Heavyflavour and quarkonium production in the LHC era: from proton–proton to heavyion collisions
Eur. Phys. J. C
Heavyflavour and quarkonium production in the LHC era: from protonproton to heavyion 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. MartinezGarcia
L. Massacrier
C. Mironov
A. Mischke
M. Nahrgang
M. Nguyen
J. Nystrand
S. Peigné
S. PorteboeufHoussais
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 LeprinceRinguet
Ecole Polytechnique
CNRS/IN
Université ParisSaclay
Palaiseau
France
Université de Savoie
AnnecyleVieux
France
Sezione di Torino
Turin
Italy
European Organization for Nuclear Research (CERN)
Geneva
Switzerland
Univ. ParisSud
CNRS/IN
Université ParisSaclay
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
CNRSIN
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
SaintPetersburg State University
Ulyanovskaya
Saint Petersburg
Russia
Departamento de Física
Centro CientíficoTecnológico de Valparaíso
Universidad Técnica Federico Santa María
Valparaiso
Chile
Laboratori Nazionali di Frascati
Frascati
Italy
Univ. ParisSud
CNRS/IN
Université ParisSaclay
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
ClermontFerrand
France
IRFU/SPhN
CEA Saclay
GifsurYvette 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
RuprechtKarlsUniversität Heidelberg
Heidelberg
Germany
Université GrenobleAlpes
CNRS/IN
Grenoble
France
Department of Theoretical Physics
Tata Institute of Fundamental Research
Mumbai
India
Department of Physics
Kent State University
IPNLyon
Université de Lyon
Université Lyon
CNRS/IN
Villeurbanne
France
Institut für Theoretische Physik
Johann Wolfgang GoetheUniversität
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 highenergy hadronic
collisions, as tools to investigate fundamental aspects of
Quantum Chromodynamics, from the proton and nucleus
structure at high energy to deconfinement and the
properties of the Quark–Gluon Plasma. Emphasis is given to the
lessons learnt from LHC Run 1 results, which are reviewed
in a global picture with the results from SPS and RHIC at
lower energies, as well as to the questions to be addressed in
the future. The report covers heavy flavour and quarkonium
production in proton–proton, proton–nucleus and nucleus–
nucleus collisions. This includes discussion of the effects of
hot and cold strongly interacting matter, quarkonium
photoproduction in nucleus–nucleus collisions and perspectives
on the study of heavy flavour and quarkonium with upgrades
of existing experiments and new experiments. The report
results from the activity of the SaporeGravis network of the
I3 Hadron Physics programme of the European Union 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
heavyflavour in proton–proton collisions . . . 4
2.1.1 Openheavyflavour production . . . . . 4
2.1.2 Quarkoniumproduction mechanism . . . 8
2.2 Recent cross section measurements at hadron
colliders . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Leptons from heavyflavour 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 multiplecharm baryons . . . . . 19
2.3 Quarkoniumpolarisation 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. MartinezGarcia, L. Massacrier, J. Nystrand, S.
