Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions

The European Physical Journal C, Feb 2016

This report reviews the study of open heavy-flavour and quarkonium production in high-energy 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.

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Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions

Eur. Phys. J. C Heavy-flavour and quarkonium production in the LHC era: from proton-proton to heavy-ion collisions A. Andronic F. Arleo R. Arnaldi A. Beraudo E. Bruna D. Caffarri Z. Conesa del Valle J. G. Contreras T. Dahms A. Dainese M. Djordjevic E. G. Ferreiro H. Fujii P.-B. Gossiaux R. Granier de Cassagnac C. Hadjidakis M. He H. van Hees W. A. Horowitz R. Kolevatov B. Z. Kopeliovich J.-P. Lansberg M. P. Lombardo C. Lourenço G. Martinez-Garcia L. Massacrier C. Mironov A. Mischke M. Nahrgang M. Nguyen J. Nystrand S. Peigné S. Porteboeuf-Houssais I. K. Potashnikova A. Rakotozafindrabe R. Rapp P. Robbe M. Rosati P. Rosnet H. Satz R. Schicker I. Schienbein I. Schmidt E. Scomparin R. Sharma J. Stachel D. Stocco M. Strickland R. Tieulent B. A. Trzeciak J. Uphoff I. Vitev R. Vogt K. Watanabe H. Woehri P. Zhuang Research Division ExtreMe Matter Institute (EMMI) GSI Helmholzzentrum für Schwerionenforschung Darmstadt Germany Laboratoire Leprince-Ringuet Ecole Polytechnique CNRS/IN Université Paris-Saclay Palaiseau France Université de Savoie Annecy-le-Vieux France Sezione di Torino Turin Italy European Organization for Nuclear Research (CERN) Geneva Switzerland Univ. Paris-Sud CNRS/IN Université Paris-Saclay Orsay Cedex France Faculty of Nuclear Sciences Physical Engineering Czech Technical University in Prague Prague Czech Republic Excellence Cluster Universe Technische Universität München Munich Germany Sezione di Padova Padua Italy Institute of Physics Belgrade University of Belgrade Belgrade Serbia Departamento de Física de Partículas IGFAE Universidad de Santiago de Compostela Santiago de Compostela Spain Institute of Physics University of Tokyo Tokyo Japan SUBATECH Ecole des Mines de Nantes Université de Nantes CNRS-IN Nantes France Department of Applied Physics Nanjing University of Science Technology Nanjing China Institute for Theoretical Physics Frankfurt Germany Department of Physics University of Cape Town Cape Town South Africa Department of High Energy Physics Saint-Petersburg State University Ulyanovskaya Saint Petersburg Russia Departamento de Física Centro Científico-Tecnológico de Valparaíso Universidad Técnica Federico Santa María Valparaiso Chile Laboratori Nazionali di Frascati Frascati Italy Univ. Paris-Sud CNRS/IN Université Paris-Saclay Orsay France Faculty of Science Institute for Subatomic Physics Utrecht University Utrecht The Netherlands National Institute for Subatomic Physics Amsterdam The Netherlands Department of Physics Duke University Durham Department of Physics Technology University of Bergen Bergen Norway Laboratoire de Physique Corpusculaire (LPC) Université Clermont Auvergne Université Blaise Pascal CNRS/IN Clermont-Ferrand France IRFU/SPhN CEA Saclay Gif-sur-Yvette Cedex France Department of Physics Astronomy Cyclotron Institute Texas A M University College Station Iowa State University Fakultät für Physik Universität Bielefeld Bielefeld Germany Physikalisches Institut Ruprecht-Karls-Universität Heidelberg Heidelberg Germany Université Grenoble-Alpes CNRS/IN Grenoble France Department of Theoretical Physics Tata Institute of Fundamental Research Mumbai India Department of Physics Kent State University IPN-Lyon Université de Lyon Université Lyon CNRS/IN Villeurbanne France Institut für Theoretische Physik Johann Wolfgang Goethe-Universität Frankfurt am Main Germany Theoretical Division Los Alamos National Laboratory Los Alamos Physics Division Lawrence Livermore National Laboratory Livermore Physics Department University of California Davis - 39Institute of Physics, University of Tokyo, Tokyo, Japan 40Key Laboratory of Quark and Lepton Physics (MOE), Institute of Particle Physics, Central China Normal University, Wuhan, China 41Physics Department, Collaborative Innovation Center of Quantum Matter, Tsinghua University, Beijing, China Abstract This report reviews the study of open heavyflavour and quarkonium production in high-energy 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 7th Framework Programme. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . 3 2 Heavy flavour and quarkonium production in proton– proton collisions . . . . . . . . . . . . . . . . . . . 4 2.1 Production mechanisms of open and hidden heavy-flavour in proton–proton collisions . . . 4 2.1.1 Open-heavy-flavour production . . . . . 4 2.1.2 Quarkonium-production mechanism . . . 8 2.2 Recent cross section measurements at hadron colliders . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Leptons from heavy-flavour decays . . . 11 2.2.2 Open charm . . . . . . . . . . . . . . . . 12 2.2.3 Open beauty . . . . . . . . . . . . . . . 13 2.2.4 Prompt charmonium . . . . . . . . . . . 15 2.2.5 Bottomonium . . . . . . . . . . . . . . . 18 2.2.6 Bc and multiple-charm baryons . . . . . 19 2.3 Quarkonium-polarisation studies . . . . . . . . 19 2.4 New observables . . . . . . . . . . . . . . . . 24 2.4.1 Production as a function of multiplicity . 24 A. Andronic, F. Arleo, R. Arnaldi, Z. C. del Valle, J. G. Contreras, T. Dahms, A. Dainese, E. G. Ferreiro, P.-B. Gossiaux, C. Hadjidakis, J.-P. Lansberg, G. Martinez-Garcia, L. Massacrier, J. Nystrand, S. Porteboeuf-Houssais, P. Robbe, P. Rosnet, R. Schicker, I. Schienbein, D. Stocco, R. Tieulent and B. A. Trzeciak are section editors. 2.4.2 Associated production . . . . . . . . . . 26 2.5 Summary and outlook . . . . . . . . . . . . . . 30 3 Cold nuclear matter effects on heavy flavour and quarkonium production in proton–nucleus collisions 31 3.1 Heavy flavour in p–A collisions . . . . . . . . 31 3.2 Theoretical models for CNM effects . . . . . . 32 3.2.1 Typical time scales . . . . . . . . . . . . 32 3.2.2 Nuclear PDFs . . . . . . . . . . . . . . . 33 3.2.3 Saturation in the colour glass condensate approach . . . . . . . . . . . . . . . . . 37 3.2.4 Multiple scattering and energy loss . . . . 37 3.2.5 Nuclear absorption . . . . . . . . . . . . 40 3.2.6 Summary of CNM models . . . . . . . . 41 3.3 Recent RHIC and LHC results . . . . . . . . . 41 3.3.1 Reference for p–A measurements at the LHC . . . . . . . . . . . . . . . . . . . . 41 3.3.2 Open heavy-flavour measurements . . . . 42 3.3.3 Quarkonium measurements . . . . . . . . 47 3.4 Extrapolation of CNM effects from p–A to AA collisions . . . . . . . . . . . . . . . . . . . . 54 3.5 Summary and outlook . . . . . . . . . . . . . . 56 4 Open heavy flavour in nucleus–nucleus collisions . . 57 4.1 Experimental overview: production and nuclear modification factor measurements . . . . . . . 59 4.1.1 Inclusive measurements with leptons . . . 59 4.1.2 D meson measurements . . . . . . . . . 61 4.1.3 Beauty production measurements . . . . 63 4.1.4 Comparison of RAA for charm, beauty and light-flavour hadrons . . . . . . . . . . . 64 4.2 Experimental overview: azimuthal anisotropy measurements . . . . . . . . . . . . . . . . . . 65 4.2.1 Inclusive measurements with electrons . . 65 4.2.2 D-meson measurements . . . . . . . . . 66 4.3 Theoretical overview: heavy flavour interactions in the medium . . . . . . . . . . . . . . . 67 4.3.1 pQCD energy loss in a dynamical QCD medium . . . . . . . . . . . . . . . . . . 68 4.3.2 A pQCD-inspired running αs energy-loss model in MC@s HQ and BAMPS . . . . 70 4.3.3 Collisional dissociation of heavy mesons and quarkonia in the QGP . . . . . . . . 72 4.3.4 T -matrix approach to heavy-quark inter actions in the QGP . . . . . . . . . . . . 73 4.3.5 Lattice-QCD . . . . . . . . . . . . . . . 73 4.3.6 Heavy-flavour interaction with medium in AdS/CFT . . . . . . . . . . . . . . . . . 75 4.4 Theoretical overview: medium modelling and medium-induced modification of heavy-flavour production . . . . . . . . . . . . . . . . . . . . 76 4.4.1 pQCD energy loss in a static fireball (Djordjevic et al.) . . . . . . . . . . . . . 76 4.4.2 pQCD embedded in viscous hydro (POWLANG and Duke) . . . . . . . . . . . . . . . . . 76 4.4.3 pQCD-inspired energy loss with running αs in a fluid-dynamical medium and in Boltzmann transport . . . . . . . . . . . 77 4.4.4 Non-perturbative T -matrix approach in a fluid-dynamic model (TAMU) and in UrQMD transport . . . . . . . . . . . . . 78 4.4.5 Lattice-QCD embedded in viscous fluid dynamics (POWLANG) . . . . . . . . . 79 4.4.6 AdS/CFT calculations in a static fireball . 79 4.5 Comparative overview of model features and comparison with data . . . . . . . . . . . . . . 79 4.6 Heavy-flavour correlations in heavy-ion collisions: status and prospects . . . . . . . . . . . 85 4.7 Summary and outlook . . . . . . . . . . . . . . 87 5 Quarkonia in nucleus–nucleus collisions . . . . . . 88 5.1 Theory overview . . . . . . . . . . . . . . . . 92 5.1.1 Sequential suppression and lattice QCD . 92 5.1.2 Effect of nuclear PDFs on quarkonium production in nucleus–nucleus collisions 94 5.1.3 Statistical (re)generation models . . . . . 96 5.1.4 Transport approach for in-medium quarkonia . . . . . . . . . . . . . . . . . . . . . 98 5.1.5 Non-equilibrium effects on quarkonium suppression . . . . . . . . . . . . . . . . 100 5.1.6 Collisional dissociation of quarkonia from final-state interactions . . . . . . . . . . 101 5.1.7 Comover models . . . . . . . . . . . . . 101 5.1.8 Summary of theoretical models for exper imental comparison . . . . . . . . . . . . 102 5.2 Experimental overview of quarkonium results at RHIC and LHC . . . . . . . . . . . . . . . . . 103 5.2.1 Proton–proton collisions as a reference for RAA at the LHC . . . . . . . . . . . . 103 5.2.2 J/ψ RAA results at low pT . . . . . . . . 103 5.2.3 J/ψ RAA results at high pT . . . . . . . . 106 5.2.4 J/ψ azimuthal anisotropy . . . . . . . . 108 5.2.5 J/ψ RAA results for various colliding systems and beam energies at RHIC . . . 109 5.2.6 Excited charmonium states . . . . . . . . 