###### Abstract

The study of angular correlations of heavy-flavor particles in hadronic
collisions can provide crucial insight into the heavy quark production,
showering, and hadronization processes. The comparison with model predictions
allows us to discriminate among different approaches for heavy quark
production and hadronization, as well as different treatments of the
underlying event employed by the models to reproduce correlation observables.
In ultra-relativistic heavy-ion collisions, where a deconfined state of
matter, the quark–gluon plasma (QGP), is created, heavy-flavor correlations
can shed light on the modification of the heavy quark fragmentation due to the
interaction between charm and beauty quarks with the QGP constituents, as well
as characterize their energy loss processes while traversing the medium.
Insight into the possible emergence of collective-like mechanisms in smaller
systems, resembling those observed in heavy-ion collisions, can also be
obtained by performing correlation studies in high-multiplicity proton–proton
and proton–nucleus collisions. In this review, the most recent and relevant
measurements of heavy-flavor correlations performed in all collision systems
at the LHC and RHIC will be presented, and the new understandings that they
provide will be discussed.

###### keywords:

heavy quarks; correlations; jet fragmentation; energy loss; collectivity in
small systems; heavy quark production; hadronization

10 1 0 Academic Editor: Firstname Lastname 15 January 2024 6 February 2024 7
February 2024 https://doi.org/ Recent Findings from Heavy-Flavor Angular
Correlation Measurements in Hadronic Collisions Recent Findings from Heavy-
Flavor Angular Correlation Measurements in Hadronic Collisions Deepa Thomas
1,* and Fabio Colamaria 2,* Deepa Thomas and Fabio Colamaria Thomas, D.;
Colamaria, F. Correspondence<EMAIL_ADDRESS>(D.T.);
<EMAIL_ADDRESS>(F.C.)

##  1 Introduction

The study of heavy quark (charm and beauty quarks) production in high-energy
hadronic collisions is an important tool to test and validate perturbative
quantum chromodynamics (pQCD) calculations Cacciari et al. (1998, 2001);
Kniehl et al. (2006, 2008), as they are produced in hard parton scattering
processes. The production cross-section of several heavy-flavor hadrons and
their decay products has been measured at different centers of mass energies
at RHIC Aggarwal et al. (2010); Adare et al. (2009, 2006), Tevatron Acosta et
al. (2003); Cacciari and Nason (2003); Acosta et al. (2005), and at the LHC
Acharya et al. (2020); Sirunyan et al. (2018); Aaij et al. (2016); Acharya et
al. (2022a, b, 2019a, 2019b, 2021); Abelev et al. (2012a, b, c); Aaij et al.
(2017); Abelev et al. (2013, 2015); Aaij et al. (2012); Aad et al. (2012,
2013); Chatrchyan et al. (2012); Khachatryan et al. (2011); Chatrchyan et al.
(2011a, b); Khachatryan et al. (2017), and are compared with pQCD calculations
Cacciari et al. (2012); Kniehl (2008); Kniehl et al. (2011, 2006); Bolzoni and
Kramer (2013, 2014). The correlated production of heavy flavors, studied as a
function of variables such as the azimuthal angle between heavy-flavor
particles, either by direct reconstruction of heavy-flavor hadrons or from
their decay products, can provide a significantly larger amount of information
than single-particle inclusive heavy-flavor production.