PorteboeufHoussais, 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 heavyflavour 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
lightflavour hadrons . . . . . . . . . . . 64
4.2 Experimental overview: azimuthal anisotropy
measurements . . . . . . . . . . . . . . . . . . 65
4.2.1 Inclusive measurements with electrons . . 65
4.2.2 Dmeson 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 pQCDinspired running αs energyloss
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 heavyquark
inter
actions in the QGP . . . . . . . . . . . . 73
4.3.5 LatticeQCD . . . . . . . . . . . . . . . 73
4.3.6 Heavyflavour interaction with medium in
AdS/CFT . . . . . . . . . . . . . . . . . 75
4.4 Theoretical overview: medium modelling and
mediuminduced modification of heavyflavour
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 pQCDinspired energy loss with running
αs in a fluiddynamical medium and in
Boltzmann transport . . . . . . . . . . . 77
4.4.4 Nonperturbative T matrix approach in
a fluiddynamic model (TAMU) and in
UrQMD transport . . . . . . . . . . . . . 78
4.4.5 LatticeQCD 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 Heavyflavour correlations in heavyion
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 inmedium
quarkonia . . . . . . . . . . . . . . . . . . . . . 98
5.1.5 Nonequilibrium effects on quarkonium
suppression . . . . . . . . . . . . . . . . 100
5.1.6 Collisional dissociation of quarkonia from
finalstate 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 fixedtarget experiments . . . . . . . . . . 127
7.3.1 Low energy projects at SPS, Fermilab and
FAIR . . . . . . . . . . . . . . . . . . . 127
7.3.2 Plans for fixedtarget experiments using
the LHC beams . . . . . . . . . . . . . . 128
8 Concluding remarks . . . . . . . . . . . . . . . . . 129
References . . . . . . . . . . . . . . . . . . . . . . . . 131
1 Introduction
Heavyflavour hadrons, containing open or hidden charm
and beauty flavour, are among the most important tools for
the study of Quantum Chromodynamics (QCD) in
highenergy hadronic collisions, from the production mechanisms
in proton–proton collisions (pp) and their modification in
proton–nucleus collisions (p–A) to the investigation of the
properties of the hot and dense strongly interacting Quark–
Gluon Plasma (QGP) in nucleus–nucleus collisions (AA).
Heavyflavour production in pp collisions provides
important tests of our understanding of various aspects of QCD.
The heavyquark mass acts as a long distance cutoff so
that the partonic hardscattering process can be calculated
in the framework of perturbative QCD down to low
transverse momenta ( pT). When the heavyquark pair forms a
quarkonium bound state, this process is nonperturbative as
it involves long distances and soft momentum scales.
Therefore, the detailed study of heavyflavour production and the
comparison to experimental data provides an important
testing ground for both perturbative and nonperturbative aspects
of QCD calculations.
In nucleus–nucleus collisions, open and hidden
heavyflavour production constitutes a sensitive probe of the hot
strongly interacting medium, because hardscattering
processes take place in the early stage of the collision on a
time scale that is in general shorter than the QGP
thermalisation time. Disentangling the mediuminduced effects
and relating them to its properties requires an accurate
study of the socalled cold nuclear matter (CNM) effects,
which modify the production of heavy quarks in nuclear
collisions with respect to proton–proton collisions. CNM
effects, which can be measured in proton–nucleus
interactions, include: the modification of the effective partonic
luminosity in nuclei (which can be described using
nuclearmodified parton densities), due to saturation of the parton
kinematics phase space; the multiple scattering of partons in
the nucleus before and after the hard scattering; the
absorption or breakup of quarkonium states, and the interaction
with other particles produced in the collision (denoted as
comovers).
The nuclear modification of the parton distribution
functions can also be studied, in a very clean environment, using
quarkonium photoproduction in ultraperipheral nucleus–
nucleus collisions, in which a photon from the coherent
electromagnetic field of an accelerated nucleus interacts with the
coherent gluon field of the other nucleus or with the gluon
field of a single nucleon in the other nucleus.
During their propagation through the QGP produced in
highenergy nucleus–nucleus collisions, heavy quarks
interact with the constituents of this medium and lose a part
of their momentum, thus being able to reveal some of the
QGP properties. QCD energy loss is expected to occur via
both inelastic (radiative energy loss, via mediuminduced
gluon radiation) and elastic (collisional energy loss)
processes. Energy loss is expected to depend on the parton
colourcharge and mass. Therefore, charm and beauty quarks
provide important tools to investigate the energyloss
mechanisms, in addition to the QGP properties. Furthermore,
low 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 inmedium dissociation probability of
these states are expected to provide an estimate of the initial
temperature reached in the collisions. At high centreofmass
energy, a new production mechanism could be at work in the
case of charmonium: the abundance of c and c quarks might
lead to charmonium production by (re)combination of these
quarks. An observation of the recombination of heavy quarks
would therefore directly point to the existence of a deconfined
QGP.