109 5.2.7 Bottomonium RAA results . . . . . . . . 110 5.3 Alternative references for quarkonium production in nucleus–nucleus collisions . . . . . . . 113 5.3.1 Proton–nucleus collisions . . . . . . . . 113 5.3.2 Open heavy flavour . . . . . . . . . . . . 113 5.4 Summary and outlook . . . . . . . . . . . . . . 115 6 Quarkonium photoproduction in nucleus–nucleus collisions . . . . . . . . . . . . . . . . . . . . . . . 115 6.1 The flux of photons from lead ions at the LHC . 116 6.2 Measurements of photonuclear production of charmonium during the Run 1 at the LHC . . . 117 6.2.1 Photonuclear production of J/ψ at RHIC 118 6.2.2 Coherent production of J/ψ in Pb–Pb UPC at the LHC . . . . . . . . . . . . . 118 6.2.3 Coherent production of ψ (2S) in Pb–Pb UPC at the LHC . . . . . . . . . . . . . 119 6.2.4 Incoherent production of J/ψ in Pb–Pb UPC at the LHC . . . . . . . . . . . . . 119 6.2.5 Coherent photonuclear production of J/ψ in coincidence with a hadronic Pb–Pb collision at the LHC . . . . . . . . . . . . . 119 6.3 Models for photonuclear production of charmonium . . . . . . . . . . . . . . . . . . . . . . . 119 6.3.1 Models based on vector dominance . . . 119 6.3.2 Models based on LO pQCD . . . . . . . 120 6.3.3 Models based on the colour dipole approach120 6.4 Photonuclear production of charmonium: comparing models to measurements . . . . . . . . . 121 6.5 Summary and outlook . . . . . . . . . . . . . . 122 7 Upgrade programmes and planned experiments . . . 122 7.1 Introduction . . . . . . . . . . . . . . . . . . . 122 7.2 Collider experiments . . . . . . . . . . . . . . 122 7.2.1 The LHC upgrade programme . . . . . . 122 7.2.2 The RHIC programme . . . . . . . . . . 126 7.3 The fixed-target experiments . . . . . . . . . . 127 7.3.1 Low energy projects at SPS, Fermilab and FAIR . . . . . . . . . . . . . . . . . . . 127 7.3.2 Plans for fixed-target experiments using the LHC beams . . . . . . . . . . . . . . 128 8 Concluding remarks . . . . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . 131 1 Introduction Heavy-flavour 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). Heavy-flavour production in pp collisions provides important tests of our understanding of various aspects of QCD. The heavy-quark mass acts as a long distance cut-off so that the partonic hard-scattering process can be calculated in the framework of perturbative QCD down to low transverse momenta ( pT). When the heavy-quark pair forms a quarkonium bound state, this process is non-perturbative as it involves long distances and soft momentum scales. Therefore, the detailed study of heavy-flavour production and the comparison to experimental data provides an important testing ground for both perturbative and non-perturbative aspects of QCD calculations. In nucleus–nucleus collisions, open and hidden heavyflavour production constitutes a sensitive probe of the hot strongly interacting medium, because hard-scattering 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 medium-induced effects and relating them to its properties requires an accurate study of the so-called 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 break-up 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 ultra-peripheral 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 high-energy 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 medium-induced gluon radiation) and elastic (collisional energy loss) processes. Energy loss is expected to depend on the parton colour-charge and mass. Therefore, charm and beauty quarks provide important tools to investigate the energy-loss mechanisms, in addition to the QGP properties. Furthermore, low- pT 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 cc (bb) state. In this scenario, quarkonium suppression should occur sequentially, according to the binding energy of each state. As a consequence, the in-medium dissociation probability of these states are expected to provide an estimate of the initial temperature reached in the collisions. At high centre-of-mass energy, a new production mechanism could be at work in the case of charmonium: the abundance of c and 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 centre-of-mass energies √s from 2.76 to 8 TeV, in p–Pb collisions at √sNN = 5.02 TeV per nucleon–nucleon interaction, in Pb–Pb collisions at √sNN = 2.76 TeV, as well as in photon-induced collisions. In the case of heavy-ion 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 heavy-flavour 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 network1 of the I3 Hadron Physics programme of the European Union 7th 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: heavy-flavour and quarkonium production in proton–proton collisions, the cold nuclear matter effects on heavy-flavour 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 photon-induced collisions. Sect. 7 presents an outlook of future heavy-flavour 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 heavy-flavour in proton–proton collisions 2.1.1 Open-heavy-flavour production Open-heavy-flavour production in hadronic collisions provides important tests of our understanding of various aspects 1 https://twiki.cern.ch/twiki/bin/view/ReteQuarkonii/SaporeGravis. 2 https://indico.cern.ch/event/247609. 3 https://indico.cern.ch/event/305164. of Quantum Chromodynamics (QCD). First of all, the heavyquark mass (m Q ) acts as a long distance cut-off so that this process can be calculated in the framework of perturbative QCD down to low pT and it is possible to compute the total cross section by integrating over pT. Second, the presence of multiple hard scales (m Q , pT) allows us to study the perturbation series in different kinematic regions ( pT < m Q , pT ∼ m Q , pT 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 heavy-flavour 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 open-heavy-flavour production is sensitive to the gluon and the heavy-quark content in the nucleon, so that LHC data in pp and p–Pb collisions can provide valuable constraints on these parton-distribution functions (PDFs) inside the proton and the lead nucleus, respectively. In addition, these cross sections in pp and p–A collisions establish the baseline for the study of heavy-quark production in heavyion collisions. This aspect is a central point in heavy-ion physics since the suppression of heavy quarks at large pT is an important signal of the QGP (see Sect. 4). Finally, let us also mention that a solid understanding of open-charm production is needed in cosmic-ray and neutrino astrophysics [ 1 ]. In the following, we will focus on pp collisions and review the different theoretical approaches to open-heavy-flavour production. Fixed-Flavour-Number Scheme Conceptually, the simplest scheme is the Fixed-Flavour-Number Scheme (FFNS) where the heavy quark is not an active parton in the proton. Relying on a factorisation theorem, the differential cross section for the inclusive production of a heavy quark Q can be calculated as follows: dσ Q+X [s, pT, y, m Q ] 1 1 i, j 0 dxi 0 or in short dσ Q+X i, j ×dσ˜i j→Q+X [xi , x j , s, pT, y, m Q , μF , μR ], f A i ⊗ f jB ⊗ dσ˜i j→Q+X , d x j fiA(xi , μF ) f jB (x j , μF ) (1) (2) where pT and y are the transverse momentum and the rapidity of the heavy quark and s is the square of the hadron centreof-mass energy. The PDFs fiA ( f jB ) give the number density of the parton of flavour ‘i ’ (‘ j ’) inside the hadron ‘ A’ (‘B’) carrying a fraction xi (x j ) of the hadron momentum at the factorisation scale μF . Furthermore, the short-distance cross section dσ˜ is the partonic cross section from which the socalled mass singularities or collinear singularities associated to the light quarks and the gluon have been removed via the mass-factorisation procedure and which therefore also depends on μF . The partonic cross section also depends on the strong-coupling constant αs , which is evaluated at the renormalisation scale μR . As a remainder of this procedure, the short-distance cross section will depend on logarithms of the ratio of μF with the hard scale. In order to avoid large logarithmic contributions, the factorisation scale μF should be chosen in the vicinity of the hard scale. Also the renormalisation scale μR is determined by the hard scale. The tilde is used to indicate that the finite collinear logarithms of the heavy-quark mass present in the partonic cross section have not been removed by the mass-factorisation procedure. These logarithms are therefore not resummed to all orders in the FFNS but are accounted for in Fixed-Order (FO) perturbation theory. The error of the approximation in Eq. (1) is suppressed by an inverse power of the hard scale which is set by the mass or the transverse momentum of the heavy quark, i.e. it is on the order of O(( /μF ) p) where ∼ 200 MeV is a typical hadronic scale, and p = 1 or 2. In Eq. (1), a sum over all possible sub-processes i + j → Q + X is understood, where i, j are the active partons in the proton: i, j ∈ {q, q = (u, u, d, d, s, s), g} for a FFNS with three active flavours (3-FFNS) usable for both charm and beauty production, and i, j ∈ {q, q = (u, u, d, d, s, s, c, 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 μF > mc) and the charm-quark mass is neglected in the hard-scattering cross section dσ˜ whereas the beauty quark mass mb is retained. At the leading order (LO) in αS, there are only two subprocesses which contribute: (i) q + q → Q + Q, (ii) g + g → Q + Q. At the next-to-leading order (NLO), the virtual one-loop corrections to these 2 → 2 processes have to be included in addition to the following 2 → 3 processes: (i) q + q → Q + Q + g, (ii) g + g → Q + Q + g, (iii) g + q → q + Q + Q and g + q → q + Q + Q. Complete NLO calculations of the integrated/total cross section and of one-particle inclusive distributions were performed in the late 80s [ 2–5 ]. 