In ultra-relativistic heavy-ion collisions, heavy quarks play an important
role in the study of the deconfined phase of strongly interacting matter, the
quark–gluon plasma (QGP), created in these collisions. Traditional observables
of heavy quarks, such as the nuclear modification factor ($R_{\rm{AA}}$) and
the elliptic flow coefficient ($v_{2}$), have been extensively studied at RHIC
Adare et al. (2011); Abelev et al. (2007); Adamczyk et al. (2017a) and at the
LHC Acharya et al. (2022); Sirunyan et al. (2018); Acharya et al. (2022, 2018,
2021). These measurements indicate that heavy quarks experience significant
in-medium energy loss at large transverse momentum ($p_{\rm{T}}$), and that
charm quarks partially thermalize within the medium at smaller $p_{\rm{T}}$.
These effects are induced by the interaction of heavy quarks with the medium
constituents, mainly made up of light partons. The interaction has two main
contributions, the purely elastic process, resulting in a collisional energy
loss Braaten and Thoma (1991); Peshier (2006); Peigne and Peshier (2008), and
gluon bremsstrahlung, producing a radiative energy loss Gyulassy and Wang
(1994); Baier et al. (1995); Gyulassy et al. (2000); Dokshitzer and Kharzeev
(2001); Armesto et al. (2005); Zhang et al. (2004). Both of these processes
depend on the parton mass; thus, studies of heavy quark in heavy-ion
collisions provide important information about the properties of the QGP.
Currently, it remains a challenge to describe $R_{\rm{AA}}$ and $v_{2}$
simultaneously from a low to high range of transverse momentum in a given
theoretical framework. More differential observables, such as angular
correlations between heavy-flavor particles, are more sensitive to the
specific interaction processes between heavy quarks and the QGP constituents,
and can thus provide further information on the propagation of heavy quarks in
the QGP medium Nahrgang et al. (2014); Cao et al. (2015).

In proton–nucleus (p–A) collisions, due to the presence of the nucleus in the
initial state, several cold-nuclear-matter effects can influence the
production, fragmentation, and hadronization of heavy quarks, such as the
impact of the nuclear parton distribution function (nPDF) Eskola et al.
(2009); de Florian and Sassot (2004); Hirai et al. (2007), the presence of a
coherent and saturated gluonic system dominated by gluons at low Bjorken-$x$
Fujii and Watanabe (2013); Tribedy and Venugopalan (2012); Albacete et al.
(2013); Rezaeian (2013), and partons undergoing multiple elastic, inelastic,
and coherent scatterings Accardi et al. (2009); Salgado et al. (2012). These
effects can also modify the heavy-flavor correlation distribution Vogt (2018,
2020); Marquet et al. (2018), and their understanding is crucial for the
interpretation of any modification of the correlation distributions in the
presence of a QGP, produced in heavy-ion collisions.

In this article, we review and summarize the latest measurements of angular
correlations of heavy-flavor particles in proton–proton (pp), proton–nucleus
(p–A), and nucleus–nucleus (A–A) collisions at the LHC and RHIC energies. The
article is structured as follows. In Section 2, studies of heavy-flavor
production mechanisms in pp and p–A collisions using angular correlation
techniques are discussed. The fragmentation of heavy quarks into final-state
jets in pp and p–A collisions was studied using angular correlations of heavy-
flavor hadrons and charged particles, as discussed in Section 3. In Section 4,
new insight into heavy quark propagation and energy loss in the QGP obtained
using correlations of heavy quarks is discussed. In recent years, questions
have been raised about the possible formation of a QGP in smaller collision
systems, such as in pp and p–A collisions, due to the observation of a “long-
range ridge structure” in two-particle azimuthal correlations of light-flavor
particles. To further investigate this possibility, measurements of azimuthal
correlations of heavy-flavor particles and charged hadrons were performed,
from which the azimuthal anisotropy of heavy-flavor particles was extracted.
These studies are reviewed in Section 5.

##  2 Study of Heavy Quark Production Mechanisms

In high-energy hadronic collisions, heavy quarks (charm and beauty) are mainly
produced in hard parton scattering processes. An inclusive production of
several heavy-flavor hadrons has been measured in experiments at the LHC and
at lower energies, and is compared with pQCD calculations such as FONLL and
GM-VFNS Cacciari et al. (2012); Kniehl (2008); Kniehl et al. (2011, 2006);
Bolzoni and Kramer (2013, 2014). Exclusive measurements of heavy-flavor
correlations, e.g., as a function of the azimuthal angle between heavy-flavor
hadrons, $\varphi$, are a stronger test of the heavy quark pair (${\rm
Q\bar{Q}}$) production mechanisms than single-particle inclusive
distributions. At leading order (LO), $2\rightarrow 2$ in $\alpha_{\rm s}$ for
the parton interaction sub-processes, the heavy quark pairs will be emitted
with a back-to-back topology in azimuth due to momentum conservation. At next-
to-leading order (NLO), $2\rightarrow 2+n~{}(n\geq 1)$, additional partons are
emitted, resulting in different topologies of the produced heavy quarks Vogt
(2018, 2020). Experimentally, correlation patterns of heavy quarks can be
assessed with measurements of angular correlations of heavy-flavor particle
pairs, as discussed below.

