###### Abstract

Reconstructed jets initiated from heavy quarks provide a powerful tool to
probe the properties of the quark-gluon plasma (QGP) and to explore the mass
hierarchy of jet quenching. In this article, we review the recent theoretical
progresses on heavy-flavour jets in high-energy nuclear collisions at the RHIC
and LHC. We focus on the yields and substructures of charm and bottom quark
jets with jet quenching effect, such as the nuclear modification factors,
transverse momentum imbalance, angular correlation, radial profiles,
fragmentation functions, the “dead-cone” effect, etc.

###### keywords:

quark–gluon plasma; jet quenching; high-energy nuclear collisions; heavy-
flavour jet

15 3 727 Academic Editors: Vasilis K. Oikonomou and Silvio Pardi31 December
2022 17 February 2023 6 March 2023 15 March 2023
https://doi.org/10.3390/sym15030727 Heavy-Flavour Jets in High-Energy Nuclear
Collisions Heavy-Flavour Jets in High-Energy Nuclear Collisions Sa Wang 1,2,
Wei Dai 3, Enke Wang 1,2, Xin-Nian Wang 4 and Ben-Wei Zhang 2* Sa Wang, Wei
Dai, Enke Wang, Xin-Nian Wang, Ben-Wei Zhang Wang, S.; Dai, W.; Wang, E.;
Wang, X.-N.; Zhang, B.-W. Correspondence<EMAIL_ADDRESS>
##  1 Introduction

High-energy nuclear collisions at the Relativistic Heavy Ion Collider (RHIC)
and Large Hadron Collider (LHC) have opened up new avenues for the search for
strongly interacting nuclear matter, the quark–gluon plasma (QGP) Wang:1992qdg
; Gyulassy:2003mc ; Mehtar-Tani:2013pia ; Qin:2015srf . Investigating the
formation of the QGP deepens our understanding of quantum chromodynamics (QCD)
under extreme conditions at high temperature and density Cunqueiro:2021wls ;
Cao:2020wlm  and the evolution of the Universe at the first microsecond
Collins:1974ky . The jet-quenching phenomena, the energy attenuation of fast
partons due to their strong interactions with the QCD medium, provide an army
of powerful tools to study the properties of the QGP, such as the yield
suppression of high-$p_{T}$ hadron/jet, the $p_{T}$ asymmetry of dijets,
$\gamma/Z^{0}$\+ jets as well as jet substructures Wang:1998bha ; Wang:2001cs
; Wang:2002ri ; Majumder:2004pt ; Zhang:2007ja ; Vitev:2009rd ; He:2020iow ;
Wang:1996yh ; Neufeld:2010fj ; Dai:2012am ; Wang:2013cia ; Li:2010ts ;
Vitev:2008rz ; Caucal:2021cfb .

In elementary proton–proton reactions, the productions of charm and bottom
quarks are perturbatively calculable, since their large masses ($M_{c}\sim
1.5$ GeV, $M_{b}\sim 4.8$ GeV) act as a natural cut-off above the
$\Lambda_{QCD}$ Andronic:2015wma . Heavy quarks are produced in the initial
hard scattering at a very early stage due to their large masses, therefore
witnessing the whole QGP evolution. Meanwhile, while their thermal production
is almost negligible with the initial conditions so far accessible in heavy-
ion programs at the RHIC and LHC Zhang:2007yoa , the productions of charm and
bottom hadron/jets make a very promising hard probe to the transport
properties of hot and dense quark matter. During the past decade, the
experimental measurements including the nuclear modification factor $R_{AA}$
Adamczyk:2014uip ; Adam:2015sza ; Sirunyan:2017xss ; PHENIX:2011img ;
STAR:2013eve ; ALICE:2014wnc ; CMS:2011all  and the collective flow (the
direct flow $v_{1}$ STAR:2019clv ; ALICE:2019sgg  and elliptical flow $v_{2}$
Abelev:2014ipa ; Adamczyk:2017xur ; Acharya:2017qps ; Sirunyan:2017plt ) of
heavy-flavour hadrons both at the RHIC and LHC have attracted much attention
from the community of high-energy nuclear physics.

A lot of theoretical studies have been performed to confront the experimental
data obtained in the high-energy heavy-ion collisions, which greatly improve
our understanding of the in-medium evolution Zhang:2003wk ; Djordjevic:2003zk
; Zhang:2004qm ; vanHees:2007me ; CaronHuot:2008uh ; Djordjevic:2015hra ;
He:2014cla ; Kang:2016ofv ; Svetitsky:1987gq ; Moore:2004tg ; Cao:2013ita ;
Alberico:2013bza ; Xu:2015bbz ; Cao:2016gvr ; Das:2016cwd ; Ke:2018tsh ;
Li:2020kax ; Liu:2009nb ; Yan:2006ve ; JETSCAPE:2022hcb ; Liu:2021dpm ;
Ding:2021ajz  and hadronization mechanisms Plumari:2017ntm ; Cao:2019iqs ;
He:2019vgs  of heavy quarks (for detailed reviews see Rapp:2009my ;
Prino:2016cni ; Rapp:2018qla ; Dong:2019unq ; Dong:2019byy ; Zhao:2020jqu ;
Apolinario:2022vzg ; He:2022ywp ; Tang:2020ame ). Specifically, the current
models treat the elastic and inelastic interactions between heavy quarks and
the QGP medium with multiple methods, consisting mainly the perturbative or
non-perturbative analytic calculations (SCET Kang:2016ofv ; Li:2018xuv , CUJET
Xu:2015bbz ; Xu:2014tda , DREENA Djordjevic:2009cr ; Djordjevic:2008iz ;
Djordjevic:2013xoa ; Zigic:2021rku ; Zigic:2018ovr ; Zigic:2018smz , WHDG
Wicks:2007am ; Wicks:2005gt , AdS/CFT (HG) Horowitz:2007su ; Horowitz:2011wm
), and the Monte Carlo transport approaches based on the Boltzmann (BAMPS
Uphoff:2010sh ; Uphoff:2011ad ; Uphoff:2013rka ; Uphoff:2014hza , MC$@_{s}$HQ
Gossiaux:2008jv ; Gossiaux:2009mk ; Nahrgang:2013saa , (Q)LBT Cao:2016gvr ;
Liu:2021dpm , LIDO Ke:2018tsh ; Ke:2020clc , Catania-pQCD/QPM Plumari:2012ep ;
Das:2015ana ; Plumari:2011mk ; Scardina:2017ipo ), the Langevin (POWL ANG
Alberico:2013bza ; Alberico:2011zy ; Beraudo:2014boa , Duke Cao:2013ita ;
Cao:2011et , UrQMD Lang:2012nqy ; Lang:2013cca ; Lang:2013wya , TAMU He:2012df
; He:2011yi ; He:2014cla , SHELL Wang:2019xey ; Dai:2018mhw ; Wang:2021jgm )
and the Kadanoff–Baym (PHSD Cassing:2008sv ; Cassing:2009vt ;
Bratkovskaya:2011wp ) equations. These phenomenological studies reveal a fact
that the elastic scattering of heavy quarks in the hot/dense nuclear matter is
important, especially at the lower $p_{T}$ region ($p_{T}^{Q}<5m_{Q}$),
different from our experience of treating light quarks or gluons. One of the
central issues of investigating the heavy-flavour production in the heavy-ion
program is extracting the diffusion coefficient $D_{s}$, which is directly
related to the transport properties of the hot QCD matter. Additionally,
different from the fragmentation hadronization of heavy quarks in a vacuum,
within the hot and dense nuclear matter, the heavy-flavour hadrons can be
produced by a combination of heavy quarks and thermal partons. Such a
coalescence hadronization mechanism plays an important role in the collective
flow Abelev:2014ipa ; Adamczyk:2017xur  and baryon-to-meson ratio STAR:2019ank
; Vermunt:2019ecg  of charmed hadron in nucleus–nucleus collisions at the RHIC
and LHC.

In recent years, the experimental measurements on heavy-flavour jet (a
reconstructed jet containing a heavy quark or a heavy-flavour hadron) have
made great strides in p+p CMS:2016wma ; ALICE:2019cbr ; CMS:2012pgw ;
ATLAS:2011chi ; ATLAS:2021agf ; STAR:2009kkp ; ATLAS:2016anw ; CMS:2020geg ,
p+A CMS:2016wma ; Khachatryan:2015sva ; ALICE:2021wct  and A+A collisions
Chatrchyan:2013exa ; Sirunyan:2018jju ; Sirunyan:2019dow ; ATLAS:2022fgb ;
Roy:2022yrw ; CMS:2021puf ; ALICE:2018lyv ; CMS:2022btc . A wealth of
information carried by heavy-flavour jets not only offers a new topic of jet
physics and the application of the perturbative QCD, but also their medium
modifications in heavy-ion collisions are also of great significance to reveal
the in-medium energy loss mechanism of heavy quarks, to address the mass
effect of jet quenching, and to extract the transport properties of the QGP.