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, GM-VFNS, POWHEG, MC@NLO). The typical range of applicability of the FFNS at NLO is roughly 0 ≤ pT 5×m Q . A representative comparison with data has been made for B+ production in [ 7 ] where it is clear that the predictions of the FFNS at NLO using the branching fraction B(b → B) = 39.8 % starts to overshoot the Tevatron data for pT 15 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 value4 μF = μR = mT with mT = m2Q + pT2. Such a kind of discrepancies at increasing pT can be attributed to the shift of the momentum between the b quark and the B meson which can be accounted for by a fragmentation function (FF). Indeed, the scope of the FFNS can be extended to slightly larger pT by convolving the differential cross section for the production of the heavy quark Q with a suitable, scale-independent, FF DQH (z) describing the transition of the heavy quark with momentum pQ into the observed heavy-flavoured hadron H with momentum pH = z pQ (see [ 7 ]): dσ H = dσ Q ⊗ DQH (z). At large transverse momenta, the differential cross section falls off with a power-like behaviour dσ Q /d pT ∝ 1/ pTn with n = 4, 5 so that the convolution with the fragmentation function (FF) effectively corresponds to a multiplication with the fourth or fifth Mellin moment of this FF which lowers the cross section and leads to an improved agreement with the data at large pT. It should be noted that this FF is included on a purely phenomenological basis and there are ambiguities on how the convolution prescription is imple T = z pQ ) leading to differences mented (E H = z E Q , p H T at pT m Q . Furthermore, at NLO, a harder FF should be used than at LO in order to compensate for the softening effects of the gluon emissions. Apart from this, it is generally believed that this scale-independent FF is universal and can be extracted from data, e.g. from e+e− data. The same conclusions about the range of applicability of the FFNS apply at the LHC where the heavy-quark production is dominated by the gg-channel (see, e.g. Figure 3 in [ 7 ]) over the qq 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. ZM-VFNS For pT m Q , the logarithms of the heavyquark mass ( 2απs ln( pT2/m2Q )) become large and will eventually have to be resummed to all orders in the perturbation theory. This resummation is realised by absorbing the large logarithmic terms into the PDFs and FFs whose scaledependence is governed by renormalisation group equations, 4 We stress here that this widespread procedure to assess theoretical uncertainties associated with pQCD computations does not provide values which should be interpreted as coming from a statistical procedure. This is only an estimate of the missing contributions at higher-order QCD corrections. dxi 0 1 dx j 0 1 dz fiA(xi , μF ) the DGLAP evolution equations. This approach requires that the heavy quark is treated as an active parton for factorisation scales μF ≥ μT where the transition scale μT is usually (for simplicity) identified with the heavy-quark mass. Such a scheme, where the number of active flavours is changed when crossing the transition scales is called a Variable-FlavourNumber Scheme (VFNS). If, in addition, the heavy-quark mass m Q is neglected in the calculation of the short-distance cross sections, the scheme is called Zero-Mass VFNS (ZMVFNS). The theoretical foundation of this scheme is provided by a well-known factorisation theorem and the differential cross section for the production of a heavy-flavoured hadron ( A + B → H + X ) is calculated as follows: × f jB (x j , μF )dσˆi j→k+X DkH (z, μF ) + O(m2Q / pT2). (4) Because the heavy-quark mass is neglected in the shortdistance cross sections (dσˆ ), the predictions in the ZM-VFNS are expected to be reliable only for very large transverse momenta. The sum in Eq. (4) extends over a large number of sub-processes i + j → k + X since a, b, c can be gluons, light quarks, and heavy quarks. A calculation of all sub-processes at NLO has been performed in the late 1980s [ 10 ]. Concerning the FFs into the heavy-flavoured hadron H = D, B, c, . . ., two main approaches are employed in the literature: • In the Perturbative-Fragmentation Functions (PFF) approach [ 11 ], the FF DkH (z, μF ) is given by a convolution of a PFF accounting for the fragmentation of the parton k into the heavy quark Q, DkQ (z, μF ), with a scaleindependent FF DQH (z) describing the hadronisation of the heavy quark into the hadron H : DkH (z, μF ) = DkQ (z, μF ) ⊗ DQH (z). (5) The PFFs resum the final-state collinear logarithms of the heavy-quark mass. Their scale-dependence 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 non-perturbative object (in the case of heavy-light-flavoured 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 scale-independent fragmentation function in Eq. (3) with the one in Eq. (5). This function describing the hadronisation process involves long-distance physics and might be modified in the presence of a QGP, whereas the PFFs (or the unresummed collinear logarithms ln pT2/m2Q in the FFNS) involve only short-distance physics and are the same in pp, p–A, and AA collisions. • In the Binnewies–Kniehl–Kramer (BKK) approach [ 12– 14 ], the FFs are not split up into a perturbative and a non-perturbative piece. Instead, boundary conditions at an initial scale μF m Q are determined from e+e− data for the full non-perturbative FFs, DkH (z, μF ), in complete analogy with the treatment of FFs into light hadrons (pions, kaons). These boundary conditions are again evolved to larger scales μF with the help of the DGLAP equations. It is also noteworthy that the BKK FFs (D(z, μF )) are directly determined as functions in z-space whereas the FFs in the PFF approach are determined in Mellin-N-space where the Nth Mellin moment of a function f (z) (0 < z < 1) is defined as f (N ) = 01 dz z N −1 f (z). GM-VFNS The FFNS and the ZM-VFNS are valid only in restricted and complementary regions of the transverse momentum. For this reason, it is crucial to have a unified framework which combines the virtues of the massive FO calculation in the FFNS and the massless calculation in the ZM-VFNS. The General-Mass VFNS (GM-VFNS) [ 15,16 ] is such a framework which is valid in the entire kinematic range from the smallest to the largest transverse momenta ( pT m Q , pT m Q , pT m Q ). It is very similar to the ACOT heavyflavour scheme [ 17,18 ] which has been formulated for structure functions in deep inelastic scattering (DIS). Different variants of the ACOT scheme exist like the S-ACOT scheme [ 19 ] and the (S)-ACOTχ scheme [ 20 ] which are used in global analyses of PDFs by the CTEQ Collaboration and the ACOT scheme has been extended to higher orders in Refs. [ 21–23 ]. The theoretical basis for the ACOT scheme has been laid out in an all-order proof of a factorisation theorem with massive quarks by Collins [ 24 ]. While the discussion in [ 24 ] deals with inclusive DIS, it exemplifies the general principles for the treatment of heavy quarks in perturbative QCD (see also [ 25,26 ]) which should be applicable to other processes as well. Therefore, it is very important to test these ideas also in the context of less inclusive observables. First steps in this direction had been undertaken in [ 27,28 ] where the ACOT scheme had been applied to inclusive D-meson production in DIS. The case of hadroproduction in the ACOT scheme had been studied for the first time in [ 29 ] taking into account the contributions from the NLO calculation in the FFNS combined with the massless contributions in the ZM-VFNS from all other sub-processes at O(αs2) resumming the collinear logarithms associated to the heavy quark at the leading-logarithmic (LL) accuracy. In contrast, the GM-VFNS has a NLO+NLL accuracy. It has been worked out for γ γ , pp, pp, e+e−, ep, and γ p collisions in a series of papers [ 7,15,30–38 ] and has been successfully compared to experimental data from LEP, HERA, (6) Tevatron and the LHC. Furthermore, inclusive lepton spectra from heavy-hadron decays have been studied for pp collisions at the LHC at 2.76 and 7 TeV centre-of-mass energy [ 39 ] and compared to data from ALICE, ATLAS and CMS. In addition, predictions have been obtained for D mesons produced at √s = 7 TeV from B decays [ 40 ]. A number of comparisons with hadroproduction data are discussed in Sect. 2.2. The cross section for inclusive heavy-flavour hadroproduction in the GM-VFNS is calculated using a factorisation formula similar to the one in Eq. (4): dxi 0 1 dx j 0 1 dz × fiA(xi , μF ) f jB (x j , μF )dσˆi j→k+X [ pT, m Q ]DkH (z, μF ). In particular, the same sub-processes as in the ZM-VFNS are taken into account. However, the finite heavy-quark-mass terms (powers of m2Q / pT2) are retained in the short-distance cross sections of sub-processes involving heavy quarks. More precisely, the heavy-quark-mass terms are taken into account in the sub-processes q + q → Q + X , g + g → Q + X , g + q → Q + X and g + q → Q + X , which are also present in the FFNS. However, in the current implementation, they are neglected in the heavy-quark-initiated subprocesses (Q + g → Q + X , Q + g → g + X , …) as is done in the S-ACOT scheme [ 19 ]. The massive hard-scattering cross sections are defined in a way that they approach, in the limit m Q / pT → 0, the massless hard-scattering cross sections defined in the MS scheme. Therefore, the GM-VFNS approaches the ZM-VNFS at large pT m Q . It can be shown that the GM-VFNS converges formally to the FFNS at small pT. However, while the S-ACOT scheme works well for the computation of DIS structure functions at NLO, this scheme causes problems in the hadroproduction case at low pT because the massless b-quark initiated cross sections diverge in the limit pT → 0. This problem can be circumvented by a suitable choice for the factorisation scale so that the heavy-quark PDF is switched off sufficiently rapidly and the GM-VFNS approaches the FFNS at small pT [ 7 ]. FONLL Similar to the GM-VFNS, the Fixed-Order plus Next-to-Leading Logarithms (FONLL) approach [ 41 ] is a unified framework which is valid in the entire kinematic range ( pT m Q , pT m Q , pT m Q ). This approach has also been applied to DIS structure