Measurements of angular correlations of heavy-flavor particles in
proton–proton (pp) collisions that fit this context were performed by the
PHENIX Collaboration at RHIC at $\sqrt{s}=200$ GeV Aidala et al. (2019), and
by ATLAS Aaboud et al. (2017), CMS Khachatryan et al. (2011), and LHCb Aaij et
al. (2012, 2017) Collaborations at the LHC at $\sqrt{s}=7$ and 8 TeV, and in
proton–lead (p–Pb) collisions at $\sqrt{s_{\rm{NN}}}=8.16$ TeV by the LHCb
Collaboration Aaij et al. (2020). The goal of these measurements is to
understand the correlated production of heavy quark pairs, and to test
theoretical calculations at higher orders.

The PHENIX Collaboration performed a study of azimuthal correlations of
$\mu\mu$ pairs from heavy-flavor hadron decays in pp collisions at
$\sqrt{s}=200$ GeV at forward and backward rapidity $(1.2<|\eta|<2.2)$ Aidala
et al. (2019). The $\mu\mu$ pairs have contributions from
$\rm{c}\bar{\rm{c}}$, $\rm{b}\bar{\rm{b}}$, Drell–Yan, and hadronic pairs
(kaons and pions), which are distinguished using template fits to opposite-
and like-sign spectra in mass and transverse momentum, $p_{\rm{T}}$. While the
decays from $\rm{c}\bar{\rm{c}}$ and Drell–Yan mechanisms contribute to
opposite-sign pairs only ($\mu^{+}\mu^{-}$), decays from $\rm{b}\bar{\rm{b}}$
pairs populate the like-sign distribution ($\mu^{\pm}\mu^{\pm}$) as well,
either because of combination of $\rm{B}\rightarrow$ $\mu$ and
$\rm{B}\rightarrow\rm{D}\rightarrow$$\mu$ decay chains or decays following
$\rm{B}^{0}\bar{\rm{B}^{0}}$ oscillations. Different components contribute
with different relative abundances to the muon pair continuum in different
mass regions of $\mu^{+}\mu^{-}$ and $\mu^{\pm}\mu^{\pm}$ pairs. The
contributions from $\rm{c}\bar{\rm{c}}$ and $\rm{b}\bar{\rm{b}}$ are separated
considering mass regions where they dominate. This corresponds to
$1.5<m_{\mu^{+}\mu^{-}}<2.5$ GeV$/c^{2}$ for charm, and
$3.5<m_{\mu^{\pm}\mu^{\pm}}<10$ GeV$/c^{2}$ for beauty. The azimuthal opening
angle distributions for $\mu\mu$ pairs from $\rm{c}\bar{\rm{c}}$ and
$\rm{b}\bar{\rm{b}}$ decays, for muons with $p>3$ GeV$/c$ and
$1.2<|\eta|<2.2$, are shown in Figure 1. The measurements are compared with
model calculations based on PYTHIA Sjostrand et al. (2006) and POWHEG Frixione
et al. (2007). The distribution from PYTHIA describes the data in the probed
kinematic range for both $\rm{c}\bar{\rm{c}}$ and $\rm{b}\bar{\rm{b}}$. While
the distribution from POWHEG simulations for $\rm{c}\bar{\rm{c}}$ are wider
compared to the one from PYTHIA, they are more similar for
$\rm{b}\bar{\rm{b}}$. Both PYTHIA and POWHEG use the PYTHIA fragmentation
scheme and very similar parton distribution functions. The differences between
the model calculation for charm–origin correlations could originate from the
different underlying correlation between $\rm{c}$ and $\bar{\rm{c}}$ quarks
emerging from the hard process. While POWHEG implements NLO calculations for
evaluating the hard parton scattering matrix elements, PYTHIA evaluates them
at LO and mimics the NLO processes in the parton shower.