The first run of the Large Hadron Collider (LHC), from
2009 to 2013, has provided a wealth of measurements in
pp collisions with unprecedented centreofmass energies
√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 photoninduced
collisions. In the case of heavyion collisions, with respect
to the experimental programmes at SPS and RHIC, the LHC
programme has not only extended by more than one order
of magnitude the range of explored collision energies, but it
has also largely enriched the studies of heavyflavour
production, with a multitude of new observables and improved
precision. Both these aspects were made possible by the
energy increase, on the one hand, and by the excellent
performance of the LHC and the experiments, on the other
hand.
This report results from the activity of the SaporeGravis
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: heavyflavour and quarkonium
production in proton–proton collisions, the cold nuclear
matter effects on heavyflavour and quarkonium production in
proton–nucleus collisions, the QGP effects on open
heavyflavour production in nucleus–nucleus collisions, the QGP
effects on quarkonium production in nucleus–nucleus
collisions, and the production of charmonium in photoninduced
collisions. Sect. 7 presents an outlook of future heavyflavour
studies with the LHC and RHIC detector upgrades and with
new experiments. A short summary concludes the report in
Sect. 8.
2 Heavy flavour and quarkonium production in
proton–proton collisions
2.1 Production mechanisms of open and hidden
heavyflavour in proton–proton collisions
2.1.1 Openheavyflavour production
Openheavyflavour production in hadronic collisions
provides important tests of our understanding of various aspects
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 cutoff 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 heavyflavour production and the
comparison to experimental data provides an important
testing ground for the theoretical ideas that deal with this class of
problems.
On the phenomenological side, the (differential) cross
section for openheavyflavour production is sensitive to the
gluon and the heavyquark content in the nucleon, so that
LHC data in pp and p–Pb collisions can provide valuable
constraints on these partondistribution functions (PDFs) inside
the proton and the lead nucleus, respectively. In addition,
these cross sections in pp and p–A collisions establish the
baseline for the study of heavyquark production in
heavyion collisions. This aspect is a central point in heavyion
physics since the suppression of heavy quarks at large pT is an
important signal of the QGP (see Sect. 4). Finally, let us also
mention that a solid understanding of opencharm
production is needed in cosmicray and neutrino astrophysics [
1
].
In the following, we will focus on pp collisions and review
the different theoretical approaches to openheavyflavour
production.
FixedFlavourNumber Scheme Conceptually, the
simplest scheme is the FixedFlavourNumber 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
centreofmass 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 shortdistance 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 massfactorisation procedure and which therefore also
depends on μF . The partonic cross section also depends on
the strongcoupling constant αs , which is evaluated at the
renormalisation scale μR . As a remainder of this procedure,
the shortdistance 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 heavyquark mass present in the partonic cross section
have not been removed by the massfactorisation procedure.
These logarithms are therefore not resummed to all orders in
the FFNS but are accounted for in FixedOrder (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 subprocesses 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 (3FFNS) 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
charmquark mass is neglected in the hardscattering 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 nexttoleading order (NLO), the
virtual oneloop 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
oneparticle 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,
GMVFNS, 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, scaleindependent, FF DQH (z)
describing the transition of the heavy quark with momentum pQ
into the observed heavyflavoured 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 powerlike 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 scaleindependent 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 heavyquark
production is dominated by the ggchannel (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.
ZMVFNS 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 higherorder
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 heavyquark mass. Such a
scheme, where the number of active flavours is changed when
crossing the transition scales is called a
VariableFlavourNumber Scheme (VFNS). If, in addition, the heavyquark
mass m Q is neglected in the calculation of the shortdistance
cross sections, the scheme is called ZeroMass VFNS
(ZMVFNS). The theoretical foundation of this scheme is provided
by a wellknown factorisation theorem and the differential
cross section for the production of a heavyflavoured 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 heavyquark mass is neglected in the
shortdistance cross sections (dσˆ ), the predictions in the ZMVFNS
are expected to be reliable only for very large transverse
momenta. The sum in Eq. (4) extends over a large number of
subprocesses i + j → k + X since a, b, c can be gluons, light
quarks, and heavy quarks. A calculation of all subprocesses
at NLO has been performed in the late 1980s [
10
].