##  2 Recent Advances of Heavy-Flavour Phenomenology in Heavy-Ion Collisions

Generally speaking, as we discussed in the last section, the reason for
treating the heavy flavours as powerful hard probes to the transport
properties of the QGP consists of at least three aspects. Firstly, the large
mass ($M_{Q}\gg\Lambda_{QCD}$) makes it available to compute the differential
cross-section of heavy quarks in the binary nucleon–nucleon collisions based
on the perturbative QCD (pQCD) scheme within the next-to-next-to-leading order
(NNLO) precision Cacciari:2005rk . Secondly, due to the large mass ($M_{Q}\gg
T_{\rm med}$), the total yield of heavy quarks in nucleus–nucleus collisions
only depends on their initial production at hard scattering. Since the
momentum transfer of the in-medium collisions $q^{2}\sim g^{2}T^{2}$($T\sim
0.4$–$0.5$ GeV) is much smaller than the creation energy of heavy quark pairs
at the current collision energy, both at the RHIC and LHC, the subsequent
contribution from the thermal creation during the QGP evolution is negligible
Zhang:2007yoa . Apart from this, according to the Heisenberg uncertainty
principle, the formation time of heavy
quarks($\tau_{0}\sim\frac{1}{2m_{Q}}<0.1$ fm/c) is shorter than the formation
time of the quark–gluon plasma ($\tau_{f}\sim 0.6$ fm/c). Therefore heavy
quarks witness the entire evolution of the hot/dense nuclear matter until the
freeze-out. In this section, we will briefly introduce the recent theoretical
advances that help us understand the heavy-flavour production in heavy-ion
collisions, including mainly the following several aspects, the initial
production, the transport approaches, the hadronization mechanisms, and the
extraction of diffusion coefficient.

###  2.1 Production of Heavy Quarks in p+p Collisions

The production of heavy quarks in proton–proton collisions establishes a
baseline to investigate the nuclear modification in high-energy nuclear
collisions both at the RHIC and LHC. The yield of heavy flavours in
nucleus–nucleus collisions generally is viewed as the sum of that in $N_{\rm
coll}$ binary nucleon–nucleon collisions while taking into account the initial
cold nuclear matter effect (usually considered by using the nuclear-modified
parton distribution function Eskola:2009uj ; Eskola:2016oht ; NNPDF:2014otw ).
In the fixed-flavour-number scheme (FFNS) Andronic:2015wma , the cross-section
of heavy quarks in p+p collisions can be expressed based on the factorization
theorem,

$\displaystyle
d{\sigma}_{Q}[s,p_{T},y,m_{Q}]\simeq\sum_{i,j}\int_{0}^{1}dx_{i}\int_{0}^{1}dx_{j}f_{i}^{A}(x_{i},\mu_{F})f_{j}^{A}(x_{j},\mu_{F})d\tilde{{\sigma}}_{ij\rightarrow
Q+X}[x_{i},x_{j},s,p_{T},y,m_{Q},\mu_{F},\mu_{R}]$ (1)

where $s$ is the square of the centre-of-mass energy of the incoming proton,
$p_{T}$ is the transverse momentum of the produced heavy quark, and $y$ is the
rapidity. $f_{i}^{A}$ ($f_{j}^{B}$) is the parton distribution function (PDF)
quantifying the probability to find a parton with flavour $i$($j$) and
carrying momentum fraction $x_{i(j)}$ in the colliding proton $A(B)$, which
relies on the factorization scale $\mu_{F}$. $\tilde{{\sigma}}_{ij\rightarrow
Q+X}$ represents the cross-section of the partonic hard process
$i+j\rightarrow Q+X$ that can be calculated relying on the pQCD. The partonic
cross-section $\tilde{{\sigma}}_{ij\rightarrow Q+X}$ also relies on the strong
coupling constant $\alpha_{s}$ determined at the renormalization scale
$\mu_{R}$. Note that Equation (1) sums all partonic hard processes
$i+j\rightarrow Q+X$, where $i,j$ are the active flavours including
($u,\bar{u},d,\bar{d},s,\bar{s},g$) but not heavy quarks. Only at the
factorization scale $\mu_{F}>m_{c}$, can charm be viewed as an active flavour,
often used for beauty production. The differential cross-section
$d{\sigma}_{Q}$ can be convolved with a scale-independent fragmentation
function $D_{Q}^{H}(z)$, such as the Peterson Peterson:1982ak  or Lund
Andersson:1983ia  forms, to obtain the cross-section of the heavy-flavour
hadron,

$\displaystyle d{\sigma}_{H}=d{\sigma}_{Q}\otimes D_{Q}^{H}(z)$ (2)

where $H$ denotes the heavy-flavour hadron and $z$ the momentum fraction
carried by $H$.

Since the FFNS is usually applicable at the low $p_{T}$ region
($0<p_{T}<5m_{Q}$), for the higher kinematic region ($p_{T}\gg m_{Q}$), the
logarithmic terms ($\frac{\alpha_{s}}{2\pi}ln(p_{T}^{2}/m_{Q}^{2})$) in the
perturbative expansion of the cross-section become large, and should be
resummed to all orders. To implement such a resummation, one has to absorb the
large logarithmic terms into the parton distribution function and
fragmentation function. This treatment requires that heavy quarks are active
flavours when the factorization scale is $\mu_{F}>m_{Q}$. In other words, such
a scheme has a variable number of active flavours when $\mu_{F}$ crosses the
heavy quark mass, hence named the variable-flavour-number scheme (VFNS). In
particular, when the heavy quark mass can be neglected in the evaluation of
the short-distance cross-section, the VFNS scheme is called the zero-mass VFNS
(ZM-VFNS). In the ZM-VFNS, the differential cross-section of a heavy-flavour
hadron based on the factorization theorem can be expressed as:

$\displaystyle
d{\sigma}_{H+X}\simeq\sum_{i,j}\int_{0}^{1}dx_{i}\int_{0}^{1}dx_{j}f_{i}^{A}(x_{i},\mu_{F})f_{j}^{A}(x_{j},\mu_{F})d\tilde{{\sigma}}_{ij\rightarrow
k+X}D_{k}^{H}(z,{\mu}_{F}^{\prime})$ (3)

where $D_{k}^{H}(z,{\mu}_{F}^{\prime})$ is given by a convolution of a
perturbative-fragmentation function (PFF) $D_{k}^{Q}(z,{\mu}_{F}^{\prime})$
describing a parton k fragmentation into heavy quark Q, with a scale-
independent one $D_{Q}^{H}(z)$ for the hadronization of a heavy quark. Note
that in Equation (3) the sum covers all possible partonic-hard processes
($i+j\rightarrow k+X$) where $i,j,k$ can be light quarks, gluons, and heavy
quarks Aversa:1988vb . Since heavy quark mass is neglected in the computation
of the cross-section, the ZM-VFNS is expected to be reliable only at very high
$p_{T}$.

To find a unified theoretical framework that combines the advantages of the
FFNS at low $p_{T}$ and the ZM-VFNS at high $p_{T}$, in recent years the
interpolation schemes have been established, such as the general-mass VFNS
(GM-VFNS) Kniehl:2004fy ; Kniehl:2005mk  and the fixed-order plus next-to-
leading logarithms (FONLL) Cacciari:2005rk ; Cacciari:2012ny . For instance,
by using an interpolating function
$G(m_{Q},p_{T})=p_{T}^{2}/(p_{T}^{2}+c^{2}m_{Q}^{2})$ where $c$ is set to
$c=5$, the FONLL scheme can well describe the heavy-flavour production in the
entire kinematic region. For more details of the interpolation schemes see
Andronic:2015wma  and the references therein.

Compared to the analytic calculation schemes discussed above, the general-
purpose Monte Carlo event generator, such as PYTHIA Sjostrand:2006za , HERWIG
Corcella:2000bw , POWHEG Frixione:2007vw  and SHERPA Gleisberg:2008ta , can
provide a more complete description of all the final-state particles at the
parton or hadron level. Especially for the studies of jet physics, the Monte
Carlo event generators can give more precise descriptions of the observations
relating to the jet substructure than analytic calculations.