Figure 1: The azimuthal opening angle distribution of $\mu\mu$ pairs from
$\rm{c}\bar{\rm{c}}$ (a) and $\rm{b}\bar{\rm{b}}$ (b) decays, measured by the
PHENIX Collaboration in pp collisions at $\sqrt{s}=200$ GeV. The data are
compared with the distributions calculated with POWHEG and PYTHIA event
generators, where the different production mechanisms are also shown
separately Aidala et al. (2019).

To study the multiple production of charm states in a single pp collision, and
understand contributions from double parton scattering (DPS) Kom et al.
(2011); Baranov et al. (2011); Novoselov (2011); Luszczak et al. (2012), the
LHCb Collaboration performed measurements of J$/\psi$ production with an
associated open-charm hadron (either $\rm{D}^{0}$, $\rm{D}^{+}$,
$\rm{D}_{s}^{+}$ or $\Lambda_{c}^{+}$), and of double open-charm hadron
production in pp collisions at $\sqrt{s}$ = 7 TeV Aaij et al. (2012). These
measurements can probe the quarkonium production mechanism Brodsky and
Lansberg (2010) and contributions from the intrinsic charm content of the
proton Brodsky et al. (1980) to the total cross-section. The J$/\psi$ and
open-charm hadron production is denoted as J$/\psi C$ and the double open-
charm hadron production as $CC$, with a control channel using
$\rm{c}\bar{\rm{c}}$ events denoted as $C\bar{C}$. The measurements are
performed in the LHCb fiducial region ($2<y^{\rm{J}}$${}^{/\psi}<4.5$,
$p_{\rm{T}}^{\rm{J}/}$${}^{\psi}<10$ GeV$/c$). The azimuthal angle and
rapidity distributions between J$/\psi$ and charm hadrons in J$/\psi C$ events
are shown in Figure 2. No significant azimuthal correlation is observed and
the $\Delta y$ distribution shows a triangular shape consistent with what is
expected when the rapidity distribution for single-charm hadrons is flat and
in the absence of physical correlations between the two particles. A similar
trend in $\Delta\varphi$ and $\Delta y$ is observed for $CC$ pairs, with a
slightly enhanced back-to-back configuration in $\Delta\varphi$, visible for
$CC$ pairs compared to J$/\psi C$ pairs. The absence of significant azimuthal
or rapidity correlations for J$/\psi C$ pairs could support the DPS
hypothesis, but no comparison with model predictions are provided in the
publication. In contrast, $C\bar{C}$ pairs show a clear enhancement in the
$\Delta\varphi$ distribution at small $|\Delta\varphi|$, consistent with
$\rm{c}\bar{\rm{c}}$ production via the gluon splitting mechanism at NLO
Norrbin and Sjostrand (2000), together with a significant contribution of
back-to-back correlations, consistent with the topology of an LO production,
as shown in Figure 3. This figure also shows a small enhancement at small
$\Delta y$, which is also consistent with gluon splitting topologies, unlike
the distributions for $CC$ and J$/\psi C$ pairs.

Figure 2: Distributions of the azimuthal angle (left) and rapidity (right)
differences between $\rm{J}/$$\psi$ and different charm mesons ($C$) for
$2<y^{\rm{J}/}$${}^{\psi},y^{C}<4$, $p_{\rm{T}}$,J/${}^{{\psi}}<12$ GeV$/c$,
and $3<p_{\rm{T}}^{C}<12$ GeV$/c$, in pp collisions at $\sqrt{s}=7$ TeV,
measured by the LHCb Collaboration. The dashed line in the right plot shows
the expected $\Delta y$ distribution for uncorrelated pairs Aaij et al.
(2012).

Figure 3: Distributions of the azimuthal angle (left) and rapidity (right)
differences between charm and anti-charm mesons ($C\bar{C}$), for $2<y^{C}<4$
and $3<p_{\rm{T}}^{C}<12$ GeV$/c$ in pp collisions at $\sqrt{s}=7$ TeV,
measured by the LHCb Collaboration. The dashed line in the right plot shows
the expected $\Delta y$ distribution for uncorrelated pairs Aaij et al.
(2012).