Concerning the FFs into the heavyflavoured hadron H =
D, B, c, . . ., two main approaches are employed in the
literature:
• In the PerturbativeFragmentation 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 finalstate collinear logarithms of the
heavyquark mass. Their scaledependence is governed
by the DGLAP evolution equations and the boundary
conditions for the PFFs at the initial scale are calculable
in the perturbation theory. On the other hand, the
scaleindependent FF is a nonperturbative object (in the case
of heavylightflavoured hadrons) which is assumed to be
universal. It is usually determined by a fit to e+e− data,
although approaches exist in the literature which attempt
to compute these functions. It is reasonable to identify the
scaleindependent fragmentation function in Eq. (3) with
the one in Eq. (5). This function describing the
hadronisation process involves longdistance physics and might
be modified in the presence of a QGP, whereas the PFFs
(or the unresummed collinear logarithms ln pT2/m2Q in
the FFNS) involve only shortdistance 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
nonperturbative piece. Instead, boundary conditions at
an initial scale μF m Q are determined from e+e−
data for the full nonperturbative 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 zspace whereas the FFs
in the PFF approach are determined in MellinNspace 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). GMVFNS The
FFNS and the ZMVFNS 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 ZMVFNS. The
GeneralMass VFNS (GMVFNS) [
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 SACOT
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 allorder 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 Dmeson 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 ZMVFNS from all other subprocesses
at O(αs2) resumming the collinear logarithms associated to
the heavy quark at the leadinglogarithmic (LL) accuracy. In
contrast, the GMVFNS 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 heavyhadron decays have been studied for pp collisions
at the LHC at 2.76 and 7 TeV centreofmass 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 heavyflavour
hadroproduction in the GMVFNS 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 subprocesses as in the ZMVFNS
are taken into account. However, the finite heavyquarkmass
terms (powers of m2Q / pT2) are retained in the shortdistance
cross sections of subprocesses involving heavy quarks. More
precisely, the heavyquarkmass terms are taken into account
in the subprocesses 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 heavyquarkinitiated
subprocesses (Q + g → Q + X , Q + g → g + X , …) as is done
in the SACOT scheme [
19
]. The massive hardscattering
cross sections are defined in a way that they approach, in the
limit m Q / pT → 0, the massless hardscattering cross
sections defined in the MS scheme. Therefore, the GMVFNS
approaches the ZMVNFS at large pT m Q . It can be
shown that the GMVFNS converges formally to the FFNS
at small pT. However, while the SACOT 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 bquark 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 heavyquark PDF is switched off sufficiently
rapidly and the GMVFNS approaches the FFNS at small
pT [
7
].
FONLL Similar to the GMVFNS, the FixedOrder plus
NexttoLeading 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 centreofmass 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 ZMVNFS (=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 powerlike mass terms can be
neglected and the cross section is dominated by the collinear
logarithm of the heavyquark 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 fixedorder
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 bquark
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 GMVFNS is given by
dσGM−VFNS = dσFO + dσRS − dσFOM0, i.e. no interpolating
factor is used.
Other differences concern the nonperturbative input. In
particular, the FONLL scheme uses fragmentation functions
in the PFF formalism whereas the GMVFNS uses
fragmentation functions which are determined in the zspace in the
BKK approach.
Monte Carlo generators The GMVFNS and FONLL
calculations are mostly analytic and provide a precise
description of the inclusive production of a heavy hadron or its
decay products at NLO+NLL accuracy. Compared to this,
generalpurpose MonteCarlo generators like PYTHIA [
45
]
or HERWIG [
46
] allow for a more complete description
of the hadronic final state but only work at LO+LL
accuracy. However, in the past decade, NLO Monte Carlo
generators have been developed using the MC@NLO [
47
] and
POWHEG [
48
] methods for a consistent matching of NLO
calculations with parton showers. They, therefore, have all the
strengths of Monte Carlo generators, which allow for a
complete modelling of the hadronic final state (parton showering,
hadronisation, decay, detector response), while, at the same
time, the NLO accuracy in the hard scattering is kept and the
soft/collinear regimes are resummed at the LL accuracy. A
comparison of POWHEG NLO Monte Carlo predictions for
heavyquark production in pp collisions at the LHC with the
ones from the GMVFNS and FONLL can be found in [
49
].