###  2.2 Transport of Heavy Quarks in the QGP

Transport approaches are wildly used in the current theoretical studies of
heavy-flavour production in high-energy nuclear collisions. At the lower
$p_{T}$ region, the elastic scattering of heavy quarks with the thermal parton
(light quark or gluon) has been proven to be the dominant mechanism of energy
loss. Generally, the kinetic theory based on the Boltzmann transport equation
is a popular treatment for in-medium heavy quark evolution. The Boltzmann
equation for the distribution function of heavy quarks can be written in a
compact form,

$\displaystyle
p_{\mu}\partial_{\mu}f_{Q}(x,p)=\mathit{C}[f_{q},f_{\bar{q}},f_{g},f_{Q}](x,p)$
(4)

where $f_{Q}(x,p)$ is the phase-space distribution of heavy quarks. In the
QGP, the phase-space distributions of light quark $f_{q}$ and gluon $f_{g}$
can be solved by the Boltzmann equation Ferini:2008he ; Ruggieri:2013ova .
Subsequently, the relativistic Boltzmann-like collision integral
$\mathit{C}[f_{Q}](x,p)$ has a simplified form Svetitsky:1987gq ;
Beraudo:2014iva ,

$\displaystyle\mathit{C}[f_{Q}]=\int
d^{3}q[\omega(\mathbf{p}+\mathbf{q},\mathbf{q})f_{Q}(\mathbf{x},\mathbf{p}+\mathbf{q},t)-\omega(\mathbf{p},\mathbf{q})f_{Q}(\mathbf{x},\mathbf{p},t)]$
(5)

where $\omega(\mathbf{p}+\mathbf{q},\mathbf{q})$ represents the transition
rate of a heavy quark from the momentum $\mathbf{p}+\mathbf{q}$ to
$\mathbf{p}$ by collisions with quasiparticles. This rate is usually
determined by the matrix elements of the $2\rightarrow 2$ QCD scattering. With
the assumption that the momentum transfer $|\mathbf{q}|$ is small compared to
the momentum of a heavy quark, we can expand
$\omega(\mathbf{p}+\mathbf{q},\mathbf{q})f_{Q}(\mathbf{x},\mathbf{p}+\mathbf{q},t)$
around $\mathbf{q}$ by utilizing the Taylor formula to obtain the
Fokker–Planck equation,

$\displaystyle\frac{\partial f_{Q}}{\partial t}=\frac{\partial}{\partial
p_{i}}\left[A_{i}(\mathbf{p})f_{Q}+\frac{\partial}{\partial
p_{j}}[B_{ij}(\mathbf{p})f_{Q}]\right]$ (6)

where two coefficients $A_{i}(\mathbf{p})=\int
d^{3}q\omega(\mathbf{p},\mathbf{q})q_{i}$ and $B_{ij}(\mathbf{p})=\int
d^{3}q\omega(\mathbf{p},\mathbf{q})q_{i}q_{j}$ are directly related to the
drag coefficient ($\eta_{D}$) and the momentum diffusion coefficient
($\kappa$), which control rate of the energy loss and the momentum broadening
of heavy quarks in the hot medium, respectively. Indeed, the Fokker–Planck
equation is equivalent to another more well-known equation, the Langevin
equation,

$\displaystyle\frac{d\vec{x}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\vec{p}}{E}$ (7) $\displaystyle\frac{d\vec{p}}{dt}$
$\displaystyle=$ $\displaystyle-\eta_{D}(p)\vec{p}+\vec{\xi}(t)$ (8)

where the stochastic term $\vec{\xi}(t)$ describes the random kicks suffered
in heavy quarks from the medium constituents, which obeys a Gaussian
distribution with a mean value $0$ and variance $\kappa$. The drag coefficient
$\eta_{D}$ and the diffusion coefficient $\kappa$ are related by the
fluctuation–dissipation theorem (FDT) $\kappa=2\eta_{D}ET$. Note that at
higher kinematic regions ($p_{T}^{Q}>5m_{Q}$), the medium-induced gluon
radiation plays an increasingly important role in the energy loss of heavy
quarks. The radiative energy loss of heavy quarks is treated with various
formalisms and at different approximations Zhang:2003wk ; Arnold:2002ja ;
Abir:2011jb ; Abir:2012pu ; Zapp:2008gi ; Armesto:2003jh , which usually
provide the radiated gluon spectra as a function of momentum fraction $x$ and
transverse momentum $k_{\perp}$. In the Langevin equation, the radiative
energy loss of heavy quarks can be coupled with the collisional one by adding
a recoil term $-\vec{p}_{g}$ caused by the radiated gluon Cao:2013ita . The
four-momentum of the radiated gluon can be easily sampled based on the
radiation spectra $d{\rm N}_{g}/d{\rm x}d{\rm k}_{\perp}^{2}$.

In many of the recently developed theoretical frameworks modelling the
production of heavy flavour in heavy-ion collisions, the Boltzmann and
Langevin equations are the two most popular choices, especially for Monte
Carlo simulations. Concerning the performance of these two approaches,
detailed comparisons have been discussed in Das:2013kea ; Li:2019wri . In
general, the implementation of the Boltzmann equation implies that the medium
consists of well-defined quasiparticles, while the Fokker–Planck (Langevin)
equation is realized in a more general way without the quasiparticle
assumption. However, the advantage of the Boltzmann equation is that it can
naturally describe the heavy quark evolution even under off-equilibrium
conditions, which may be the case of the early pre-equilibrium stage in heavy-
ion collisions Das:2017dsh .

###  2.3 Hadronization: Fragmentation and Coalescence

Studying the yield suppression and collective flow of heavy-flavour hadrons
also deepens our understanding of heavy quark hadronization in nucleus–nucleus
collisions, which shows different mechanisms with that in a vacuum. As
discussed in Section 2.1, fragmentation functions describe the non-
perturbative hadronization process of heavy quarks into heavy-flavour hadron
in a vacuum. The most commonly used fragmentation function is the Peterson
form Peterson:1982ak ,

$\displaystyle D_{H/Q}(z)=\frac{N}{z[1-\frac{1}{z}-\frac{\epsilon_{Q}}{1-z}]}$
(9)

where $z$ denotes the momentum fraction carried by the heavy hadron from the
heavy quark in the fragmentation process ($0<z<1$), which implies that the
heavy hadron must have smaller energy than the heavy quark. The only tunable
parameter in Equation (9) is $\epsilon_{Q}$ that can be determined by fitting
to the measured spectra of the heavy-flavour hadrons. $N$ is the normalization
factor to guarantee $\int_{0}^{1}dzD_{H/Q}(z)=1$.

Measurements on the collective flow Abelev:2014ipa ; Adamczyk:2017xur  and
baryon-to-meson ratio STAR:2019ank ; Vermunt:2019ecg  of charmed hadron A+A
collisions suggest the existence of a new hadronization mechanism, coalescent
of heavy quarks. The basic idea behind the coalescence mechanism is that a
heavy quark can combine with a light anti-quark from the medium when they have
enough small distance in the coordinate-momentum space. It means that the
heavy-flavour meson has larger energy than the parent heavy quark, differing
from the mechanism of fragmentation. The distribution function of the formed
heavy-flavour meson usually can be obtained by a convolution with the
following schematic form.

$\displaystyle f_{M}\sim g_{M}f_{Q(\bar{Q})}\otimes
f_{\bar{q}(q)}\otimes\phi_{M}$ (10)

where $g_{M}$ denotes the degeneracy of the heavy-flavour meson in spin and
isospin, $f_{Q(\bar{Q})}$ and $f_{\bar{q}(q)}$ are the distribution functions
of the heavy and light quarks in the coordinate-momentum space, respectively.
$\phi_{M}$ represents the Wigner transform of the wave function of the heavy-
flavour meson, commonly approximated by the ground state one of the simple
harmonic oscillators Cao:2016gvr .

In the realistic implementation of the heavy quark hadronization in nuclear
collisions, the first step is to determine the probability of coalescence by
integrating the distribution function of Equation (10). If coalescence occurs,
one can sample a light anti-quark based on the thermal equilibrium
distribution, otherwise Equation (9) is used to fragment the heavy quark into
a hadron. At least in the lower $p_{T}$ region, the experimental results
favour the coalescence mechanism Scardina:2017ipo . The coalescence of heavy
quarks seems to decrease the suppression factor and enhance the collective
flow of heavy-flavour hadrons, especially at $p_{T}<6$ GeV. The recent studies
Cao:2019iqs ; He:2019vgs  show that the coalescence mechanism is important in
the description of the $\Lambda_{c}/D^{0}$ ratio measured by the STAR
STAR:2019ank  and ALICE Vermunt:2019ecg  collaborations. Additionally, the
hadronic scattering between the D meson and light-flavour hadrons ($D-\pi$,
$D-\rho$) has also been studied in Lin:2000jp , but its influence on the D
meson $R_{AA}$ was found to be very limited Cao:2015hia .

###  2.4 Extraction of the Diffusion Coefficient of Heavy Quarks

One of the most important goals of the heavy-ion collision experiment is to
investigate the transport properties of the QCD matter under extremely hot and
dense conditions. As discussed above, due to the large mass ($m_{Q}\gg T_{\rm
med}$), heavy quarks are believed to be powerful tools for exploring the
transport properties of the QGP. Phenomenological studies of heavy-flavour
production in high-energy nuclear collisions provide a unique opportunity to
extract the transport coefficient of the QGP, such as the momentum diffusion
coefficient $\kappa$ of heavy quarks, whose longitudinal and transverse
components can be convenient to define as,

$\displaystyle\kappa_{||}\equiv-\frac{d\left\langle(\Delta
p_{||})^{2}\right\rangle}{dt}$ (11)
$\displaystyle\kappa_{\perp}\equiv\frac{1}{2}\frac{d\left\langle(\Delta
p_{\perp})^{2}\right\rangle}{dt}$ (12)

where $\Delta p_{||}$ and $\Delta p_{\perp}$ momentum changes parallel and
perpendicular to the heavy quark formulation. By definition, $\kappa_{\perp}$
can be directly related to the jet transport coefficient $\hat{q}$ which
quantifies all the transverse momentum broadening of hard partons as
traversing the QGP medium. Assuming that the $\kappa$ is isotropic, namely,
$\kappa_{\perp}=\kappa_{||}=\kappa$, one can obtain a simplified relation
$\hat{q}=2\kappa$. This relation has been employed in the modified Langevin
equation to balance the two parts of the contribution from the collisional and
radiative energy loss of heavy quarks Cao:2013ita ; Li:2019lex . Here we only
overview the recent advances of the $\kappa$ extraction by different model
calculations. A more detailed and profound discussion about this topic can be
found in Rapp:2018qla .