functions and is used in the global analyses of PDFs by the NNPDF Collaboration [ 42,43 ]. Predictions for c and b quark production at the LHC with a centre-of-mass energy of 7 TeV have been presented in [44]. The FONLL scheme is based on the matching of the massive NLO cross section in the FFNS (=FO) with the massless NLO calculation in the ZM-VNFS (=RS) according to the prescription dσFONLL = dσFO + G(m Q , pT) × (dσRS − dσFOM0) (7) where dσFOM0 is the cross section dσFO in the asymptotic limit pT m Q where the finite power-like mass terms can be neglected and the cross section is dominated by the collinear logarithm of the heavy-quark mass. The condition dσFONLL → dσRS for pT m Q implies that the matching function G(m Q , pT ) has to approach unity in this limit. Furthermore, in the limit of small transverse momenta dσFONLL has to approach the fixed-order calculation dσFO. This can be achieved by demanding that G(m Q , pT ) → 0 for pT → 0, which effectively suppresses the contribution from the divergent b-quark initiated contributions in dσRS. In the FONLL, the interpolating function is chosen to be G(m Q , pT) = pT2/( pT2 + a2m2Q ) where the constant is set to a = 5 on phenomenological grounds. In this language the GM-VFNS is given by dσGM−VFNS = dσFO + dσRS − dσFOM0, i.e. no interpolating factor is used. Other differences concern the non-perturbative input. In particular, the FONLL scheme uses fragmentation functions in the PFF formalism whereas the GM-VFNS uses fragmentation functions which are determined in the z-space in the BKK approach. Monte Carlo generators The GM-VFNS 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, general-purpose Monte-Carlo 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 heavy-quark production in pp collisions at the LHC with the ones from the GM-VFNS and FONLL can be found in [ 49 ]. 2.1.2 Quarkonium-production mechanism The theoretical study of quarkonium-production processes involves both pertubative and non-perturbative aspects of QCD. On one side, the production of the heavy-quark pair, Q 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 open-heavy-flavour production discussed in the previous section. On the other side, the evolution of the Q Q pair into the physical quarkonium state is non-perturbative, over long distances, with typical momentum scales such as the momentum of the heavy-quarks in the bound-state rest frame, m Q v and their binding energy m Q v2, v being the typical velocity of the heavy quark or antiquark in the quarkonium rest frame (v2 ∼ 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 heavy-quark-pair production. In the following, we briefly describe three of them which can be distinguished in their treatment of the non-perturbative part: the Colour-Evaporation Model (CEM), the Colour-Singlet Model (CSM), the Colour-Octet Mechanism (COM), the latter two being encompassed in an effective theory referred to as Non-Relativistic QCD (NRQCD). The Colour-Evaporation 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 Q pair in an invariant-mass 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 open-heavy-flavour 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 FQ related to a process-independent probability that the pair eventually hadronises into this state. One assumes that a number of non-perturbative-gluon emissions occur once the Q Q pair is produced and that the quantum state of the pair at its hadronisation is essentially decorrelated – at least colour-wise – with that at its production. From the reasonable assumption [ 52 ] that one ninth – one colour-singlet Q 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 JQ of the quarkonium Q, FQ = 1/9 × (2 JQ + 1)/ i (2 Ji + 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 J/ψ hadroproduction data of the late 1990s and, in fact, is comparable to the fit value in [ 53 ]. Mathematically, one has σ Q(N)LO = FQ 2m Q 2m H dσ (N)LO Q Q dm Q Q dm Q Q (8) In the latter formula, a factorisation between the shortdistance Q Q-pair production and its hadronisation is the quarkonium state is of course implied although it does not rely on any factorisation proof. In spite of this, this model benefits – as some figures will illustrate it in the next section – from a successful phenomenology but for the absence of predictions for polarisation observables and discrepancies in some transverse momentum spectra. The Colour-Singlet Model (CSM) The second simplest model to describe quarkonium production relies on the rather opposite assumption that the quantum state of the pair does not evolve between its production and its hadronisation, neither in spin, nor in colour [ 54–56 ] – gluon emissions from the heavy-quark are suppressed by powers of αs (m Q ). In principle, they are taken into account in the (p)QCD corrections to the hard-scattering part account for the Q Q-pair production. If one further assumes that the quarkonia are nonrelativistic bound states with a highly peaked wave function in the momentum space, it can be shown that partonic cross section for quarkonium production should then be expressed as that for the production of a heavy-quark pair with zero relative velocity, v, in a colour-singlet state and in the same angular-momentum and spin state as that of the to-be produced quarkonium, and the square of the Schrödinger wave function at the origin in the position space. In the case of hadroproduction, which interests us most here, one should further account for the parton i, j densities in the colliding hadrons, fi, j (x ), in order to get the following hadronic cross section: dσ [Q + X ] = dxi dx j fi (xi , μF ) f j (x j , μF ) i, j ×dσˆi+ j→(Q Q)+X (μR , μF )|ψ (0)|2 (9) In the case of P-waves, |ψ (0)|2 vanishes and, in principle, one should consider its derivative and that of the hard scattering. In the CSM, |ψ (0)|2 or |ψ (0)|2 also appear in decay processes and can be extracted from decay-width measurements. The model then becomes fully predictive but for the usual unknown values of the non-physical factorisation and renormalisation scales and of the heavy-quark mass entering the hard part. A bit less than ten years ago, appeared the first evaluations of the QCD corrections [ 57–61 ] to the yields of J/ψ and ϒ (also commonly denoted Q) in hadron collisions in the CSM. It is now widely accepted [ 62–64 ] that αs4 and αs5 corrections to the CSM are significantly larger than the LO contributions at αs3 at mid and large pT and that they should systematically be accounted for in any study of their pT spectrum. 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 [ 65–67 ], e+e− data at B factories [ 68–70 ] and photoproduction data at HERA [ 71 ]. Taking into account NLO – one loop – corrections and approximate NNLO contributions (dubbed NNLO in the following) has reduced the most patent discrepancies in particular for pT up to a couple of mQ [ 72–75 ]. A full NNLO computation (i.e. at αs5) is, however, needed to confirm this trend. It is, however, true that the CSM is affected by infra-red divergences in the case of P-wave 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 colour-octet states. The Colour-Octet 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 long-distance matrix elements (LDMEs). In addition to the usual expansion in powers of αs , NRQCD further introduces an expansion in v. It is then natural to account for the effect of higher-Fock states (in v) where the Q Q pair is in an octet state with a different angular-momentum 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 non-perturbative transitions between these coloured states and the physical meson. One of the virtues of this is the consideration of 3 S[ 8 ] states in 1 P-wave 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 non-physical scale μ . As compared to the Eq. (9), one has to further consider additional quantum numbers (angular momentum, spin and colour), generically denoted n, involved in the production mechanism: dσ [Q + X ] = dxi dx j fi (xi , μF ) f j (x j , μF ) i, j,n n . ×dσˆi+ j→(Q Q)n+X (μR , μF , μ ) OQ (10) Instead of the Schrödinger wave function at the origin squared, the former equation involves the aforementioned n , which cannot be fixed by decay-width meaLDMEs, OQ surements nor lattice studies5 – but the leading CSM ones of course. Only relations based on Heavy-Quark Spin Symmetry (HQSS) can relate some of them. 5 We, however, note that a first attempt to evaluate colour octet decay LDMEs was made in [ 79 ]. In principle they can be related by crossing symmetry to the production LDMEs which are relevant for the present discussion. Quantitative results are, however, still lacking. Three groups (Hamburg [ 80 ], IHEP [ 81 ] and PKU [ 82 ]) have, in recent years, carried out a number of NLO studies6 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 J/ψ , the studies of the Hamburg group, which is the only one to fit low pT 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- pT region where a description in terms of fragmentation functions is justified, the infra-red 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 αs is probably not optimal since higher QCD corrections receive contributions which are enhanced by powers of pT/mQ. It may therefore be expedient to organise the quarkonium-production cross section in powers of pT/mQ before performing the αs -expansion of the short-distance coefficients for the Q Q production. This is sometimes referred to as the fragmentation-function approach (see [ 86,87 ]) which offers new perspectives in the theoretical description of quarkonium hadroproduction especially at mid and large pT. Complementary information could also be obtained from similar studies based on Soft Collinear Effective Theory (SCET); see [88]. At low pT, it was recently emphasised in [ 67 ] that oneloop results show an intriguing energy dependence which might hint at a break-down of NRQCD factorisation in this kinematical region. In any case, as for now, past claims that colour-octet transitions are the dominant source of the lowpT J/ψ and ϒ cannot be confirmed at one loop accuracy. Approaches such as the kT factorisation based on the Lipatov action in the quasi-multi Regge kinematics (see [ 89,90 ] for quarkonium studies), the TMD factorisation (see [ 91,92 ] for recent applications to quarkonium production) or the combined use of the CGC formalism and NRQCD [ 93,94 ] may therefore bring their specific bricks in the building of a consistent theory of quarkonium production. Finally, let us mention the relevance of the colour-transfer mechanism [95], beyond 6 A recent LO study has also been performed including LHC data in the used sample [ 83 ]. NRQCD, in the case of production of a quarkonium in the vicinity of another heavy quark. 