2.1.2 Quarkoniumproduction mechanism
The theoretical study of quarkoniumproduction processes
involves both pertubative and nonperturbative aspects of
QCD. On one side, the production of the heavyquark
pair, Q Q, which will subsequently form the quarkonium,
is expected to be perturbative since it involves momentum
transfers at least as large as the mass of the considered heavy
quark, as for openheavyflavour production discussed in the
previous section. On the other side, the evolution of the Q Q
pair into the physical quarkonium state is nonperturbative,
over long distances, with typical momentum scales such as
the momentum of the heavyquarks in the boundstate rest
frame, m Q v and their binding energy m Q 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 heavyquarkpair production. In the
following, we briefly describe three of them which can be
distinguished in their treatment of the nonperturbative part:
the ColourEvaporation Model (CEM), the ColourSinglet
Model (CSM), the ColourOctet Mechanism (COM), the
latter two being encompassed in an effective theory referred to
as NonRelativistic QCD (NRQCD).
The ColourEvaporation Model (CEM) This approach is
in line with the principle of quark–hadron duality [
50,51
]. As
such, the production cross section of quarkonia is expected
to be directly connected to that to produce a Q Q pair in an
invariantmass region where its hadronisation into a
quarkonium is possible, that is between the kinematical threshold
to produce a quark pair, 2m Q , and that to create the lightest
openheavyflavour hadron pair, 2m H .
The cross section to produce a given quarkonium state is
then supposed to be obtained after a multiplication by a
phenomenological factor FQ related to a processindependent
probability that the pair eventually hadronises into this state.
One assumes that a number of nonperturbativegluon
emissions occur once the Q Q pair is produced and that the
quantum state of the pair at its hadronisation is essentially
decorrelated – at least colourwise – with that at its
production. From the reasonable assumption [
52
] that one ninth
– one coloursinglet Q 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 Qpair 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 ColourSinglet 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 heavyquark are suppressed by powers of αs (m Q ). In
principle, they are taken into account in the (p)QCD
corrections to the hardscattering part account for the Q Qpair
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 heavyquark pair with zero
relative velocity, v, in a coloursinglet state and in the same
angularmomentum and spin state as that of the tobe
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 Pwaves, 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 decaywidth
measurements. The model then becomes fully predictive but for the
usual unknown values of the nonphysical factorisation and
renormalisation scales and of the heavyquark 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 infrared
divergences in the case of Pwave decay at NLO, which were
earlier regulated by an ad hoc binding energy [
76
]. These can
nevertheless be rigorously cured [
77
] in the more general
framework of NRQCD which we discuss now and which
introduce the concept of colouroctet states.
The ColourOctet Mechanism (COM) and NRQCD Based
on the effective theory NRQCD [
78
], one can express in a
more rigorous way the hadronisation probability of a
heavyquark pair into a quarkonium via longdistance matrix
elements (LDMEs). In addition to the usual expansion in
powers of αs , NRQCD further introduces an expansion in v. It
is then natural to account for the effect of higherFock states
(in v) where the Q Q pair is in an octet state with a different
angularmomentum and spin states – the sole consideration
of the leading Fock state (in v) amounts to the CSM, which is
thus a priori the leading NRQCD contribution (in v).
However, this opens the possibility for nonperturbative
transitions between these coloured states and the physical meson.
One of the virtues of this is the consideration of 3 S[
8
] states in
1
Pwave productions, whose contributions cancel the
aforementioned divergences in the CSM. The necessity for such
a cancellation does not, however, fix the relative importance
of these contributions. In this precise case, it depends on a
nonphysical scale 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 decaywidth
meaLDMEs, OQ
surements nor lattice studies5 – but the leading CSM ones of
course. Only relations based on HeavyQuark 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 infrared
poles at NNLO could be absorbed in the NRQCD LDMEs,
provided that the NRQCD production operators were
modified to include nonabelian phases.