The momentum diffusion coefficient $\kappa$ can be easily converted to the
spatial one $D_{s}$ with the relation $\kappa=2T^{2}/D_{s}$. In recent years,
the temperature dependence of the dimensionless quantity $2\pi TD_{s}$ has
been estimated by a lot of theoretical frameworks, such as the lattice QCD
(lQCD) Ding:2015ona ; Banerjee:2011ra ; Kaczmarek:2014jga , LO pQCD
Moore:2004tg ; vanHees:2004gq , QPM calculations Das:2015ana , T-matrix
He:2014cla , PHSD Song:2015sfa , MC$@_{s}$HQ Andronic:2015wma , Ads/CFT
Horowitz:2015dta , duke (Bayesian analysis) Xu:2017obm , and hadronic matter
He:2011yi ; Tolos:2013kva , as shown in Figure 1.

The estimates by the lQCD from the first principles provide a valuable
reference for the model extractions of $2\pi TD_{s}$. As one can see, with
relatively large uncertainties, the lQCD calculations in the quenched
approximation give $D_{s}\sim$ 3.7–7.0 Kaczmarek:2014jga  over the temperature
range from $T_{pc}$ to 2$T_{pc}$. However, it is difficult to extract
meaningful information about the temperature dependence of $2\pi TD_{s}$ from
the current lQCD results. Furthermore, except for the pQCD calculations at the
leading-order which show obvious larger values than others, these extractions
of $2\pi TD_{s}$ based on the recently developed models are consistent with
the lQCD data, as well as previous studies presented in Rapp:2018qla  which
give $2\pi TD_{s}\sim$ 2–4 near the critical temperature. Although these
calculations give different values of $2\pi TD_{s}$ versus $T/T_{pc}$, most
estimations show that $D_{s}$ slightly increases with $T$. It implies that the
interactions between a charm quark and the QCD medium have the strongest
strength near the critical temperature. However, no direct evidence has been
found in the experiment to verify this upward trend of $D_{s}$ so far because
it is hard to find an observation only sensitive to the in-medium interactions
at the late stage of the QGP evolution. Fortunately, the data-driven analysis
utilizing Bayesian inferences seems to shed new light on this issue. The
temperature and momentum dependence of $D_{s}$ has been extracted from the
available experimental data ($R_{AA}$ and $v_{2}$ of a D meson both at the
RHIC and LHC) Xu:2017obm  based on the Duke–Langevin transport model, which
indeed shows an upward trend of $2\pi TD_{s}$. More recently, this approach of
Bayesian inference has been improved with the help of information field theory
Bialek:1996kd ; Lemm:1999kd  in Xie:2022fak . Therefore, one can now extract
model parameters without relying on an explicit form of parametrization,
leading to a robust determination by such a model–data fit.

Figure 1: Spatial diffusion coefficient ($2\pi TD_{s}$) of charm quark in the
quark–gluon plasma calculated by different approaches versus the reduced
temperature ($T/T_{pc}$). The lattice QCD calculations in the quenched
approximation Ding:2015ona ; Banerjee:2011ra ; Kaczmarek:2014jga  are compared
with the estimations based on different models Moore:2004tg ; vanHees:2004gq ;
Das:2015ana ; He:2014cla ; Song:2015sfa ; Andronic:2015wma ; Horowitz:2015dta
; Xu:2017obm ; He:2011yi ; Tolos:2013kva . The figure is from Dong:2019byy .

##  3 Production of Heavy-Flavour Jets in Heavy-Ion Collisions

###  3.1 Nuclear Modification Factors of Production Yields

To address the nuclear effect in relativistic heavy-ion collisions, the
nuclear modification factor $R_{AA}$ is conventionally utilized to quantify
the yield suppression of hadron/jet in A+A collisions per binary
nucleon–nucleon collision relative to p+p STAR:2003pjh ,

$\displaystyle R_{AA}=\frac{1}{\left\langle N_{\rm bin}^{\rm
AA}\right\rangle}\frac{d\sigma^{\rm AA}/\rm dydp_{T}}{d\sigma^{\rm pp}/\rm
dydp_{T}}$ (13)

where the scaling factor $\left\langle N_{\rm bin}^{\rm AA}\right\rangle$
denotes the number of binary nucleon–nucleon collisions in A+A Miller:2007ri .
It has been observed that the values of $R_{AA}$ of hadrons and jets are
smaller than one in nucleus–nucleus collisions both at the RHIC STAR:2003wqp ;
PHENIX:2005nhb ; PHENIX:2010bqp  and LHC ATLAS:2018gwx ; ATLAS:2014ipv , and
these measurements could be explained by the mechanism of partonic energy
loss, which in turns serve as convincing evidence for the formation of QGP in
such extremely hot and dense conditions. Meanwhile, the jet transport
parameter $\hat{q}\equiv d\left\langle p^{2}_{\perp}\right\rangle/dL$
Baier:2002tc  representing the strength of in-medium partonic interactions
could be extracted from the available $R_{AA}$ data by various theoretical
models JET:2013cls ; Xie:2019oxg ; Ru:2019qvz ; JETSCAPE:2021ehl ; Xie:2022fak
.

Additionally, to test the mass dependence of jet quenching, the $R_{AA}$ has
also been used in a comparison of the yield suppression between heavy-flavour
jets and inclusive jets. Benefiting from the fact that heavy-flavour jets are
produced abundantly as the centre-mass energy increases in hadronic collisions
at the LHC, the exploration of a heavy quark-tagged jet produced in heavy-ion
collisions has gradually attracted much attention. The first experimental
effort focused on the production of a b-jet was implemented by the CMS
collaboration Chatrchyan:2013exa  in 2013, as shown in the left plot of Figure
2, where a b-jet is defined as jets containing at least one B hadron inside
the jet-cone. The red points are the CMS data and the coloured bands are the
theoretical calculations. This measurement accounts for the b-jet samples in
minimum bias collisions (0–100%). We note that even with large experimental
uncertainties, the b-jet $R_{AA}$ slightly increases with jet $p_{T}$ and
varies from 0.4 to 0.8. Significant suppression of the b-jet yield in Pb+Pb
collisions at $\sqrt{s_{NN}}$ = 2.76 TeV relative to the p+p baseline was
observed for the first time, which indicates that bottom quarks strongly
interact with the hot/dense nuclear matter. Furthermore, within experiment
uncertainties, the results were found to be consistent with the pQCD-based
calculations conducted in Huang:2013vaa  when the coupling factor $g^{med}$
varied from 1.8 to 2.2.

Figure 2: Left: the measured nuclear modification factor $R_{AA}$ of the
inclusive b-jet versus b-jet $p_{T}$ by the CMS collaboration in Pb+Pb at
$\sqrt{s_{NN}}$ = 2.76 TeV for minimum bias collisions Chatrchyan:2013exa .
Right: the comparison of $R_{AA}$ between the inclusive jets and b-jets versus
jet $p_{T}$ in central Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
Dai:2018mhw . The figures are from Chatrchyan:2013exa ; Dai:2018mhw .

To address the difference of the yield suppression between the b-jet and
inclusive jet (mainly initiated by a massless light quark or gluon), a direct
comparison of their $R_{AA}$ in the right plot of Figure 2 was presented by
the SHELL approach, which applies a Langevin transport model to describe heavy
quark propagation in the QGP Dai:2018mhw  in central Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 2.76 TeV, as well as the next-to-leading order pQCD
calculations matched with the parton shower effect for the p+p baseline
Gleisberg:2008ta ; Frixione:2002ik . In the model, the jet transport parameter
$\hat{q}$ was extracted by the production of an identified hadron in A+A
collisions Ma:2018swx , and then the spatial diffusion coefficient $D_{s}$ of
heavy quarks can be determined by the D meson $R_{AA}$ data Sirunyan:2017xss ;
ALICE:2018lyv . The measured $R_{AA}$ of the inclusive jet with the centrality
of 0–5% CMS:2016uxf  and b-jet with 0–10% Chatrchyan:2013exa  are also
illustrated in the plot of Figure 2. Although the $R_{AA}$ of the b-jet seems
to be slightly smaller than that of the inclusive jet, the CMS collaboration
claims that no clear difference of $R_{AA}$ between the inclusive jet and
b-jet was found, because the current uncertainties of the b-jet data are too
large. However, the theoretical calculations in Dai:2018mhw  suggest that
b-jet $R_{AA}$ may be larger than inclusive jet $R_{AA}$, due to the “dead-
cone” effect of the bottom quarks, which suppresses the medium-induced gluon
radiation of massive heavy quarks within a cone $\theta\sim M/E$
Dokshitzer:2001zm . A more precise measurement is necessary to resolve the
tension between the experimental data and theoretical calculations. It is very
exciting the fact that recently, the ATLAS collaboration reported preliminary
results by simultaneously measuring the $R_{AA}$ of the inclusive jet and
b-jet in 0–20% Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV ATLAS:2022fgb ,
which shows a clear weaker suppression of the b-jet, and the features can be
described by the theoretical calculations Dai:2018mhw ; Ke:2018tsh . Although
the mass hierarchy of jet quenching at the particle level has been confirmed
by a lot of experimental data PHENIX:2022wim ; STAR:2021uzu , it is
indisputable that the ATLAS measurement makes a crucial step towards finding
the mass effect at the jet level. The comparison of the c- and b-jet $R_{AA}$
has been presented in Li:2018xuv  with the SCET model Ovanesyan:2011kn ;
Sievert:2018imd , which shows no significant difference at $p_{T}>50$ GeV.
More recently, some exploratory estimates indicate that the $R_{AA}$ of the
c-jet may be stronger than that of the inclusive jet at higher jet $p_{T}$ due
to their different constituents Ke:2020nsm ; wang:2022aaa , an interesting
finding to be investigated further in detail.

