhadron decays) in the low \(p_{\mathrm {T}}\) region, down to \(p_{\mathrm {T}} = 0\). STAR measures \(\mathrm {J}/\psi \) reconstructed from their \(e^{+}e^{}\) decay at midrapidity (\(y <\) 1), while PHENIX detects charmonia in two rapidity ranges: at midrapidity (\(y<0.35\)) in the \(e^{+}e^{}\) decay channel and at forward rapidity (\(1.2<y<2.2\)) in the \(\mu ^{+}\mu ^{}\) decay channel. Similarly, ALICE studies the inclusive \(\mathrm {J}/\psi \) production in the \(e^{+}e^{}\) decay channel at midrapidity (\(y<0.9\)) and in the \(\mu ^{+}\mu ^{}\) decay channel at forward rapidity (\(2.5<y<4\)). A summary of the main experimental results, together with their kinematic coverage and references, is given in Tables 13 and 14. The experiments have investigated the centrality dependence of the \(\mathrm {J}/\psi \) nuclear modification factor measured in AA collisions, i.e. Au–Au at \(\sqrt{s_{\mathrm{NN}}}\) = 200\(~\text {GeV}\) for PHENIX [668] and STAR [676] and Pb–Pb at \(\sqrt{s_{\mathrm{NN}}}\) = 2.76 \(\text {TeV}\) in the ALICE case [679, 680]. As an example, PHENIX and ALICE results are shown in Fig. 83 for the forward (left) and the midrapidity (right) regions. While the RHIC results show an increasing suppression towards more central collisions, the ALICE \(R_{\mathrm {AA}}\) has a flatter behaviour both at forward and at midrapidity. In the two y ranges there is clear evidence for a smaller suppression at the LHC than at RHIC.
A similar behaviour is expected by the statistical model [754], discussed in Sect. 5.1.3, where the \(\mathrm {J}/\psi \) yield is completely determined by the chemical freezeout conditions and by the abundance of \(c\overline{c}\) pairs. In Figs. 84 and 85, the statistical model predictions are compared to the ALICE \(R_{\mathrm {AA}}\) in the two covered rapidity ranges. As discussed in Sect. 5.1.3, a crucial ingredient in this approach is the \({c\overline{c}}\) production cross section: the error band in the figures stems from the measurement of the \({c\overline{c}}\) cross section itself and from the correction introduced to take into account the \(\sqrt{s}\) extrapolation to evaluate the cross section at the Pb–Pb energy (\(\sqrt{s_{\mathrm{NN}}}\) = 2.76 \(\text {TeV}\)). In Fig. 85 the RHIC data [668] and the corresponding statistical model calculations are also shown. Inspecting Fig. 85, for central collisions a significant increase in the \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) at LHC as compared to RHIC is visible and well reproduced by the statistical hadronisation model. In particular, as a characteristic feature of the model, the shape as a function of centrality is entirely given by the charm cross section at a given energy and rapidity and is well reproduced both at RHIC and LHC. This applies also to the maximum in \(R_{\mathrm {AA}}\) at midrapidity due to the peaking of the charm cross section there.
The (re)combination or the statistical hadronisation process are expected to be dominant in central collisions and, for kinematical reasons, they should contribute mainly at low \(p_{\mathrm {T}}\), becoming negligible as the \(\mathrm {J}/\psi \) \(p_{\mathrm {T}}\) increases. This behaviour is investigated by further studying the \(R_{\mathrm {AA}}\) \(p_{\mathrm {T}}\)dependence. In Fig. 86, the ALICE \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) (\(p_{\mathrm {T}}\)), measured at forward rapidity (left) or at midrapidity [480] (right), are compared to corresponding PHENIX results obtained in similar rapidity ranges. The forwardrapidity result has been obtained in the centrality class 0–20 %, while the midrapidity one in 0–40 %. In both rapidity regions, a striking different pattern is observed: while the ALICE \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) shows a clear decrease from low to high \(p_{\mathrm {T}}\), the pattern observed at low energies is rather different, being almost flat versus \(p_{\mathrm {T}}\), with a suppression up to a factor four (two) stronger than at LHC at forward rapidity (midrapidity).