As mentioned above, it seems that the mere expansion
of the hard matrix elements in 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 quarkoniumproduction cross section
in powers of pT/mQ before performing the αs expansion
of the shortdistance coefficients for the Q Q production.
This is sometimes referred to as the fragmentationfunction
approach (see [
86,87
]) which offers new perspectives in the
theoretical description of quarkonium hadroproduction
especially at mid and large 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 breakdown of NRQCD factorisation in this
kinematical region. In any case, as for now, past claims that
colouroctet 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 quasimulti 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 colourtransfer 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 openheavyflavour 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 nonprompt mesons are therefore supposed
to come from bdecay only.
3. Detection of leptons from these decays. This can be done
(i) by subtracting a cocktail of known/measured sources
(photon conversions, Dalitz decays of 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
invariantmass 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 bjets. 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 btagging 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 (btagged jets give access to very large
pT whereas exclusivedecay channels allow for differential
studies down to pT equal 0). A fifth method based on the
indirect extraction of the total charm and beautyproduction
from dileptons – as opposed to single leptons – (see e.g. [
97
])
is not discussed in this review.
Hiddenheavyflavour, i.e. quarkonia, are also analysed
through their decay products. The triplet Swaves are the
most studied since they decay up to a few per cent of the
time in dileptons. This is the case for J/ψ , ψ (2S), ϒ (1S),
ϒ (2S), ϒ (3S). The triplet Pwaves, 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 heavyflavour decay leptons in pp collisions: a
electrons at midrapidity for √s = 200 GeV from PHENIX [112], b
electrons at midrapidity 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). GMVFNS [
15,16
] and kTfactorisation [
122
] calculations
are also drawn in b
Swave. For other states, such as the singlet Swave, 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 pTintegrated
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
pTintegrated results and nearly all yintegrated 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 heavyflavour decays
The first openheavyflavour measurements in heavyion
collisions were performed by exploiting heavyflavour decay
leptons at RHIC by the PHENIX and STAR Collaborations.
These were done both in pp and AA collisions [
112–116
].
At the LHC, the ATLAS and ALICE Collaborations have
also performed such studies in heavyion collisions [
117–
121
]. A selection of the pTdifferential production cross
sections of heavyflavour 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 kTfactorisation [
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
/bG102
(
m
[)
y
dpT
dpT104
π
2
cc/)(
2 σ106
d
( 0 1
2
D0 / 0.565
D* / 0.224
FONLL
powerlaw fit
3
(a)
4
5 6
pT (GeV/c)
Fig. 3 dσ/d pT for Dmeson 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 heavyflavourdecay leptons are
well described by pQCD calculations.
In addition, leptons from open charm and beauty
production can be separated out via: (i) a cut on the lepton impact
parameter, i.e. the distance between the origin of the
lepton and the collision primary vertex, (ii) a fit of the lepton
impactparameter distribution using templates of the
different contributions to the inclusive spectra, (iii) studies of the
azimuthal angular correlations between heavyflavour decay
leptons and charged hadrons (see e.g. [
107,123
]). These
measurements are also described by pQCD calculations.
2.2.2 Open charm
Recently, Dmeson 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. Dmeson 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 Dmeson 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 Dmeson
yields. Prompt yields include both the direct production and
the feeddown from excited charmed resonances. The
secondaries contribution to the Dmeson yields is evaluated and
subtracted by: (i) either scrutinising the Dmeson candidates
impactparameter distribution, exploiting the larger lifetime
of b than cflavoured hadrons [
102,106,124
], which requires
stat. unc.
syst. unc.
FONLL
GMVFNS
102 ± 3.5% lumi,± 2.1% BR norm. unc. (not shown)
0 5 10 15
ALICE
D+, pp s = 7 TeV, Lint = 5 nb1
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 pQCDbased calculations [
104,125,126
],
advantageous strategy for smaller data samples but limited by the
theoretical uncertainties.