The production of heavy quarks via single parton scattering (SPS) or via DPS
can be affected by cold-nuclear-matter effects Eskola et al. (2009); de
Florian and Sassot (2004); Hirai et al. (2007); Fujii and Watanabe (2013);
Tribedy and Venugopalan (2012); Albacete et al. (2013); Rezaeian (2013);
Accardi et al. (2009); Salgado et al. (2012); Vogt (2018, 2020) in proton–ion
collisions. In particular, DPS production can be very sensitive to the nuclear
PDF in p–A as it involves two parton pairs, including possible dependence on
the position of the partons inside the nucleus Shao (2020). Based on the
Glauber model Miller et al. (2007), in the absence of nuclear effects, the SPS
production cross-section is expected to scale with the ion mass number. DPS
production, on the other hand, is enhanced compared to a mass number scaling
due to collisions of partons from two different nucleons in the ion, and the
enhancement factor is about three in proton–lead (p–Pb) collisions Luszczak et
al. (2012); Cazaroto et al. (2018); Helenius and Paukkunen (2020). This
feature was studied by the LHCb Collaboration by measuring the production of D
meson pairs ($\rm{D}^{0},\rm{D}^{+},\rm{D}^{+}_{s}$ mesons), as well as
J/$\psi\rm{D}$ meson pairs in p–Pb collisions at $\sqrt{s_{\rm{NN}}}=8.16$ TeV
Aaij et al. (2020). Like-sign (LS) pairs, where two hadrons have the same
charm quark charge, and opposite-sign (OS) pairs, where they have opposite
charm charge, are considered. Pairs of OS charm hadrons are dominantly
produced from a $\rm{c}\bar{\rm{c}}$ pair via SPS; thus, the kinematics of the
two hadrons are correlated, while DPS leads to both correlated and
uncorrelated OS pairs. The kinematic correlation between the two charm hadrons
was investigated using the two-charm invariant mass ($m_{\rm{DD}}$) and their
relative azimuthal angle $\Delta\varphi$. The $\Delta\varphi$ distribution for
LS $\rm{D}^{0}\rm{D}^{0}$ pairs and OS $\rm{D}^{0}\bar{\rm{D}^{0}}$ pairs for
all $p_{\rm{T}}$ and when requiring a $p_{\rm{T}}^{\rm{D}^{0}}>2$ GeV$/c$
condition is shown in Figure 4. Without the $p_{\rm{T}}$ condition, the
$\Delta\varphi$ distribution is almost uniform for both LS and OS pairs,
similar to that predicted by PYTHIA8 simulation. However, with the
$p_{\rm{T}}^{\rm{D}^{0}}>2$ GeV$/c$ condition, the
$\rm{D}^{0}\bar{\rm{D}^{0}}$ pair distribution shows an enhancement at
$\Delta\varphi\sim 0$, while the $\rm{D}^{0}\rm{D}^{0}$ distribution is
consistent with being flat; both are inconsistent with the predictions from
PYTHIA8 simulations. The flat $\Delta\varphi$ behavior for
$\rm{D}^{0}\rm{D}^{0}$ is qualitatively consistent with a large DPS
contribution in LS pair production. The effective cross-section and nuclear
modification factor for LS charm hadron pairs were also measured and were
reported to be compatible with the expected enhancement by a factor of 3 for a
DPS-over-SPS production ratio from pp to p–Pb collisions Aaij et al. (2020).

Figure 4: The $\Delta\varphi$ distribution for $\rm{D}^{0}\rm{D}^{0}$ (red)
and $\rm{D}^{0}\bar{\rm{D}^{0}}$ (blue) pairs measured by the LHCb
Collaboration in p–Pb collisions at $\sqrt{s_{\rm{NN}}}=8.16$ TeV, compared
with PYTHIA8 simulations (magenta dashed line), without (left) and with
(right) a $p_{\rm{T}}^{\rm{D}^{0}}>2$ GeV$/c$ requirement. Vertical bars
(filled box) represent statistical (systematic) uncertainties Aaij et al.
(2020).