As discussed, the ALICE results are for inclusive \(\mathrm {J}/\psi \), therefore including two contributions: the first one from \(\mathrm {J}/\psi \) direct production and feeddown from higher charmonium states and the second one from \(\mathrm {J}/\psi \) originating from bhadron decays. Beauty hadrons decay mostly outside the fireball, hence the measurement of nonprompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) is mainly connected to the b quark inmedium energy loss, discussed in Sect. 4.1.3. Nonprompt \(\mathrm {J}/\psi \) are, therefore, expected to behave differently with respect to the prompt ones. In the low\(p_{\mathrm {T}}\) region covered by ALICE the fraction of nonprompt \(\mathrm {J}/\psi \) is smaller than 15 % [755] (slightly depending on the y range). Based on this fraction, the ALICE Collaboration has estimated the influence of the nonprompt contribution on the measured inclusive \(R_{\mathrm {AA}}\). At midrapidity the prompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) can vary within \(7\) and \(+17\) % with respect to the inclusive \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) assuming no suppression (\(R_{\mathrm {AA}} ^{\mathrm{nonprompt}}=1\)) or full suppression (\(R_{\mathrm {AA}} ^{\mathrm{nonprompt}}=0\)) for beauty, respectively. At forwardy, the prompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) would be 6 % lower or 7 % higher than the inclusive result in the two aforementioned cases [679].
5.2.3 \(\mathrm J/\psi \) \(R_{\mathrm {AA}}\) results at high \(p_{\mathrm {T}}\)
The CMS experiment is focussed on the study of the \(\mathrm {J}/\psi \) production at high \(p_{\mathrm {T}}\). The limit in the charmonium acceptance at low\(p_{\mathrm {T}}\) is due to the fact that muons from the charmonium decay need a minimum momentum (\(p\approx 3\text {}5~\text {GeV}/c \)) to reach the muon tracking stations, overcoming the strong CMS magnetic field (3.8 T) and the energy loss in the magnet and its return yoke. The CMS vertex reconstruction capabilities allow for the separation of nonprompt \(\mathrm {J}/\psi \) from bhadron decays from prompt \(\mathrm {J}/\psi \), using the reconstructed decay vertex of the \(\mu ^{+}\mu ^{}\) pair. The prompt \(\mathrm {J}/\psi \) include directly produced \(\mathrm {J}/\psi \) as well as those from decays of higher charmonium states (e.g. \(\psi \text {(2S)}\) and \(\chi _c\)), which cannot be removed because their decay lengths are orders of magnitude smaller compared to those from b decays, and not distinguishable in the analysis of the Pb–Pb data.
The \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) was evaluated in the Pb–Pb data sample collected in 2010, corresponding to \(\mathcal {L} _{\text {int}} = 7.3~\mu \text {b}^{1} \). The nuclear modification factor, integrated over the rapidity range \(y<2.4\) and \(p_{\mathrm {T}}\) range \(6.5<p_{\mathrm {T}} <30~\text {GeV}/c \), was measured in six centrality bins [482], starting with the 0–10 % bin (most central), up to the 50–100 % bin (most peripheral). The \(R_{\mathrm {AA}}\) obtained for prompt \(\mathrm {J}/\psi \), when integrating over the \(p_{\mathrm {T}}\) range \(6.5<p_{\mathrm {T}} <30~\text {GeV}/c \) and \(y<2.4\), is shown in Fig. 89 ( left). The same centrality dependence, with a smooth decrease towards most central collisions, is observed also for inclusive \(\mathrm {J}/\psi \), even if the suppression is slightly more important for prompt \(\mathrm {J}/\psi \). In both cases the \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) is still suppressed even in the (rather wide) most peripheral bin. A more recent analysis, based on the larger 2011 Pb–Pb data sample (\(\mathcal {L} _{\text {int}} = 150 ~\mu \text {b}^{1} \)), has allowed one to study the \(R_{\mathrm {AA}}\) in a much narrower centrality binning (12 centrality bins) and confirms the observed pattern [494].