Figure 3 presents a selection of the Dmeson
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, GMVFNS and kTfactorisation
calculations agree with data. The agreement among the FONLL
and POWHEG calculations is better for heavyflavour decay
leptons than for charmed mesons, which seems to be related
to the larger influence of the fragmentation model on the
latter. The Ds+ pTdifferential cross section is compared to
calculations in Fig. 3c. The Ds+ measurements are also
reproduced by FONLL, GMVFNS and kTfactorisation
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 Bfactories and fixedtarget 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 pTdifferential cross
section compared to GMVFNS 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 GMVFNS [
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
Openbeauty production is usually measured by looking for
bjets or for beauty hadrons via their hadronic decays,
similarly to D mesons. They have been traditionally studied at
the e+e− Bfactories (see e.g. [
134,135
]), where, despite the
small bquark 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 Bfactories. They are, however, studied at
Tevatron and at the LHC hadron colliders. The higher
collision energy increases their production cross section, although
their reconstruction is more difficult at hadron colliders due
to the larger combinatorics compared to the e+e−
environment. It should also be kept in mind that the experiments
optimised to study the high 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 bhadron
production cross section at hadron colliders. bjet
measurements have the advantage to be the least dependent on the
bquark 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 pseudoproper decay length
distribution of the charmonium candidates, L x y (m/ pT)J/ψ . Figure
5 presents a selection of the LHC results: the nonprompt
pTdifferential 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 GMVFNS [
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 nonprompt charmonium in a purely hadronic
decay channel at hadron colliders, shows a similar
transversemomentum spectrum for nonprompt singlet and triplet
Swave charmonia.
Studies of openbeauty 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 1011
 )
+π 102
 πμ) 103
+ μ→104
(
ψ/J 105
→)S106
2
(
ψ( 107 10
B
]V 10
e
G
/
b
μ[
.)
4
2
< 1
y
;
+ XB
→pp101
(pT
d
σ/
d
102
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 crosssection measurement
in Ref. [154]. In particular, the doubly strange bbaryon
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
pTdistribution 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
pTspectrum at midrapidity 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 pTdifferential cross sections for nonprompt 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 GMVFNS [
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
]
Nonprompt 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 bhadron taking
about 70 % of the parton momentum on average at the
Zpole [
167
]. Identification of jets from beauty quark
fragmentation or “btagging” 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 impactparameter 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 btagging, 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 bjet 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 fb1
y b < 2.0
)
V
e
/bG101
n
(
)
X
Λ
Ψ
/J102
→
X
b
Λ
→
p
p
(
pT103
d
/
σ
d
Data
PYTHIA
POWHEG
POWHEG uncertainty
10
20
40pTΛb (GeV)
50
CMS s = 7 TeV
30 40 50
bhadron pT (GeV)
(c)
20 40 60 80 100 120 140 160 180 200
bjet pT (GeV)
200 300
Jet pT [GeV]
Fig. 8 bjet cross section as a function of pT in pp collisions at
√s = 7 TeV: a dσ/d pT from the lifetimebased and muonbased
analyses by CMS [
149
] and ATLAS [
150
] compared to the MC@NLO
calculation, and b d2σ/d pTdy by ATLAS from the lifetimebased
analysis [
150
] compared to the predictions of PYTHIA, POWHEG (matched
to PYTHIA) and MC@NLO (matched to HERWIG) [
46,47,151–153
]
the bjet 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
leptonbased 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
signaloverbackground 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 heavyion studies are carried out – none of the models
can simply be ruled out owing to their theoretical
uncertainties (heavyquark mass, scales, nonperturbative parameters,
unknown QCD and relativistic corrections, ...). Second, we
show the fraction of 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 Promptηc transversemomentum
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 Pwave state above
the opencharm 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 baryonantibaryon decay.
Indeed, all n = 1 charmonia can decay in this channel with
a similar branching ratio, which is small, i.e. on the order
of 10−3. LHCb is a pioneer in such a study with the first
measurement of 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 heavyquark 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 feeddown 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 pTintegrated 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 onshell 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 feeddown 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 Pwave quarkonia per se; these may not be
the same as that of Swave 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 coloursinglet and
colouroctet 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 feeddown.