The production of beauty quark pairs was studied by the ATLAS Aaboud et al.
(2017), CMS Khachatryan et al. (2011), and LHCb Aaij et al. (2017)
Collaborations by measuring correlations of particles with beauty content. The
ATLAS experiment measured the production of two b-hadrons, where one decays to
J$/\psi(\rightarrow\mu\mu)$ \+ X and the other to $\mu$+Y, resulting in three
muons in the final state in pp collisions at $\sqrt{s}=8$ TeV Aaboud et al.
(2017). To probe the b-hadron production, several differential cross-sections
were measured, such as $\Delta\varphi(\rm{J}$$/\psi,\mu)$, $\Delta
y(\rm{J}$$/\psi,\mu)$, separation between the J$/\psi$ and the third muon in
azimuth–rapidity plane $\Delta R(\rm{J}$$/\psi,\mu)$, mass of the three muon
system $m(\rm{J}$$/\psi,\mu)$, etc. Since this review focuses on the angular
correlation measurements, the $\Delta\varphi$ and $\Delta y$ distributions
compared with predictions from PYTHIA8.2 Sjostrand et al. (2008), HERWIG++
Bahr et al. (2008), MADGRAPH5-AMC@NLO+PYTHIA8 Alwall et al. (2014), and SHERPA
Gleisberg et al. (2009); Schumann and Krauss (2008) Monte Carlo generators are
shown in Figure 5. The $\Delta\varphi$ prediction from HERWIG++ provides the
best description of the data compared to others. The trends seen in $\Delta
R(\rm{J}$$/\psi,\mu)$ are similar to $\Delta\varphi(\rm{J}$$/\psi,\mu)$. For
$\Delta y(\rm{J}$$/\psi,\mu)$ distribution, the MADGRAPH5-AMC@NLO+PYTHIA8 and
SHERPA predictions provide a good description of the data, while PYTHIA8 and
HERWIG++ fail to describe data at high $\Delta y(\rm{J}$$/\psi,\mu)$.
Different kinematic correlation observables can thus provide enhanced
sensitivity to the underlying model differences and allow us to discriminate
among them.

Figure 5: Normalized differential cross-section of J$/\psi$ and $\mu$ pairs as
a function of $\Delta\varphi(\rm{J}$$/\psi,\mu)$ (left) and $\Delta
y(J/\psi,\mu)$ (right) in pp collisions at $\sqrt{s}=8$ TeV, measured by the
ATLAS Collaboration. Data are compared with predictions from PYTHIA8.2,
HERWIG++, MADGRAPH5-AMC@NLO+PYTHIA8, and SHERPA Monte Carlo generators. The
model-to-data ratios are shown in the bottom panels Aaboud et al. (2017).

The azimuthal angular correlations between beauty hadron pairs, called here “B
hadrons”, were measured by the CMS Collaboration in pp collisions at
$\sqrt{s}=7$ TeV, where the B hadrons were identified by the presence of
displaced secondary vertices (SVs) from their decays Khachatryan et al.
(2011). The flight direction of the B hadron is defined by the direction
connecting the primary and the secondary vertices. The angular correlation
variables between B and $\bar{\rm{B}}$ hadrons, in events with two SVs, are
calculated using their flight directions. While a back-to-back configuration
is expected for LO processes, the B$\bar{\rm{B}}$ production at small opening
angles directly relates to collinear emission processes at higher order (g
$\rightarrow$ b$\bar{\rm{b}}$). The B$\bar{\rm{B}}$ pair production cross-
section as a function of $\Delta\varphi$ for three different energy scales,
characterized by the leading jet $p_{\rm{T}}$, is presented in Figure 6 (left
plot). A significantly large cross-section is observed at small angles, with
values higher than at large angles, whose relative contribution increases with
increasing jet $p_{\rm{T}}$. At higher energy scales, larger contributions
from higher-order processes, for example, gluon radiation, are expected,
resulting in more gluon splitting into B$\bar{\rm{B}}$ pairs. PYTHIA
predictions Sjostrand et al. (2006) are normalized to the region
$\Delta\varphi>3/4\pi$, where the theoretical calculation is more reliable as
the cross-section is expected to be dominated by the LO diagrams. Data and
theory predictions from MADGRAPH Maltoni and Stelzer (2003); Alwall et al.
(2007), MC@NLO Frixione and Webber (2002); Frixione et al. (2003); Frixione
and Webber (2008), and CASCADE Jung and Salam (2001) models are compared, with
respect to the PYTHIA prediction, as shown in Figure 6 (right plot). It is
observed that none of the predictions describe the data particularly well, in
particular for the collinear region. The data lie between MADGRAPH and PYTHIA
curves.