2.2 Recent cross section measurements at hadron colliders Due to their short lifetimes (at most a picosecond), the production of open-heavy-flavour particles is studied through their decay products. Four main techniques are used: 1. Fully reconstruction of exclusive decays, such as B0 → J/ψ KS0 or D0 → K− π +. 2. Selection of specific (semi-)inclusive decays. For beauty production, one looks for a specific particle, for example J/ψ , and imposes it to point to a secondary vertex displaced a few hundred7 µm from the primary vertex. Such displaced or non-prompt mesons are therefore supposed to come from b-decay 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 π 0 and η in the case of electrons, light hadron, Drell–Yan pair, J/ψ ,…) to the lepton yield. Alternatively, the photon conversion and Dalitz decay contribution can be evaluated via an invariant-mass analysis of the e+e− pairs. (ii) By selecting displaced leptons with a track pointing to a secondary vertex separated by few hundred µm 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 ]. Different methods are used in different contexts, depending on the detector information available, the trigger strategy, the corresponding statistics (hadronic decays are less abundant than leptonic ones), the required precision (only exclusive decay channels allow for a full control of the kinematics), the kinematical range (b-tagged jets give access to very large pT whereas exclusive-decay channels allow for differential studies down to pT equal 0). A fifth method based on the indirect extraction of the total charm- and beauty-production from dileptons – as opposed to single leptons – (see e.g. [ 97 ]) is not discussed in this review. Hidden-heavy-flavour, i.e. quarkonia, are also analysed through their decay products. The triplet S-waves are the most studied since they decay up to a few per cent of the time in dileptons. This is the case for J/ψ , ψ (2S), ϒ (1S), ϒ (2S), ϒ (3S). The triplet P-waves, such as the χc and χb, are usually reconstructed via their radiative decays into a triplet 7 For larger pT or y, such a distance can significantly be larger. rapidity in pp and pp collisions [ 107–111 ]. Results are compared to pQCD calculations, NLO MNR [ 6 ] and FONLL [ 44,99 ] for cc and bb, respectively Fig. 2 dσ/d pT for heavy-flavour decay leptons in pp collisions: a electrons at mid-rapidity for √s = 200 GeV from PHENIX [112], b electrons at mid-rapidity for √s = 2.76 TeV [ 118 ] and c muons at forwardrapidity for √s = 2.76 TeV from ALICE [ 119 ]. FONLL [ 44,99 ] predictions are also shown; in a and c the calculations for leptons from charm and beauty decays are shown separately (without theoretical uncertainty bands in a). GM-VFNS [ 15,16 ] and kT-factorisation [ 122 ] calculations are also drawn in b S-wave. For other states, such as the singlet S-wave, studies are far more complex. The very first inclusive hadroproduction study of ηc was just carried out this year in the pp decay channel by the LHCb Collaboration [ 98 ]. A compilation of the measurements of the pT-integrated cc and bb cross section, σcc and σbb, is shown in Fig. 1 from SPS to LHC energies. Let us stress that most of the pT-integrated results and nearly all y-integrated ones are based on different extrapolations, which significantly depend on theoretical inputs and which are not necessarily identical in the presence of nuclear effects. The results are described within the uncertainties by pQCD calculations, NLO MNR [ 6 ] and FONLL [ 44, 99 ] for the cc and bb, respectively. Note that most of the experimental results for σcc, in particular at high energies, lie on the upper edge of the NLO MNR uncertainties. 2.2.1 Leptons from heavy-flavour decays The first open-heavy-flavour measurements in heavy-ion collisions were performed by exploiting heavy-flavour decay leptons at RHIC by the PHENIX and STAR Collaborations. These were done both in pp and AA collisions [ 112–116 ]. At the LHC, the ATLAS and ALICE Collaborations have also performed such studies in heavy-ion collisions [ 117– 121 ]. A selection of the pT-differential production cross sections of heavy-flavour decay leptons in pp collisions at different rapidities and energies is presented in Fig. 2. The measurements are reported together with calculations of FONLL [ 44, 99 ] for √s = 0.2 and 2.76 TeV, GMVFNS [ 15, 16 ] and kT-factorisation [ 122 ] at √s = 2.76 TeV. The POWHEG predictions [ 49 ], not shown in this figure, show a remarkable agreement with the FONLL ones. The 2 ])c 1 p+p 200 GeV / V e /bG10-2 ( m [) y dpT dpT10-4 π 2 cc/)( 2 σ10-6 d ( 0 1 2 D0 / 0.565 D* / 0.224 FONLL power-law fit 3 (a) 4 5 6 pT (GeV/c) Fig. 3 dσ/d pT for D-meson production in pp collisions at different energies. a D0 and D∗+ measurements at √s = 200 GeV with the STAR detector [ 103 ]. b D+ data at √s = 7 TeV with the ALICE detector [ 125 ]. c Ds+ data at √s = 7 TeV with the LHCb detector [ 106 ], differential cross sections of heavy-flavour-decay 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 impact-parameter distribution using templates of the different contributions to the inclusive spectra, (iii) studies of the azimuthal angular correlations between heavy-flavour decay leptons and charged hadrons (see e.g. [ 107,123 ]). These measurements are also described by pQCD calculations. 2.2.2 Open charm Recently, D-meson production has been studied at RHIC, Tevatron and LHC energies [ 102–104,106,124–126 ]. The measurements were performed by fully reconstructing the hadronic decays of the D mesons, e.g. D0 → K−π + and charge conjugates. D-meson candidates are built up of pairs or triplets of tracks with the appropriate charge sign combination. The analyses exploit the detector particle identification abilities to reduce the combinatorial background, which is important at low pT. For the measurements at Tevatron and LHC, the background is also largely reduced by applying topological selections on the kinematics of the secondary decay vertices, typically displaced by few hundred µm from the interaction vertex. The results at RHIC energies report the inclusive D-meson yields [ 103 ], i.e. those from both c and b quark fragmentation. The former are called prompt, and the later secondary D mesons. The measurements at Tevatron and LHC energies report prompt D-meson yields. Prompt yields include both the direct production and the feed-down from excited charmed resonances. The secondaries contribution to the D-meson yields is evaluated and subtracted by: (i) either scrutinising the D-meson candidates impact-parameter distribution, exploiting the larger lifetime of b- than c-flavoured hadrons [ 102,106,124 ], which requires stat. unc. syst. unc. FONLL GM-VFNS 10-2 ± 3.5% lumi,± 2.1% BR norm. unc. (not shown) 0 5 10 15 ALICE D+, pp s = 7 TeV, Lint = 5 nb-1 LHCb s = 7 TeV 2.0 <y< 2.5, m=0 2.5 <y< 3.0, m=1 large statistics, (ii) or evaluating the beauty hadron contribution using pQCD-based calculations [ 104,125,126 ], advantageous strategy for smaller data samples but limited by the theoretical uncertainties. Figure 3 presents a selection of the D-meson measurements compared to pQCD calculations. The D0, D+ and D∗+ dσ/d pT 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 pT, where the quark mass effects are important, the FONLL and POWHEG predictions show a better agreement with data. At intermediate to high pT, where the quark mass effects are less important, all the FONLL, POWHEG, GM-VFNS and kT-factorisation calculations agree with data. The agreement among the FONLL and POWHEG calculations is better for heavy-flavour decay leptons than for charmed mesons, which seems to be related to the larger influence of the fragmentation model on the latter. The Ds+ pT-differential cross section is compared to calculations in Fig. 3c. The Ds+ measurements are also reproduced by FONLL, GM-VFNS and kT-factorisation predictions, but POWHEG calculations predict a lower production cross section than data. Charmed baryon production measurements in hadron colliders are scarce. The properties and decay branching ratios of the c, c and c states have been studied at the charmand B-factories and fixed-target experiments; see e.g. [ 132– 135 ]. An example are the results by Fermilab E791 [128], FOCUS [ 129 ], and CLEO [ 130 ] Collaborations. The CDF Collaboration measured charmed baryons in pp collisions at √s = 1.96 TeV; see for example [ 127 ]. For illustration, a compilation of the c+ and c+ mass difference is shown in Fig. 4a. The LHCb Collaboration measured the pT and y differential production cross section of c in pp collisions at √s = 7 TeV [ 106 ]. Figure 4b shows the pT-differential cross section compared to GM-VFNS calculations. No dedicated ises the world average value and its uncertainty taken from Ref. [ 131 ]. b dσ/d pT for c+ production in pp collisions at √s = 7 TeV [ 106 ] compared to GM-VFNS [ 15,16 ] calculations FONLL calculation is available for c production due to the lack of knowledge of the fragmentation function. The GMVFNS predictions include the fragmentation functions resulting from a fit to e+e− collider data [34], where the prompt and secondary contributions to the measurements were not separated. 2.2.3 Open beauty Open-beauty 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− B-factories (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 Bs , Bc or c cannot be produced at the B-factories. 