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 openbeauty threshold offers
a wider variety of states that can be detected in the
dilepton decay channel – this, however, introduces a complicated
feeddown 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 – associatedproduction
channels, which we discuss in Sect. 2.4, are probably more
promising. Figure 12c shows ratios of different Swave
bottomonium yields. These are clearly not constant as one might
anticipate following the idea of the CEM. Simple mass effects
through feeddown decays can induce an increase of these
ratios [
74,199
], but these are likely not sufficient to explain
the observed trend if all the direct yields have the same pT
dependence. The χb feeddown, 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 openbeauty 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 spinstate 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) pTdifferential cross
section show that the feeddown 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 feeddown 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
heavyion 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 feeddown 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
feeddown 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 feeddown
fraction of 40 % with a significant uncertainty (up to 15 %). The
situation is schematically summarised on Fig. 14.
2.2.6 Bc and multiplecharm 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 fixedtarget experiment
SELEX at Fermilab [
222, 223
].
2.3 Quarkoniumpolarisation studies
Measurements of quarkonium polarisation can shed more
light on the longstanding puzzle of the quarkonium
hadroproduction. Various models of the quarkonium production,
described in the previous Sect. 2.1.2, are in reasonable
agreement with the cross section measurements but they usually
fail to describe the measured polarisation.
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 orderαs4 calculation – as
opposed to fragmentationfunctionbased 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 crosscheck
between measurements done in different reference frames.
Different methods have been used to extract the polarisation
parameter(s) from the angular dependence of the yields. In
the following, we divide them into two groups: (i) 1D
technique: fitting cos 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)
2D technique: fitting a twodimensional 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, physicswise.14
Indeed, as we discussed in the previous section, a given
quarkonium yield can come from different sources, some of
which are not of specific interests for data–theory
comparisons. The most obvious one is the nonprompt charmonium
yield, which is expected to be the result of quite different
mechanism that the prompt yield. Nowadays, the majority of
the studies are carried out on a prompt sample thanks to a
precise vertexing of the events. Yet, a further complication
also comes from feeddown from the excited states in which
case vertexing is of no help. As for now, no attempt of
removing it from e.g. prompt 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 feeddown
could be expected in the analysed sample.
In view of this, Tables 1, 2, 3, 4 and 5 contain, in addition
to the information on the colliding systems and the
kinematical coverages, information on the fit technique and a
short reminder of the expected feeddown. For each
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 highmultiplicity
proton–nucleus collisions in the lightflavour sector could
manifest also for heavyflavour production. This question
could become accessible with future higherstatistics proton–
nucleus data samples at RHIC and LHC.
The strong electromagnetic field of lead ions circulating
in the LHC is an intense source of quasireal 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 datataking 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 inmedium interaction mechanisms, such as
radiative, collisional energy loss and inmedium
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
feeddown contributions) make the situation not satisfactory
enough.
The next steps in the study of heavyflavour hadron
production in heavyion 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 setups 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
ECOSConicyt 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
ECOSConicyt 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:
68047232) and Projectruimte grants from the Dutch Foundation for
Fundamental Research (project numbers: 10PR2884 and 12PR3083).
The work of M. Nahrgang was supported by the PostdocProgram of the
German Academic Exchange Service (DAAD) and the U.S. Department
of Energy under Grant DEFG0205ER41367. The work of S. Peigné
was supported by the ECOSConicyt Grant No. C12E04 between France
and Chile. The work of R. Rapp was supported by the USNSF Grant
No. PHY1306359. The work of B. Trzeciak was supported by the
European social fund within the framework of realizing the project, Support
of intersectoral 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.1320841S. The
work of I. Vitev was supported by Los Alamos National Laboratory
DOE Office of Science Contract No. DEAC5206NA25396 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 DEAC5207NA27344 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
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
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to the original author(s) and the source, provide a link to the Creative
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Funded by SCOAP3.
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