Beyond $R_{AA}$, another observation $I_{AA}$ Neufeld:2010fj  has also been
utilized to study the yield suppression of b-jets tagged by $Z^{0}$ bosons in
high-energy nuclear collisions Wang:2020qwe . Similar to $R_{AA}$, $I_{AA}$ is
defined as follows,

$\displaystyle I_{AA}=\frac{1}{\left\langle N^{\rm AA}_{\rm
bin}\right\rangle}\frac{\frac{d\sigma^{\rm AA}}{dp^{\rm jet}_{\rm T}}|_{p_{\rm
T}^{\rm min}<p_{\rm T}^{Z}<p_{\rm T}^{\rm max}}}{\frac{d\sigma^{\rm
pp}}{dp^{\rm jet}_{\rm T}}|_{p_{\rm T}^{\rm min}<p_{\rm T}^{Z}<p_{\rm T}^{\rm
max}}}$ (14)

One can see that $I_{AA}$ quantifies the yield variation of the jet in A+A
collisions per binary nucleon–nucleon collision relative to the p+p baseline,
after integrating the $Z^{0}$ boson $p_{T}$. It has been proposed that the
associated production $Z^{0}\,+\,$b-jet may be helpful in addressing the mass
dependence of the jet-quenching effect, since the $Z^{0}\,+\,$jet processes
significantly exclude the contamination of gluon-initiated jets
Kartvelishvili:1995fr . Thereby the comparison of $I_{AA}$ between the
$Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet can provide direct features of the mass
effect of heavy quark jets compared to light-quark jets. Figure 3 shows the
comparisons of the calculated $I_{AA}$ of the $Z^{0}\,+\,$b-jet and
$Z^{0}\,+\,$jet in central 0–10% Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02
TeV, where the jets are reconstructed by the anti-$k_{T}$ algorithm with a
cone size R = 0.3 and pseudorapidity $|\eta^{\rm jet}|<$ 1.6. The calculations
are presented within different $p_{T}$ windows of $Z^{0}$ bosons in the three
panels, namely 40–60, 60–80, and 80–100 GeV. We note that the shapes of
$I_{AA}$ are flat at the panel of $40<p_{T}^{Z}<60$ GeV but have downward
trends at that of $80<p_{T}^{Z}<100$, and the $I_{AA}$ in the right panel even
shows enhancement at$p_{T}^{\rm jet}$. This is because if one constrains the
$Z^{0}$ meson momentum in the event selection, such as $80<p_{T}^{Z}<120$ GeV,
then the selected jet distribution with $p_{T}$ would fall steeper
at$p_{T}^{\rm jet}<$ 80 GeV, which naturally leads to a relatively large value
of the nuclear modification factors at $p_{T}^{\rm jet}<$ 80 GeV, even larger
than one. Additionally, one observes that at each panel the model calculations
show that $I_{AA}$ of the $Z^{0}\,+\,$b-jet is visibly higher than that of the
$Z^{0}\,+\,$jet in nucleus–nucleus collisions, which indicates that the
$Z^{0}$-tagged light-quark jets lose more energy than the $Z^{0}$-tagged
b-jets traversing the QGP. These comparisons would be helpful to directly test
the mass effect of jet energy loss in heavy-ion collisions at the LHC from a
new perspective.

Figure 3: Nuclear modification factor $I_{AA}$ as a function of the transverse
momentum of the tagged jet within three $p_{T}^{Z}$ windows: 40–60, 60–80,
80–120 GeV in central 0–10% Pb+Pb collisions at $\sqrt{s_{NN}}~{}=~{}5.02$ TeV
Wang:2020qwe .

###  3.2 Transverse Momentum Imbalance

The transverse momentum imbalance ($x_{J}=p_{\rm T,2}/p_{\rm T,1}$) is another
useful observation, describing the momentum asymmetry of the dijet system in
the transverse plane, where $p_{\rm T,1}$ and $p_{\rm T,2}$ denote the leading
and sub-leading jet $p_{T}$. It is noted that in the fixed-leading-order QCD
calculations the two outgoing hard partons should be strictly back-to-back in
the transverse ($x_{J}$ = 1), but the higher-order corrections and vacuum
shower may break the symmetry which leads to $x_{J}<$ 1\. In heavy-ion
collisions, the smaller $x_{J}$ of the $\gamma$+jet CMS:2017ehl  and
$Z^{0}$+jet CMS:2017eqd  systems have been observed in Pb+Pb collisions
compared to p+p, which results from the energy loss of the tagged jet. The CMS
collaboration reports the measurement on $x_{J}$ of the inclusive and
$b\bar{b}$ dijets in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV
Sirunyan:2018jju . In their measurements, the biggest challenge was how to
select the $b\bar{b}$ dijet events initiated by the hard heavy-quark pairs,
because it is crucial to address the mass effect by directly comparing such
heavy-quark dijets with inclusive dijets. On the theoretical side, the
production mechanisms of heavy quarks can be categorized into three classes:
flavour creation (FCR), flavour excitation (FEX), and gluon splitting (GSP)
Nason:1989zy ; Beenakker:1990maa ; Mangano:1991jk ; Norrbin:2000zc , only FCR
represents the dijets initiated by heavy-quark pairs originating from the hard
process. The CMS collaboration suggests a strategy to separate the FCR
processes by selecting $b\bar{b}$ dijets that have a large opening angle
($|\Delta\phi|>2\pi/3$) in azimuth, which could significantly suppress the
contributions of the other two. This method has also been used in theoretical
studies Huang:2015mva ; Dai:2018mhw ; Kang:2018wrs . Figure 4 shows a
comparison of the averaged $x_{J}$ of the inclusive and $b\bar{b}$ dijets in
both p+p and Pb+Pb collisions with different centrality bins at
$\sqrt{s_{NN}}$ = 5.02 TeV, as well as the experimental data Sirunyan:2018jju
, where $\left\langle x_{J}\right\rangle$ was estimated as follows.

Figure 4: Averaged $x_{J}$ value as a function of the number of participants
calculated in p+p and Pb+Pb collisions $\sqrt{s_{NN}}$ = 5.02 TeV within
different centrality bins compared with the experimental data, both for
inclusive (left) and $b\bar{b}$ (right) dijets. Figures are from Dai:2018mhw .

$\displaystyle\left\langle
x_{J}\right\rangle=\frac{1}{\sigma}\int_{0}^{1}\frac{d\sigma}{dx_{J}}dx_{J}$
(15)

The black triangle points are the CMS data in Pb+Pb collisions, and the black
star points are the p+p reference used in their measurements. The blue and red
rhombus points are the theoretical calculations, while the p+p reference is
provided by the Monte Carlo event generator SHERPA Gleisberg:2008ta  which
matches the next-to-leading order QCD matrix elements and the parton shower
effect in a vacuum Gleisberg:2008fv ; Schumann:2007mg . The $\left\langle
x_{J}\right\rangle$ points of the inclusive (left panel) and $b\bar{b}$ (right
panel) dijets are listed within three centrality bins which correspond to the
different numbers of the participant in Pb+Pb collisions. In Figure 4, the
theoretical calculations based on the SHELL model Dai:2018mhw  show an overall
decrease in $\left\langle x_{J}\right\rangle$ in Pb+Pb collisions relative to
the p+p baseline both for the inclusive and $b\bar{b}$ dijets, consistent with
the CMS data and indicates that the asymmetry between these two leading jets
is amplified in A+A collisions. The reduction in $\left\langle
x_{J}\right\rangle$ is centrality-dependent since the in-medium interaction is
sensitive to the temperature and size of the QGP. It is even more important
that the calculations show that the decrease in $\left\langle
x_{J}\right\rangle$ of the $b\bar{b}$ dijets is slightly smaller than that of
the inclusive dijets within the same centrality bins. These results suggest
that dijets initiated by bottom quarks may suffer smaller energy loss compared
to those initiated by light quarks or gluons. Furthermore, another study on
the $b\bar{b}$ dijet in heavy-ion collisions Kang:2018wrs  proposed that the
invariant mass $m_{jj}$ of the dijet system could be a novel observation
sensitive to mass effects of jet quenching.