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- pT probes, like top production, are not as good for low- pT 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. In addition, measurements of the B cross section via a displaced charmonium have been performed multiple times ] ) c / eV102 G ( / b μ [pT d / σ d 10 1 0 LHCb s = 7 TeV at Tevatron and at LHC. Charmonia from beauty decays are selected by fitting the pseudo-proper decay length distribution of the charmonium candidates, L x y (m/ pT)J/ψ . Figure 5 presents a selection of the LHC results: the non-prompt pT-differential cross section of J/ψ , ψ (2S), ηc, χc1 and χc2 in pp collisions at √s = 7 TeV [ 98,136,137 ]. The results at intermediate to low pT are well reproduced by the FONLL [ 44,99 ], NLO GM-VFNS [ 15,16 ] and NLO [ 6 ] with FONLL fragmentation calculations. At high- pT the predictions tend to overestimate data. This could be related to the usage of the e+e− fragmentation functions in an unexplored kinematic range. Figure 5c, which reports the first measurement of non-prompt charmonium in a purely hadronic decay channel at hadron colliders, shows a similar transversemomentum spectrum for non-prompt singlet and triplet Swave charmonia. Studies of open-beauty production have also been performed in exclusive channels at Tevatron and at the LHC, e.g. in the case of B±, B0 and Bs0 [ 139,140,142–148 ]. As example, Fig. 6 presents the B+ pT and y differential cross section in pp collisions at √s = 7 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 mb and by the fact that for pT mb, 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 mb and the renormalisation and factorisation scales, and describe the data within uncertainties. The FONLL prediction provides a good description of the measurements within uncertainties. Measurements of the beauty and charm baryon production in pp collisions at √s = 1.96 TeV are summarised ]eV 104 /bG 103 n [ /yd 102 ) /pdT 10 ψ(2S2⋅σd 10-11 - ) +π 10-2 - πμ) 10-3 + μ→10-4 ( ψ/J 10-5 →)S10-6 2 ( ψ( 10-7 10 B ]V 10 e G / b μ[ .) 4 2 |< 1 y |; + XB →pp10-1 (pT d σ/ d 10-2 5 Fig. 6 B+differential cross sections dσ/d pT (a, b) and dσ/dy (c) in pp collisions at √s = 7 TeV compared to theory predictions [ 139,140 ]. In c the error bars correspond to the differential cross-section measurement in Ref. [154]. In particular, the doubly strange b-baryon b− and measurement of the b− and b− properties can be found in Refs. [ 155,156 ]. Such measurements have also been performed at the LHC in pp collisions at √s = 7 and 8 TeV. For example, the observation of the b baryon states was reported in Refs. [ 157,158 ]. The measured mass and width of the b baryon states is consistent with theoretical expectations [ 133,159–166 ]. The b+ pT and y differential production cross section in pp collisions at √s = 7 TeV by CMS [ 141 ] is reported in Fig. 7. The b dσ/d pT and dσ/dy are not reproduced by neither PYTHIA (Z2 tune) nor POWHEG calculations: PYTHIA expects a harder pT-distribution and flatter y distribution than data, while POWHEG underestimates its production cross section, particularly at low pT; see Fig. 7a and b. The measured b pT-spectrum at mid-rapidity seems to fall more steeply than the B0 and B+ ones, see Fig. 7c, and falls also faster than predicted by PYTHIA and POWHEG. As discussed for the nonprompt charmonium measurements, this could be influenced Fig. 5 pT-differential cross sections for non-prompt charmonia – assumed to come from b decays – for a ψ(2S) [ 136 ] d2σ/d pTdy by ATLAS compared to NLO [ 6 ], FONLL [ 44,99 ] and GM-VFNS [ 15,16 ] calculations, b χc1 and χc2 dσ/d pT by ATLAS [ 137 ] compared to FONLL [ 44 ], c ηc and J/ψ dσ/d pT by LHCb [ 98 ] Non-prompt |yJ/ψ| < 0.75 Data χc1 Data χc2 (b) 20 by the lack of data to extract the fragmentation functions in this kinematic region. The fragmentation of the b quark is relatively hard compared to that of lighter flavours, with the b-hadron taking about 70 % of the parton momentum on average at the Zpole [ 167 ]. Identification of jets from beauty quark fragmentation or “b-tagging” can be achieved by direct reconstruction of displaced vertices, although the efficiency for doing so is limited. Higher efficiency can be achieved by looking for large impact-parameter tracks inside jets, or by a combination of the two methods, which are collectively known as lifetime tagging. Leptons inside jets can also be used for b-tagging, but, due to the branching fraction, are usually only used as a calibration of the lifetime methods. At the LHC, both ATLAS and CMS have performed measurements of the b-jet cross section [ 149,150 ]. Theoretical comparisons can be made to models which calculate fully exclusive final states, which can be achieved by matching NLO calculations to parton showers [168]. Figure 8 shows CMS s = 7 TeV L =Λ 1.9 fb-1 |y b| < 2.0 ) V e /bG10-1 n ( ) X Λ Ψ /J10-2 → X b Λ → p p ( pT10-3 d / σ d Data PYTHIA POWHEG POWHEG uncertainty 10 20 40pTΛb (GeV) 50 CMS s = 7 TeV 30 40 50 b-hadron pT (GeV) (c) 20 40 60 80 100 120 140 160 180 200 b-jet pT (GeV) 200 300 Jet pT [GeV] Fig. 8 b-jet cross section as a function of pT in pp collisions at √s = 7 TeV: a dσ/d pT from the lifetime-based and muon-based analyses by CMS [ 149 ] and ATLAS [ 150 ] compared to the MC@NLO calculation, and b d2σ/d pTdy by ATLAS from the lifetime-based analysis [ 150 ] compared to the predictions of PYTHIA, POWHEG (matched to PYTHIA) and MC@NLO (matched to HERWIG) [ 46,47,151–153 ] the b-jet cross section measurement by ATLAS and CMS in pp collisions at √s = 7 TeV. The measurements are shown as a function of pT and in several bins of rapidity. Calculations from POWHEG [ 152 ] (matched to PYTHIA [ 151 ]) and MC@NLO [ 47, 153 ] (matched to HERWIG [46]), are found to reproduce the data. Measurements from both lifetime- and lepton-based tagging methods are shown. 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 J/ψ and ψ (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 signalover-background ratio and a purer triggering. There are many recent experimental studies. In Fig. 9, we show only two of these. First we show dσ/d pT for prompt J/ψ at √s = 7 GeV as measured by LHCb compared to a few predictions for the prompt yield from the CEM and from NRQCD at NLO8 as well as the direct yield9 compared to a NNLO CS evaluation. Our point here is to emphasise the precision of the data and to illustrate that at low and mid pT– which is the region where heavy-ion studies are carried out – none of the models can simply be ruled out owing to their theoretical uncertainties (heavy-quark mass, scales, non-perturbative parameters, unknown QCD and relativistic corrections, ...). Second, we show the fraction of J/ψ from b decay for y close to 0 at √s 8 Let us stress that the NRQCD band in Fig. 9a is not drawn for pT lower than 5 GeV because such a NLO NRQCD fit overshoots the data in this region and since data at low pT are in fact not used in this fit. For a complete discussion of NLO CSM/NRQCD results for the pTintegrated yields; see [ 67 ]. As regards the CEM curves, an uncertainty band should also be drawn (see for instance [ 169 ]). 9 The expected difference between prompt and direct is discussed later on. Fig. 9 a Prompt J/ψ yield as measured by LHCb [ 172 ] at √s = 7 TeV compared to different theory predictions referred to as “prompt NLO NRQCD”[ 173 ], “Direct NLO CS”[ 57,58 ], “Direct NNLO CS” [ 61,62 ] and “Prompt NLO CEM” [ 174 ]. b Fraction of J/ψ from B as measured by ALICE [ 108 ], ATLAS [ 170 ] and CMS [ 171 ] at √s = 7 TeV in the central rapidity region LHCb 20 pt GeV (b) Fig. 10 a ATLAS ψ(2S) differential cross section [ 136 ] compared to different theoretical curves. b Prompt X (3872) production cross section measured by the CDF [ 175,176 ], CMS [177], and LHCb [ 178 ] Collaborations compared with NLO NRQCD allowing the CS contribution to differ from that from HQSS [ 179 ]. c Prompt-ηc transverse-momentum cross section in pp collisions at √s = 8 TeV measured by LHCb [ 98 ] compared to the CS contribution following HQSS and fitted CO contributions at NLO [ 180 ] = 7 TeV as function of pT as measured by ALICE [ 108 ], ATLAS [ 170 ] and CMS [ 171 ]. At low pT, the difference between the inclusive and prompt yield should not exceed 10 % – from the determination of the σbb, it is expected to be a few percent at RHIC energies [ 111 ]. It, however, steadily grows with pT. At the highest pT reached at the LHC, the majority of the inclusive J/ψ is from b decays. At pT 10 GeV, which could be reached in future quarkonium measurements in Pb–Pb collisions, it is already three times higher than at low pT: 1 J/ψ out of three comes from b decays. For excited states, there is an interesting alternative to the sole dilepton channel, namely J/ψ +π π . This is particularly relevant since more than 50 % of the ψ (2S) decay in this channel. The decay chain ψ (2S) → J/ψ + π π → μ+μ− +π π is four times more likely than ψ (2S) → μ+μ−. The final state J/ψ + π π is also the one via which the X (3872) was first seen at pp colliders [ 175,181 ]. ATLAS released [136] the most precise study to date of ψ (2S) production up to pT of 70 GeV at √s = 7 TeV, precisely in this channel. The measured differential cross section is shown for three rapidity intervals in Fig. 10a with four theoretical predictions. Along the same lines, the CDF, CMS and LHCb Collaborations measured the prompt X (3872) yields at different values of pT (see Fig. 10b). In the NRQCD framework, these measurements tend to contradict [ 179 ] a possible assignment of the X (3872) as a radially excited P-wave state above the open-charm threshold. Such a statement should, however, be considered with care owing to the recurrent issues in understanding prompt quarkonium production. In addition, LHCb determined the X (3872) quantum numbers to be J PC = 1++, excluding explanation of the X (3872) as a conventional ηc2(11 D2) state [ 182 ]. A brief survey of the new charmonium states above he D D¯ threshold and their interpretation can be found in Ref. [ 131 ]. Ultimately the best channel to look at all n = 1 charmonium yields at once is that of baryon-antibaryon 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 J/ψ into pp, made along that of the ηc. 