In Fig. 89 (right) the prompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) centrality dependence is compared with the predictions of the TAMU transport model. The observed suppression, increasing as a function of centrality, is due to the melting of primordial \(\mathrm {J}/\psi \). The TAMU model provided a reasonable description of the ALICE low\(p_{\mathrm {T}}\) \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) (see Fig. 84), with a significant recombination contribution. On the contrary, no recombination component is needed to describe the high\(p_{\mathrm {T}}\) \(\mathrm {J}/\psi \) results.
In Fig. 90 (left), the prompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) is compared to shadowing calculations. As already discussed for low\(p_{\mathrm {T}}\) \(\mathrm {J}/\psi \) results, shadowing, here considered as the only cold nuclear matter effect, cannot account for the observed suppression, clearly indicating that other cold or hot matter effects are needed to describe the experimental results.
In Fig. 90 (right), the centrality dependence of the prompt \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\) is compared to the collisional dissociation model, discussed in Sect. 5.1.6. The model describes the more peripheral events, but underestimates the suppression for the most central events. It also underestimate the \(p_{\mathrm {T}}\) dependence of the \(\mathrm {J}/\psi \) \(R_{\mathrm {AA}}\).
The CMS Collaboration also measured the \(R_{\mathrm {AA}}\) of nonprompt \(\mathrm {J}/\psi \), presented in Sect. 4.1.3.
5.2.4 \(\mathrm J/\psi \) azimuthal anisotropy
Further information on the \(\mathrm {J}/\psi \) production mechanism can be accessed by studying the azimuthal distribution of \(\mathrm {J}/\psi \) with respect to the reaction plane. As discussed in Sect. 4, the positive \(v_2\) measured for D mesons at LHC and heavyflavour decay electrons at RHIC suggests that charm quarks participate in the collective expansion of the medium and do acquire some elliptic flow as a consequence of the multiple collisions with the medium constituents. \(\mathrm {J}/\psi \) produced through a recombination mechanism, should inherit the elliptic flow of the charm quarks in the QGP and, as a consequence, \(\mathrm {J}/\psi \) are expected to exhibit a large \(v_{2}\). Hence this quantity is a further signature to identify the charmonium production mechanism.
ALICE measured the inclusive \(\mathrm {J}/\psi \) elliptic flow in Pb–Pb collisions at forward rapidity [681], using the eventplane technique. For semicentral collisions there is an indication of a positive \(v_{2}\), reaching \(v_{2} = 0.116\pm 0.046\,\text {(stat.)}\pm 0.029\,\text {(syst.)}\) in the transverse momentum range \(2<p_{\mathrm {T}} <4~\text {GeV}/c \), for events in the 20–40 % centrality class. In Fig. 91 (left), the \(\mathrm {J}/\psi \) \(v_{2}\) in the 20–60 % centrality class is compared with the TAMU and THU transport model calculations, which also provide a fair description of the \(R_{\mathrm {AA}}\) results, discussed in Sect. 5.2.2. Both models, which reasonably describe the data, include a fraction (\(\approx \)30 % in the centrality range 20–60 %) of \(\mathrm {J}/\psi \) produced through (re)generation mechanisms, under the hypothesis of thermalisation or nonthermalisation of the bquarks. More in detail, charm quarks, in the hot medium created in Pb–Pb collisions at the LHC, should transfer a significant elliptic flow to regenerated \(\mathrm {J}/\psi \). Furthermore, primordial \(\mathrm {J}/\psi \) might acquire a \(v_{2}\) induced by a pathlength dependent suppression due to the fact that \(\mathrm {J}/\psi \) emitted outofplane traverse a longer path through the medium than those emitted inplane. Thus, outofplane emitted \(\mathrm {J}/\psi \) will spend a longer time in the medium and have a higher chance to melt. The predicted maximum \(v_{2}\) at \(p_{\mathrm {T}}\) = 2.5\(~\text {GeV}/c\) is, therefore, the result of an interplay between the regeneration component, dominant at low \(p_{\mathrm {T}}\) and the primordial \(\mathrm {J}/\psi \) component which takes over at high \(p_{\mathrm {T}}\) (see Fig. 82). The \(v_{2}\) measurement complements the \(R_{\mathrm {AA}}\) results, favouring a scenario with a significant fraction of \(\mathrm {J}/\psi \) produced by (re)combination in the ALICE kinematical range.