In addition to the dijet system, the transverse momentum imbalances of the
$Z^{0}\,+\,$jet ($x_{jZ}=p_{T}^{\textit{\rm jet}}/p_{T}^{Z}$) and
$Z^{0}\,+\,$b-jet ($x_{bZ}=p_{T}^{\textit{\rm b-jet}}/p_{T}^{Z}$) have also
been investigated Wang:2020qwe . It was found that the $Z^{0}$-tagging
requirement considerably decreased the contribution of gluon-jets by 40% in
$Z^{0}\,+\,$jets compared to the dijet sample, especially at a lower jet
$p_{T}$. The comparison of the medium modification on the $x_{J}$ of
$Z^{0}\,+\,$jet and $Z^{0}\,+\,$b-jet may be suitable to address the mass
effect of jet quenching. Figure 5 shows the distributions of the $x_{jZ}$
(left) and $x_{bZ}$ (right) both in p+p and 0–10% Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV. In the calculations, the selected $Z^{0}$ bosons
are required to have $p_{T}^{Z}>$ 60 GeV. The tagged jets (b-jets) are
reconstructed with the anti-$k_{T}$ algorithm with a cone-size R = 0.3 and
pseudorapidity $|\eta^{\rm jet}|<$ 1.6, required to have $p_{T}^{\rm jet}>$ 30
GeV. In particular, to guarantee that the $Z^{0}$ bosons and the tagged jets
are back-to-back in the transverse plane, the $Z^{0}\,+\,$jet or
$Z^{0}\,+\,$b-jet pairs are required to have a large opening angle in azimuth,
$\Delta\phi_{jZ}(\Delta\phi_{bZ})>7\pi/8$. The differences of $x_{jZ}$
($x_{bZ}$) distributions in p+p and Pb+Pb collisions are also shown in the
lower panels. Due to the jet energy loss, the $x_{jZ}$ and $x_{bZ}$
distributions shift towards smaller $x_{J}$ values in Pb+Pb collisions
relative to p+p. Furthermore, one can find in the lower panel that the
variations of $x_{bZ}$ are slightly smaller than that of $x_{jZ}$. More
intuitive comparisons between the averaged $x_{jZ}$ and $x_{bZ}$ are listed in
Table 1. Within the statistical errors, the results show that
$\Delta\left\langle x_{jz}\right\rangle\sim$ 0.136 is considerably larger than
$\Delta\left\langle x_{bz}\right\rangle\sim$ 0.092, consistent with the
expectation that bottom jets lose less energy than light-quark jets.

Figure 5: Distributions of $x_{jZ}$ ($x_{bZ}$) of $Z^{0}\,+\,$jet (left) and $Z^{0}\,+\,$b-jet (right) both in p+p and 0–10% Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV. The differences of $x_{jZ}$ ($x_{bZ}$) distributions in p+p and Pb+Pb collisions (green line) are also shown in the lower panels. Figures are from Ref. Wang:2020qwe . Table 1: The averaged $x_{J}$ of $Z^{0}$ \+ jet and $Z^{0}$ \+ b-jet both in p+p and Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV, as well as their variations $\Delta x_{J}={\left\langle x_{J}\right\rangle}_{\rm pp}-{\left\langle x_{J}\right\rangle}_{\rm PbPb}$. The statistical errors of $x_{J}$ in the simulations are also presented. Table is from Ref. Wang:2020qwe . |  $Z^{0}$ \+ jet |  $Z^{0}$ \+ b-jet
---|---|---
${\left\langle x_{J}\right\rangle}_{pp}$ |  0.987 $\pm$ 0.0047 |  0.941 $\pm$ 0.0056
${\left\langle x_{J}\right\rangle}_{PbPb}$ |  0.851 $\pm$ 0.0061 |  0.849 $\pm$ 0.0064
$\Delta\left\langle x_{J}\right\rangle$ |  0.136 $\pm$ 0.0108 |  0.092 $\pm$ 0.012

###  3.3 Angular Correlation

Jet angular correlations, such as $\Delta\phi$ distribution of dijets
Mueller:2016gko ; Jia:2019qbl  and $\gamma/Z^{0}$ +jet Luo:2018pto ;
Zhang:2018urd , are useful observable to address the medium-induced transverse
momentum effect. In this context, estimating the medium modification on the
angular distribution of heavy quark dijets in nucleus–nucleus collisions may
also be of interest from the theoretical point of view. As shown in the left
plot of Figure 6, medium modification of the azimuthal angular correlations
($\Delta\phi=|\phi_{b1}-\phi_{b2}|$) of the $b\bar{b}$ dijet system in Pb+Pb
collisions with different centralities at $\sqrt{s_{NN}}$ = 5.02 TeV are
calculated Wang:2018gxz . One can observe suppression at $\Delta\phi\sim$0 and
enhancement at $\Delta\phi\sim\pi$ in Pb+Pb collisions compared to the p+p,
and the modifications are centrality dependent. Since the distributions are
self-normalized, it implies that $b\bar{b}$ dijets with a larger opening angle
(back-to-back) suffer relatively weaker yield suppression compared to that
with a smaller one (collinear). It can be noted that the main contribution of
$b\bar{b}$ dijet production at smaller $\Delta\phi$ is from the GSP process
while larger $\Delta\phi_{bb}$ from the FCR process. The two b-jets from the
former process share the energy of the gluon and then usually have lower
$p_{T}$ than that from the latter process. As a result, the yield at the
smaller $\Delta\phi$ region is more sensitive to the selection cut $p_{T}^{\rm
jet}>$ 20 GeV. Actually, in another study on the angular correlations of
$Z^{0}\,+\,$b-jet Wang:2020qwe , it’s found that initial average b-jet $p_{T}$
distribution versus $\Delta\phi$ play a critical role, as shown in the right
plot of Figure 6. We see that the ratio of PbPb/pp in the middle panel is
flat, and the average b-jet $p_{T}$ distribution is also flat. It’s reasonable
to guess that in Pb+Pb the azimuthal angle between b-jet and $Z^{0}$ has not
been modified compared to p+p, and the overall suppression occurs at whole
$\Delta\phi_{\rm bZ}$ region. Of course, we can imagine that it is more
difficult for high-$p_{T}$ ($>$30 GeV) jets to be significantly deflected by
the scattering with thermal parton.

Figure 6: Left: normalized azimuthal angular distributions of $b\bar{b}$ dijet
system in p+p and Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV. Results for
different centrality bins, 0–10%, 10–30%, 30–100%, are presented. Right: the
azimuthal angular distribution of $Z^{0}\,+\,$b-jet in p+p and 0–10% Pb+Pb
collisions at $\sqrt{s_{NN}}$ = 5.02 TeV in the upper panel, while the ratio
of PbPb/pp (green solid line) was shown in the middle panel and the averaged
b-jet $p_{T}$ (blue band) in the lower panel. Figures are from Refs.
Wang:2018gxz ; Wang:2020qwe .

To probe the angular deflection caused by the in-medium $p_{T}$-broadening,
observables accessible to lower $p_{T}$ region are needed. For this reason,
it’s proposed in Ref. Wang:2021jgm  that the heavy-flavour meson tagged by
direct photon ($\gamma+$HF) may provide a promising channel, with several
advantages: (1) the transverse momentum resolution of $D^{0}$ meson can be low
down to $\sim$1 GeV ALICE:2018lyv  where the angular deflection is
significant, (2) the photon gauges the initial momentum of heavy quarks,
therefore, it’s easy to quantify the direction change, (3) the selection bias
effect can be suppressed by constraining the photon energy to
citeCunqueiro:2021wls. In this way, the considerable angular de-correlations
between the heavy quarks and photons are predicted both in central Au+Au
collisions at the RHIC and Pb+Pb collisions at the LHC. Furthermore, by
constructing the 2-dimensional ($\Delta\phi,x_{J}$) correlation diagram of
$\gamma+$HF, it’s argued that the two aspects of jet quenching, energy loss,
and $p_{T}$-broadening, can be well displayed simultaneously. Additionally,
it’s noted that another measurement on the angular correlations of
$D^{0}$+hadron in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV may reflect
the medium modification of the charm+jet correlation in the $\eta-\phi$ plane
STAR:2019qbf , that awaits further detailed investigations.