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 Fig. 11 a Typical source of prompt J/ψ at low and high pT. b Ratio of χc1 to χc2 as measured by the LHCb experiment in pp collisions at √s = 7 TeV compared to results from other experiments [ 185,190,191 ] and about constraints [ 180,183,184 ] on the existing global fits of NRQCD LDMEs by virtue of heavy-quark spin symmetry (HQSS) which is an essential property of NRQCD. As for now, it seems that the CS contributions to ηc 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 χc1,2 production at hadron colliders corresponds to the studies involving P waves decaying into J/ψ and a photon. Very recently the measurement of χc0 relative yield was performed by LHCb [ 185 ] despite the very small branching ratio χc0 → J/ψ + γ of the order of one percent, that is 30 (20) times smaller than that of χc1 (χc2). LHCb found that σ (χc0)/σ (χc2) is compatible with unity for pT >4 GeV/c, in striking contradiction with statistical counting, 1/5. Currently, the experimental studies are focusing on the ratio of the χc J yields which are expected to be less sensitive to the photon acceptance determination. They bring about constraints on production mechanism but much less than the absolute cross section measurements which can also be converted into the fraction of J/ψ from χc J . This was the first measurement of this fraction at the Tevatron by CDF in 1997 [ 186 ] which confirmed that our understanding of quarkonium production at colliders was incorrect (for reviews see e.g. [ 187,188 ]). It showed that the J/ψ yield at Tevatron energies was mostly from direct J/ψ and not from χc J decays. The latter fraction was found to be at most 30 %. Similar information are also fundamental to use charmonia as probes of QGP, especially for the interpretation of their possible sequential suppression. It is also very important to understand the evolution of such a fraction as function of √s, y and pT. Fig. 11a shows the typical size of the feed-down fraction of the χc and ψ (2S) into J/ψ at low and high pT, which are different. One should therefore expect differences in these fraction between pT-integrated yields and yields measured (c) 20 30 pJT/ψ [GeV] NRQCD calculations [ 189,192 ]. c χc1 dσ/d pT [137] as compared to LO CSM12, NLO NRQCD [ 173,189,193 ] and kT factorisation [ 194,195 ] at pT = 10 GeV/c and above. Figure 11b shows the ratio of the χc2 over χc1 yields as measured10 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 pT as small as 2 GeV/c, where the ratio seems to strongly increase. This increase is in line with the Landau–Yang theorem according to which χc1 production from collinear and on-shell gluons at LO is forbidden. Such an increase appears in the LO NRQCD band, less in the NLO NRQCD one. At larger pT, 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 χc feed-down fractions which are of paramount importance to interpret inclusive J/ψ results. One can of course also derive absolute cross section measurements which are interesting to understand the production mechanism of the P-wave quarkonia per se; these may not be the same as that of S-wave quarkonia. Figure 11c shows the pT dependence of the yield of the χc1 measured by ATLAS (under the hypothesis of an isotropic decay), which is compared to predictions from the LO CSM,11 NLO NRQCD and kT 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 χc2. 10 The present ratio depends on the polarisation of the χc since it induces different acceptance correction. 11 For the P wave case, the distinction between colour-singlet and colour-octet transition is not as clear that for the S wave. In particular the separation between CS and CO contribution depends on the NRQCD factorisation scale μ . 12 As encoded in ChiGen: https://superchic.hepforge.org/chigen.html. 2.5 ) 2 b n ( rxB 1.5 y d / itrced ()1Sϒ 1 σ d (c) The study of bottomonium production at LHC energies offers some advantages. First, there is no beauty feed-down. Second, owing to their larger masses, their decay products – usually leptons – are more energetic and more easily detectable (detector acceptance, trigger bandwidth, ...). Third, the existence of three sets of bottomonia with their principal quantum number n = 1, 2, 3 below the open-beauty threshold offers a wider variety of states that can be detected in the dilepton decay channel – this, however, introduces a complicated feed-down pattern which we discuss later on. Fourth, at such high energies, their production rates with respect to those of charmonia are not necessarily much lower. It was for instance noticed [ 196 ] that, for their production in association with a Z boson, the cross sections are similar. Fig. 12a shows the rapidity dependence of the ϒ (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 ϒ (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 pT 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 pT would provide striking evidence in favour of one or another mechanism – associated-production channels, which we discuss in Sect. 2.4, are probably more promising. Figure 12c shows ratios of different S-wave bottomonium yields. These are clearly not constant as one might anticipate following the idea of the CEM. Simple mass effects through feed-down 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 pT dependence. The χb feed-down, which we discuss in the following, can also affect these ratios. Since the discovery of the χb(3 P) by ATLAS [ 200 ], we know that the three n = 1, 2, 3 families likely completely lie under the open-beauty threshold. This means, for instance, that we should not only care about m S → n S and n P → n S+γ transitions but also of m P → n S+γ ones. Obviously, the n = 1 family is the better known of the three. Figure 13a shows the ratio of the production cross section of χb2(1 P) over that of χb1(1 P) measured by CMS and LHCb. Although the experimental uncertainties are significant, one does not observe the same trend as the LO NRQCD, i.e. an increase at low pT due to the Landau–Yang theorem. Besides, the ratio is close to unity which also seems to be in contradiction to the simple spin-state counting. Recently, LHCb performed a thorough analysis [ 203 ] of all the possible m P → n S+γ transitions in the bottomonium system. These new measurements along with the precise measurements of ϒ (2S) and ϒ (3S) pT-differential cross section show that the feed-down structure is quite different from that commonly accepted ten years ago based on the CDF measurement [ 209 ]. The latter, made for pT > 8 GeV/c [ 209 ], suggested that the χ (n P) → ϒ (1S) + γ feed-down could be as large as 40 % (without excluding values of the order of 25 %) and that only 50 % of the ϒ (1S) were direct. Based on the LHC results, one should rather say that, at low pT, where heavy-ion measurements are mostly carried out, 70 % of the ϒ (1S) are direct; the second largest source is from χb(1 P) – approximately two thirds from χb1(1 P) and one third from χb2(1 P) [ 201,202 ]. At larger pT (above 20 GeV/c, say), the current picture is similar to the old one, i.e. less than half of the ϒ (1S) are direct and each of the feed-down is nearly doubled. For the ϒ (2S), there is no χb(2 P) → ϒ (2S) + γ measurement at pT lower than 20 GeV/c. Above, it is measured to be about 30 % with an uncertainty of 10 %. The feed-down from χb(3 P) is slightly lower than from ϒ (3S). slightly displaced from the bin centres. The inner error bars represent statistical uncertainties, while the outer error bars indicate statistical and systematic uncertainties added in quadrature (a) (b) (c) Fig. 14 Typical sources of ϒ (n S) at low and high pT. These numbers are mostly derived from LHC measurements [ 197–199,203–208 ] assuming an absence of a significant rapidity dependence. a ϒ (1S); b ϒ (2S); c ϒ (3S) Taken together they may account for 10–15 % of the ϒ (2S) yield. For the ϒ (3S), the only existing measurement [203] is at large pT and also shows (see Fig. 13c) a feed-down fraction of 40 % with a significant uncertainty (up to 15 %). The situation is schematically summarised on Fig. 14. 2.2.6 Bc and multiple-charm baryons After a discovery phase during which the measurement of the mass and the lifetime of the Bc was the priority, the first measurement of the pT and y spectra of promptly produced Bc+ was carried out by the LHCb Collaboration [ 210 ]. Unfortunately, as for now, the branching B+ → J/ψ π + is not yet c known. This precludes the extraction of σpp→Bc++X and the comparison with the existing theoretical predictions [ 213– 220 ]. Aside from this normalisation issue, the pT 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 pp collisions at 7 and 8 TeV. As for now, no analysis could confirm the signals seen by the fixed-target experiment SELEX at Fermilab [ 222, 223 ]. 