CMS has investigated the prompt \(\mathrm {J}/\psi \) \(v_{2}\) as a function of the centrality of the collisions and as a function of transverse momentum [685]. Preliminary results indicate a positive \(v_{2}\). The observed anisotropy shows no strong centrality dependence when integrated over rapidity and \(p_{\mathrm {T}}\). The \(v_{2}\) of prompt \(\mathrm {J}/\psi \), measured in the 10–60 % centrality class, has no significant \(p_{\mathrm {T}}\) dependence either, whether it is measured at low \(p_{\mathrm {T}}\), \(3<p_{\mathrm {T}} <6.5~\text {GeV}/c \), in the forwardrapidity interval \(1.6<y<2.4\), or at high \(p_{\mathrm {T}}\), \(6.5<p_{\mathrm {T}} <30~\text {GeV}/c \), in the rapidity interval \(y<2.4\). The preliminary CMS result supports the presence of a small anisotropy over the whole \(p_{\mathrm {T}}\) range, but the present level of precision does not allow for a definitive answer on whether this anisotropy is constant or not. In the rapidity interval \(y<2.4\), for \(p_{\mathrm {T}} >8~\text {GeV}/c \), the anisotropy is similar to that observed for charged hadrons, the latter being attributed to the pathlength dependence of the partonic energy loss [758].
5.2.5 \(\mathrm J/\psi \) \(R_{\mathrm {AA}}\) results for various colliding systems and beam energies at RHIC
A unique feature of RHIC is the possibility of accelerating various symmetric or asymmetric ion species, allowing for the study of charmonium suppression as a function of the system size. Furthermore, since at RHIC it is possible to collect data at various \(\sqrt{s_{\mathrm{NN}}}\), the charmonium production beamenergy dependence was also investigated from the top energy \(\sqrt{s_{\mathrm{NN}}}\) = 200\(~\text {GeV}\) down to \(\sqrt{s_{\mathrm{NN}}}\) = 39\(~\text {GeV}\).
The PHENIX Collaboration measured \(\mathrm {J}/\psi \) production from asymmetric Cu–Au heavyion collisions at \(\sqrt{s_{\mathrm{NN}}}\) = 200 \(~\text {GeV}\) at both forward (Cugoing direction) and backward (Augoing direction) rapidities [672]. The nuclear modification of \(\mathrm {J}/\psi \) yields in Cu–Au collisions in the Augoing direction is found to be comparable to that in Au–Au collisions when plotted as a function of the number of participating nucleons, as shown in Fig. 92 (left). In the Cugoing direction, \(\mathrm {J}/\psi \) production shows a stronger suppression. This difference is comparable to expectation from nPDF effects due to stronger lowx gluon suppression in the larger Au nucleus.
Moreover, the PHENIX Collaboration measured nuclear modification factors also by varying the collision energies, studying Au–Au data at \(\sqrt{s_{\mathrm{NN}}}\) = 39 and 62.4\(~\text {GeV}\) [674]. The observed suppression patterns follow a trend very similar to those previously measured at \(\sqrt{s_{\mathrm{NN}}}\) = 200\(~\text {GeV}\), as shown in Fig. 92 (right). Similar conclusions can be drawn also from preliminary STAR results [678].
In spite of the large uncertainties associated to these results, up to now, this similarity presents a challenge to theoretical models that contain competing hot and cold matter effects with possibly different energy dependencies. For example, in the TAMU transport model [720], the larger