###  3.4 Radial Profile

The radial profile of the heavy-flavour jet represents the distribution of the
angular distance $r=\sqrt{(\phi_{\rm Q}-\phi_{\rm jet})^{2}+(\eta_{\rm
Q}-\eta_{\rm jet})^{2}}$ between the heavy-flavour meson and the jet-axis in
the $\eta-\phi$ plane. Systematic studies with a focus on the radial profiles
of D-jet and B-jet in heavy-ion collisions are performed in Refs. Wang:2019xey
; Wang:2020bqz . As shown in the left panel of Figure 7, the model calculated
radial profiles of D-jets both in p+p and 0–100% Pb+Pb collisions at
$\sqrt{s_{NN}}=$ 5.02 TeV compared to the CMS measurements Sirunyan:2019dow .
The black and red triangle points represent the measured data. The D-jets are
reconstructed with anti-$k_{T}$ algorithm with R = 0.3 and $|\eta^{\rm jet}|<$
1.6. All selected D-jets must satisfy $p_{T}^{\rm jet}>$ 60 GeV and contain at
least one $D^{0}$ meson in jet-cone with $4<p_{T}^{D}<20$ GeV. The blue solid
line is the p+p baseline provided by SHERPA Gleisberg:2008ta , and the red
dashed line denotes the calculations based on the SHELL model. One can observe
that the model calculations show the radial profile of D-jets in Pb+Pb
collisions shifts towards larger radii relative to that of p+p, which is
consistent with the diffusion trend observed by the CMS collaboration. These
results show a clear physics picture, that charm quarks change their moving
direction when scattering with the thermal partons in the hot and dense QCD
matter. The studies argue that the diffusion behavior of D meson is closely
related to the $p_{T}$-broadening when charm quarks scatter with the thermal
partons in the medium. It should be noted that in such an estimate the jets
are required to have $p_{T}>60$ GeV while D meson $p_{T}<20$ GeV, which makes
that the higher $p_{T}$ jets can be viewed as a reference to probe the moving
direction changes of charm quarks. It is found that the angular deviation
$\Delta
r=\sqrt{(\phi_{c}^{f}-\phi_{c}^{i})^{2}+(\eta_{c}^{f}-\eta_{c}^{i})^{2}}$ of
charm quarks from their initial position in the $\eta-\phi$ plane is $p_{T}$
dependent, as shown in the right plot of Figure 7. The charm quarks with lower
$p_{T}$ are more likely to change their traveling direction via the in-medium
scattering, and this feature also explains why no visible modification is
observed in the CMS data for $p_{T}^{D}>20$ GeV Sirunyan:2019dow . The angular
deviation at lower $p_{T}$ ($<$5 GeV) is dominated by elastic scattering,
whereas at higher $p_{T}$ by inelastic reactions. These investigations may
cast light on the in-medium energy loss mechanisms and constrain the transport
coefficients of heavy quarks from a new perspective. We notice that a
preliminary result of the D-jet radial profile in Au+Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV has been reported by the STAR collaboration in Ref.
Roy:2022yrw . This result shows a similar diffusion effect of charm quark in
jets in mid-central 10–40% collisions.

To test the mass effect reflected in the radial profile, an additional
comparison of the medium modification between D-jet and B-jet has been
presented in Refs. Wang:2020ffj ; Wang:2020ukj , where an inverse modification
pattern on the radial profile of B-jets compared to D-jets is observed. The
jet quenching effect seems to narrow the jet radial profiles of B-jets while
broadening those of D-jets. It’s demonstrated that the selection bias effect
Renk:2012ve  in A+A collisions may play a pivotal role. Heavy quark jets with
higher $p_{T}$ have narrower initial radial distributions, and would naturally
lead to narrower modifications when they fall into the lower $p_{T}$ domain
due to jet energy loss. This reveals the fact that the final-state
modification of the jet is not only influenced by the pure medium effect, but
also by the other factors, such as the initial spectra and the selection bias
Cunqueiro:2021wls .

Figure 7: Left: radial profile of D-jet in p+p and Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV. Right: angular deviation of charm quark as a
function of initial $p_{T}$. Figures are from Refs. Wang:2019xey ;
Wang:2020bqz .

###  3.5 Fragmentation Function

The jet fragmentation function $D(z)=(1/N_{\rm jet})dN_{\rm ch}(z)/dz$ is one
of the most well-explored jet substructure observable Sjostrand:1986hx ;
Webber:1983if ; Procura:2009vm , which usually refers to the longitudinal
momentum distribution of charged hadrons inside the jet-cone CMS:2014jjt ;
ATLAS:2014dtd ; ATLAS:2017nre ; ATLAS:2019dsv . For heavy-flavour jets, the
corresponding observable is the longitudinal momentum distribution of heavy-
flavour mesons in jets, defined as in ALICE:2019cbr .

$\displaystyle D(z_{||})=\frac{1}{N_{\rm jet}}\frac{dN_{\rm
HQ}(z_{||})}{dz_{||}},\quad{\rm where}\ z_{||}=\frac{\vec{p}_{\rm
HQ}\cdot\vec{p}_{\rm jet}}{\vec{p}_{\rm jet}\cdot\vec{p}_{\rm jet}}\,\,.$ (16)

On the one hand, the $D(z_{||})$ distribution may provide useful information
to reveal the production mechanisms and substructure of heavy quark jets
ALICE:2022mur . On the other hand, since $z_{||}$ denotes the momentum
projection of the heavy-flavour hadron on the jet axis, the medium
modification of the $D(z_{||})$ distribution in nucleus–nucleus collisions is
closely related to the interplay of the partonic energy loss between the
massive heavy quarks and the massless light partons Li:2022tcr . Figure 8
shows the first theoretical investigation of the medium modification on the
$D(z_{||})$ distributions of both D-jets and B-jets in Pb+Pb collisions at
$\sqrt{s_{NN}}$ = 5.02 TeV. In these calculations, two jet $p_{T}$ ranges are
chosen, $5<p_{T}^{\rm jet}<15$ GeV and $15<p_{T}^{\rm jet}<50$ GeV.
Respectively the selected $D^{0}$ ($B^{0}$) mesons are also required to have
$p_{T,D^{0}(B^{0})}>$ 2 GeV and $p_{T,D^{0}(B^{0})}>$ 5 GeV. The black solid
lines represent the p+p baseline of $D(z_{||})$ distributions calculated by
the POWHEG+PYTHIA8 event generator Nason:2004rx ; Frixione:2007vw ;
Alioli:2010xd ; Sjostrand:2007gs , and the orange dash lines are the
theoretical calculations in Pb+Pb collisions based on the SHELL model. The
upper and middle panels correspond to the $D(z_{||})$ distributions of
$D$-jets and $B$-jets, while the lower panels are their nuclear modification
$D(z_{||})_{PbPb}/D(z_{||})_{pp}$ (green is D-jet and yellow B-jet). One can
observe that the initial $D(z_{||})$ distributions in p+p are sensitive to the
kinematic region of jet and heavy-flavour hadron, especially for D-jets.
Moreover, even within the same kinematic region, a B-jet has an evident harder
fragmentation pattern compared to a D-jet. The difference could be relevant to
the fact that the stronger “dead-cone” effect suffered in heavier bottom
quarks, in other words, the bottom quarks radiate less gluon and carry more
energy fraction of jets than charm quarks. Besides, the contribution of the
GSP process may also play different roles in the production of B-jets and
D-jets, which may lead to additional differences in their $z_{||}$
distributions Li:2022tcr . In nuclear collisions, the main finding is that the
jet quenching effect results in softer fragmentation patterns of heavy-flavour
jets in the QGP compared to that in a vacuum. It’s different from what one
could naively argue, that is, the energy fraction of heavy quarks in jets may
increase because heavy quarks lose less energy than light partons. The
modification of $D(z_{||})$ reveals the different energy loss mechanisms
between the single parton and the full jet. Critically, the lost energy from
the jet constituents may be partially brought back to the jet energy by the
reconstruction procedures. This is an essential difference in energy loss
mechanisms between the full-jet and the single parton, which leads to less
energy loss of full-jet compared to heavy quarks. Therefore, stronger medium
modification of $D(z_{||})$ can be obtained with larger R, which may be
related to the R-dependence of jet energy loss ATLAS:2012tjt ; Bossi:2022fpc .
Furthermore, stronger medium modification of $D(z_{||})$ is observed for
B-jets compared to D-jets, due to their different initial spectra.

Figure 8: $D(z_{||})$ distributions of D-jet and B-jet within two $p_{T}$
windows both in p+p and 0–10% Pb+Pb collisions, as well as the medium
modifications (PbPb/pp). Figure is from Ref. Li:2022tcr .

###  3.6 The “Dead-Cone” Effect and Other Observables

Until now, there are a few other heavy-flavour jet observables accessible in
the current experimental measurements at the LHC, which have also attracted
attention from the high-energy nuclear physics community. We briefly discuss
them in the following.

  * •

The Cambridge-Aachen (CA) declustering techniques Dokshitzer:1997in  which can
help to obtain the angular-ordered pairwise tree of subjets Larkoski:2014wba
and the Soft Drop condition mentioned above enable us to expose the most basic
heavy quark splitting structure by measuring the splitting-angle distributions
in D0 meson jets in p+p collisions at $\sqrt{s_{NN}}=13$ TeV ALICE:2021aqk .
It has been measured in three different energy intervals of the radiators:
$5\leq E_{\rm Radiator}\leq 10$ GeV, $10\leq E_{\rm Radiator}\leq 20$ GeV and
$20\leq E_{\rm Radiator}\leq 30$ GeV and constrain the transverse momentum of
the D0 meson in jet to be $2<p^{\rm D^{0}}_{\rm T}<36$ GeV/c. The ALICE
collaboration directly observed for the first time a clear distribution
suppression at the splitting angle smaller than the ratio of quark mass and
the energy of such quark radiator: $\theta\leq M_{\rm charm}/E_{\rm
radiator}$, known as the “dead-cone” effect Dokshitzer:1991fd ;
Cunqueiro:2018jbh . Such a heavy quark jet and its substructure measurement
reveal and confirm this most basic property of a fast quark interacting with
the vacuum described by the QCD theory.