2.3 Quarkonium-polarisation studies Measurements of quarkonium polarisation can shed more light on the long-standing 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. s0.16 d l ie0.14 y ed0.12 z i la 0.1 m ro0.08 N 0.06 Fig. 15 Bc+ meson production in pp collisions at √s = 8 TeV as measured by the LHCb Collaboration in its Bc+ → J/ψ π+ decay [ 210 ] within 0 < pT < 20 GeV/c and 2.0 < y < 4.5. The solid histogram is a theory evaluation based on the complete order-αs4 calculation – as opposed to fragmentation-function-based computations – implemented in the Bc generator BCVEGPY [ 211,212 ] We have collected in this section all results of polarisation measurements performed by different experiments at different collision energies √sNN and in different kinematic regions. The results for J/ψ and ψ (2S) can be found in Tables 1 and 2 for 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 ϒ (1S), ϒ (2S) and ϒ (3S) in pp collisions. Polarisation of a vector quarkonium state is analysed experimentally via the angular distribution of the leptons from the quarkonium dilepton decay, that is parametrised by: d2 N d(cos θ )dφ ∝ 1 + λθ cos2 θ + λφ sin2 θ cos 2φ + λθφ sin 2θ cos φ, θ is the polar angle between the positive lepton in the quarkonium rest frame and the chosen polarisation axis and φ angle is the corresponding azimuthal angle defined with respect to the plane of colliding hadrons. The angular decay coefficients, λθ , λφ and λθφ , are the polarisation parameters. In the case of an unpolarised yield, one would have (λθ , λφ , λθφ ) = (0, 0, 0) for an isotropic decay angular distribution, whereas (1, 0, 0) and (−1, 0, 0) refer to fully transverse and fully longitudinal polarisation, respectively. It is, however, very important to bear in mind that the angular distribution of Eq. (11) is frame dependent as the polarisation parameters. All experimental analyses have been carried in a few specific reference frames, essentially defined by their polarisation axis,13 namely: the helicity (H X ) frame, the Collins–Soper (C S) [ 225 ] frame, the Gottfried–Jackson (G J ) [ 226 ] frame as well as the perpendicular helicity ( P X ) [ 227 ] frame. (11) 13 See [ 224 ] for the definition of the corresponding axes. In spite of the frame dependence of λθ , λφ , λθφ , there exist some combinations which are frame invariant [ 224,228 ]. An obvious one is the yield, another one is λ˜ = (λθ + 3λφ )/(1 − λφ ) [224]. As such, it can be used as a good cross-check 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) 1-D technique: fitting cos θ distribution with the angular distribution, Eq. (11), averaged over the azimuthal φ angle, and fitting the φ distribution, Eq. (11), averaged over the polar θ angle (ii) 2-D technique: fitting a two-dimensional cos θ vs φ 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, physics-wise.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 non-prompt 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 feed-down from the excited states in which case vertexing is of no help. As for now, no attempt of removing it from e.g. prompt J/ψ and inclusive ϒ (1S) samples has been made owing its intrinsic complication. We have therefore found it important to specify what kind of feed-down 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 feed-down. For each mea14 Irrespective of the experimental techniques used to extract it, a sample of inclusive low pT ψ(2S) at energies around 100 GeV is essentially purely direct. nucleus–nucleus data in terms of a combination of cold and hot medium effects is yet to be fully understood. In addition, it is still an open question whether the possible signals of collective behaviour observed in high-multiplicity proton–nucleus collisions in the light-flavour sector could manifest also for heavy-flavour production. This question could become accessible with future higher-statistics proton– nucleus data samples at RHIC and LHC. The strong electromagnetic field of lead ions circulating in the LHC is an intense source of quasi-real photons, which allows for the study of γ γ , γ p and γ Pb reactions at unprecedented high energies. The coherent and the incoherent photoproduction of J/ψ and ψ (2S) is a powerful tool to study the gluon distribution in the target hadron and the first data from the LHC using Run 1 already set strong constraints to shadowing models. The statistical precision is one of the main, and in some cases the dominant, sources of uncertainty of the current measurements. The large increase in statistics expected for Run 2 and other future data-taking periods, as well as improvements in the detectors, the trigger and the data acquisition systems, will allow for a substantial reduction of the uncertainties. These future measurements will then shed a brighter light on the phenomena of shadowing and the gluon structure of dense sources, like lead ions. The measurements of open heavy flavour production in nucleus–nucleus, proton–proton and proton–nucleus collisions at RHIC and the LHC allow us to conclude that heavy quarks experience energy loss in the hot and dense QGP. A colour charge dependence in energy loss is not clearly emerging from the data, but it is implied by the fair theoretical description of the observed patterns. A quark mass ordering is suggested by the data (some of them still preliminary, though) and the corresponding model comparisons. However, this observation is still limited to a restricted momentum and centrality domain. The important question of thermalisation of heavy quarks appears to be partly answered for charm: the positive elliptic flow observed at both RHIC and LHC indicates that charm quarks take part in the collective expansion of the QGP. This is consistent with thermalisation, but the degree of thermalisation is not yet constrained. For the beauty sector, thermalisation remains an open issue entirely. The role of the different in-medium interaction mechanisms, such as radiative, collisional energy loss and in-medium hadronisation, is still not completely clarified, although the comparison of data with theoretical models suggests the relevance of all these effects. For the quarkonium families, the LHC data demonstrated the presence of colour screening for both charmonium and bottomonium. In the case of J/ψ , the LHC data implies the presence of other production mechanisms, generically called (re)generation. Whether production takes place throughout the full (or most of the) lifetime of the deconfined state or rather suddenly at the confinement transition (crossover) cannot be disentangled using the existing measurements. The ϒ production seems to exhibit a sequential pattern, but several assumed quantities in this interpretation (e.g. the feed-down contributions) make the situation not satisfactory enough. The next steps in the study of heavy-flavour hadron production in heavy-ion collisions will lead to a stage of quantitative understanding of the data, towards the extraction of the charm and beauty quarks transport coefficients and the temperature history of the deconfined state, including the temperature of the confinement crossover. An incremental, but nevertheless important, progress is expected with the existing experimental set-ups at RHIC and the LHC (where in particular the increased collision energy enhances the relevance of the data in the next three years). The ultimate goal can only be achieved with upgraded or new detectors, which will allow for the extension of the set of observables and the precision of the measurements over a broad range of collision energies. This experimental effort needs to be matched on the theory side. Even though the field of study of extreme deconfined matter with heavy quarks seems to be driven by experiment, the contribution of theory is of crucial importance. In particular, accurate theoretical guidance and modelling are required to interpret the measurements in terms of the QGP properties mentioned in the previous paragraph. Ultimately, the quantitative stage can only be reached in a close collaboration of experiment and theory. Acknowledgments The SaporeGravis network was supported by the European Community Research Infrastructures Integrating Activity “Study of strongly interacting matter” (acronym HadronPhysics3) – Grant Agreement No. 283286 – within the Seventh Framework Programme (FP7) of EU. The Work of F. Arleo was supported by the ECOS-Conicyt grant No. C12E04 between France and Chile. The work of T. Dahms was supported by the DFG cluster of excellence “Origin and Structure of the Universe”. The work of E. Ferreiro was supported by the Ministerio de Economia y Competitividad of Spain. The work of P.B. Gossiaux was supported by the Region Pays de la Loire through the TOGETHER project. The work of R. Granier de Cassagnac was supported by the European Research Council under the FP7 Grant Agreement no. 259612. The work of B. Kopeliovich was supported by the Fondecyt (Chile) Grants 1130543, 1130549, 1100287,and ECOS-Conicyt Grant No. C12E04. The work of L. Massacrier was supported by the European Community Research Infrastructures Integrating Activity “Study of strongly interacting matter (acronym HadronPhysics3) – Grant Agreement No. 283286 – within the Seventh Framework Programme (FP7) of EU, and by the P2IO Excellence Laboratory. The work of A. Mischke was supported by the Vidi grant from the Netherlands Organisation for Scientific Research (project number: 680-47-232) and Projectruimte grants from the Dutch Foundation for Fundamental Research (project numbers: 10PR2884 and 12PR3083). The work of M. Nahrgang was supported by the Postdoc-Program of the German Academic Exchange Service (DAAD) and the U.S. Department of Energy under Grant DE-FG02-05ER41367. The work of S. Peigné was supported by the ECOS-Conicyt Grant No. C12E04 between France and Chile. The work of R. Rapp was supported by the US-NSF Grant No. PHY-1306359. The work of B. Trzeciak was supported by the European social fund within the framework of realizing the project, Support of inter-sectoral mobility and quality enhancement of research teams at Czech Technical University in Prague, CZ.1.07/2.3.00/30.0034 and by Grant Agency of the Czech Republic, Grant No.13-20841S. The work of I. Vitev was supported by Los Alamos National Laboratory DOE Office of Science Contract No. DE-AC52-06NA25396 and the DOE Early Career Program. The work of R. Vogt was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344 and supported in part by the JET Collaboration, the U.S. Department of Energy, Office of Science, Office of Nuclear Physics (Nuclear Theory). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Funded by SCOAP3. 1. A. Bhattacharya , R. Enberg , M.H. Reno , I. Sarcevic , A. 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A. Andronic, F. Arleo, R. Arnaldi, A. Beraudo, E. Bruna. Heavy-flavour and quarkonium production in the LHC era: from proton–proton to heavy-ion collisions, The European Physical Journal C, 2016, 107, DOI: 10.1140/epjc/s10052-015-3819-5