A subsequent phenomenology study exposed the “dead-cone” effect of the medium-
induced gluon radiation of jet queching Dokshitzer:2001zm ; Zhang:2003wk ;
Armesto:2003jh , by calculating the emission angle distribution of the heavy-
flavour quark initiated splittings in a $\rm D^{0}$ meson tagged jet and that
of the light parton initiated splittings with the existence of the QGP in
Pb+Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV Dai:2022sjk , as demonstrated in
Figure 9. Very interestingly, they find the collisional energy loss mechanism
will not obscure the observation of the “dead-cone” effect in the medium-
induced radiation. Such a proposal has also been verified by an analytical
study that proposes a new jet substructure groomer that selects the most
collinear splitting in a QCD jet above a certain transverse momentum cutoff
Cunqueiro:2022svx . It’s also found in another study that the “dead-cone”
domain would be partially filled by the medium-induced emission as heavy
quarks traversing QGP Armesto:2003jh .

Figure 9: The splitting-angle distributions for $\rm D^{0}$ meson tagged jets,
inclusive jets and also light-quark jets normalized to the number of jets in
Pb+Pb collisions at $\rm\sqrt{s}=5.02$ TeV (upper plots) and also the $D^{0}$
meson tagged jets/inclusive jets (light-quark jets/inclusive jets) ratios
(bottom plots) calculated for three energy intervals of the radiators: $\rm
5<E_{\text{Radiator}}<10$ GeV (left panel), $\rm 10<E_{\text{Radiator}}<20$
GeV (middle panel) and $\rm 20<E_{\text{Radiator}}<30$ GeV (right panel). The
shaded areas correspond to the angles at which the radiation is suppressed due
to the “dead-cone” effect. Figure is from Ref. Dai:2022sjk .

  * •

The jet shape $\rho(r)$ describes the transverse energy profile of charged
hadrons as a function of the angular distance from the jet axis. This
observable has been well-studied for light flavor jets Chang:2019sae ;
Luo:2018pto  to search the medium response effect as energetic parton
dissipating energy to the medium Cao:2020wlm . The measurement of the medium
modification on the b-jet shape has been reposted in Refs. CMS:2020geg ;
CMS:2022btc  by the CMS collaboration. On the ond hand, the comparison of jet
shapes of b-jets in Pb+Pb and p+p collisions shows the presence of the QGP
modifies the energy distributions around the jet axis of b-jets. On the other
hand, their measurements indicate a stronger jet energy redistribution of
b-jets at larger radii compared to that of inclusive jets. Generally speaking,
the bottom quarks are expected to dissipate less energy in nuclear collisions
compared to light quarks and gluons due to the “dead-cone” effect. However, at
larger jet radii, the medium response effect plays the dominant role in the
enhancement of jet energy distribution in Pb+Pb collisions compared to the p+p
baseline. Therefore, these interesting results may suggest that the heavier
quark, like the bottom, may drive a stronger medium response effect than a
massless parton. In this context, the heavy-flavour jets can serve as
promising sensitive probes to the quasi-particle excitation of the quark soup.

  * •

The Soft Drop (SD) grooming procedures reveal the two-prong structure of a
jet, described by the momentum sharing $z_{g}$ and opening angle $R_{g}$
Larkoski:2014wba , which establishes the connection between the final state
observable to the parton splitting function. The splitting history could be
helpful to identify the production mechanisms of heavy-flavour jets
Goncalves:2015prv ; Ilten:2017rbd . Heavy quark jets from the gluon splitting
process usually tend to have more balanced $z_{g}$ and larger $R_{g}$ compared
to that from the FEX and FCR. The first measurement of the D-jet splitting
function is performed by ALICE ALICE:2022phr , and some theoretical efforts
which focus on the medium modifications of $z_{g}$ and $R_{g}$ of c- and
b-jets are presented in Refs. Li:2017wwc ; Zhang:2023jpe . The medium effects
result in more imbalanced $z_{g}$ distribution and larger opening angles
between the two subjets in the heavy quark jets, similar to the medium
modification of inclusive jets observed by the CMS Sirunyan:2017bsd  and ALICE
Acharya:2019djg  collaboration.

##  4 Summary and Conclusions

This review covers the current development of theoretical studies on heavy-
flavour jets in ultra-relativistic heavy-ion collisions. We introduce the
recent theoretical advances of heavy-flavour production in heavy-ion
collisions and then give a comprehensive discussion of several recent
investigations relating to the heavy-flavour jet observables.

  * •

We briefly overview the recent theoretical advances that help us understand
the heavy-flavour production in heavy-ion collisions, mainly focusing on the
initial production, transport approaches, hadronization mechanism, and
diffusion coefficient extraction. These phenomenological studies based on the
transport models reveal a fact that the elastic scattering of heavy quarks is
dominant at lower $p_{T}$ region ($p_{T}^{Q}<5m_{Q}$), while the inelastic one
dominate the high $p_{T}$ regions. Besides, different from the fragmentation
hadronization of heavy quarks in a vacuum, within the hot and dense nuclear
matter, the coalescence mechanism plays an important role in explaining the
large collective flow and the enhancement of baryon-to-meson ratio of a
charmed hadron in nucleus–nucleus collisions at the RHIC and the LHC. The
diffusion coefficient of heavy quarks in the QGP has been extracted by various
theoretical frameworks, which implies that $2\pi TD_{s}$ slightly increases
with temperature. The newly developed Bayesian inference approach may be
promising to implement a robust determination of the transport coefficient of
heavy quarks by a model-data fit.

  * •

The studies on yield suppression and momentum imbalance of heavy-flavour jets
are dedicated to addressing the mass effect of jet energy loss. Theoretical
investigations predict stronger yield suppression of light quark jets compared
to heavy-flavour jets, which is preliminarily proven by the recent ATLAS
measurement of b-jet $R_{AA}$. However, the dijet asymmetry shows a reduced
sensitivity to the jet quenching effect, therefore the difference of the
medium modification on $x_{J}$ between inclusive and $b\bar{b}$ dijets seems
to be moderate. We have to say the nuclear modification factor is still an
effective and powerful observable to test the mass effect of energy loss in
QGP. On the other hand, the strategy to isolate the jets initiated by heavy
quarks is also crucial to address the mass effect, since GSP processes indeed
have a large contribution to the production of heavy quark jets but suffer
stronger suppression in nucleus–nucleus collisions.

  * •

An observable related to angular correlation aims at the deflection of the jet
axis caused by the medium-induced $p_{T}$-broadening of jet quenching. It’s
found that the angular deviation caused by the in-medium scattering is hard to
be observed for high-$p_{T}$ jets, both for $b\bar{b}$ dijets and
$Z^{0}\,+\,$(b-)jet. That makes sense because higher $p_{T}$ jets are more
difficult to be changed by the in-medium scattering with the thermal parton in
QGP. Meanwhile, medium modification on the radial profiles of jets containing
lower-$p_{T}$ D meson can well capture the angular de-correlation of the charm
quark and the jet axis. This suggests that heavy flavors may be more suitable
to address the medium-induced $p_{T}$-broadening of jet quenching since they
are experimentally accessible to the low-$p_{T}$ domain where the angular
deviation is visible.

  * •

The substructure observable can reveal a wealth of information about the inner
configuration of heavy-flavour jets. In the vacuum case, declustering
techniques provide an inventive way to reestablish the splitting history of
hard partons which helps us unlock the “dead-cone” effect of charm quark in
the experiment. For heavy-flavour jets, the substructure observable also
provides a unique opportunity to identify their production mechanisms.
Furthermore, jet substructure, such as jet shape, seems more sensitive to the
induced medium excitation in nucleus–nucleus collisions than full-jet
observables. Much theoretical effort should be made to address the interplay
of the “dead-cone” effect of medium-induced radiation and the medium response
of heavy quarks. From the current perspective, the studies of substructures of
heavy-flavour jets could play an increasingly important role in high-energy
nuclear physics.

  * •

The initial jet spectra and the “selection bias” play important roles in the
medium modifications of jet substructure in nuclear collisions. Normally when
we focus on the mass effect of the yield or substructure modification of heavy
quark jets, it is apriori to believe that bottom jets should have a weaker
medium modification in heavy-ion collisions compared to charm jets under the
same conditions. However, in the studies of radial profile and fragmentation
function of heavy-flavour jets, it’s found that b-jets have very different
initial substructure compared to that of c-jets event within the same
kinematic constraints, which eventually leads to stronger medium modification
of b-jets at the final-state compared to c-jets. On the other hand, the
“selection bias” poses a challenge to the theoretical studies that aim at the
nuclear modification mechanism of heavy-flavour jets in the hot and dense QCD
medium. It brings additional “modifications” to the ratio PbPb/pp of jet
substructure distributions, nevertheless, these “modifications” do not exactly
reflect the change of jet substructure but only the decrease of jet energy
from the higher kinematic region in Pb+Pb collisions.

Conceptualization, S.W. and B.-W.Z.; methodology, W.D. and B.-W.Z.;
investigation, S.W. and W.D.; writing—original draft preparation, S.W. and
W.D.; writing—review and editing, B.-W.Z., E.W. and X.-N.W.; supervision,
B.-W.Z., E.W. and X.-N.W. All authors have read and agreed to the published
version of the manuscript.

This research is supported by the Guangdong Major Project of Basic and Applied
Basic Research No. 2020B0301030008, and the Natural Science Foundation of
China with Project Nos. 11935007, 12035007, 12247127. S. Wang is also
supported by China Postdoctoral Science Foundation under project No.
2021M701279.

Not applicable.

The authors declare no conflict of interest.

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