The azimuthal and rapidity correlations in $\rm{b}\bar{\rm{b}}$ production in
the forward rapidity region were investigated by LHCb experiment in pp
collisions at $\sqrt{s}=7$ and 8 TeV by correlating pairs of beauty hadrons
Aaij et al. (2017). The beauty hadrons were reconstructed via their inclusive
decays into J$/\psi$ mesons (b$\rightarrow$J$/\psi X$). The
$|\Delta\varphi^{*}|$ and $|\Delta\eta^{*}|$ variables, i.e., the difference
in azimuthal angle $\varphi^{*}$ and pseudorapidity $\eta^{*}$ between the two
beauty hadrons, estimated from the direction of the vector from the primary
vertex to the decay vertex of the J$/\psi$ meson, were measured. The
normalized differential production cross-sections as a function of
$|\Delta\varphi^{*}|$ for $p_{\rm{T}}^{\rm{J}}$${}^{/\psi}>3$ and $>7$ GeV$/c$
are shown in Figure 7. No significant enhancement is observed at small
$|\Delta\varphi^{*}|$ at low $p_{\rm{T}}^{\rm{J}}$/ψ, but a peak starts to
appear in that region at higher $p_{\rm{T}}^{\rm{J}}$/ψ, due to higher
contribution from NLO processes. This observation is similar to the CMS
measurement Khachatryan et al. (2011) from B hadron pairs. Compared to open-
charm mesons Aaij et al. (2012), the small angle enhancement is observed at
higher $p_{\rm{T}}$ since the contribution from processes like gluon splitting
requires higher energy scales to produce higher-mass b quarks. The
$|\Delta\varphi^{*}|$ distributions are compared with LO and NLO expectations
from PYTHIA Sjostrand et al. (2008) and POWHEG Gauld et al. (2015) MC
simulations, respectively. The prediction from an artificial data-driven model
assuming uncorrelated $\rm{b}\bar{\rm{b}}$ production is also shown. At lower
$p_{\rm{T}}^{\rm{J}}$/ψ, the PYTHIA prediction describes the data well,
suggesting that NLO effects in $\rm{b}\bar{\rm{b}}$ production in this
kinematic region are small compared with the experimental precision. At higher
$p_{\rm{T}}^{\rm{J}}$/ψ, data are instead better described by POWHEG
calculations.

Figure 6: (Left) Differential $\rm{B}\bar{\rm{B}}$ production cross-section as
a function of $\Delta\varphi$ for three leading jet $p_{\rm{T}}$ regions in pp
collisions at $\sqrt{s}=7$ TeV, measured by the CMS Collaboration, and
compared to PYTHIA predictions. For the data points, the error bars show the
statistical (inner bars) and the total (outer bars) uncertainties. (Right)
Ratio of the cross-section as a function of $\Delta\varphi$ for data,
MADGRAPH, MC@NLO, and CASCADE models, with respect to PYTHIA predictions, for
the three leading jet $p_{\rm{T}}$ regions. The simulations (shaded bands) are
normalized to the region $\Delta\varphi>\frac{3}{4}\pi$. The widths of the
shaded bands indicate the statistical uncertainties of the predictions
Khachatryan et al. (2011).

Figure 7: Normalized differential production cross-sections of B hadron pairs
as a function of $|\Delta\varphi^{*}|$ for $p_{\rm{T}}^{\rm{J}}$${}^{/\psi}>3$
GeV/$c$ (left) and $p_{\rm{T}}^{\rm{J}}$${}^{/\psi}>7$ GeV/$c$ (right) in pp
collisions at $\sqrt{s}=7$ and 8 TeV, measured by the LHCb Collaboration. The
data are compared with POWHEG and PYTHIA predictions. The expectations for
uncorrelated $\rm{b}\bar{\rm{b}}$ production are shown by the dashed magenta
line. The uncertainties in the model predictions due to the choice of
factorization and re-normalization scales are shown as solid bands Aaij et al.
(2017).

##  3 Characterizing the Fragmentation of Heavy Quarks into Jets

In the absence of a surrounding nuclear medium, the process of hadronization
of a heavy quark into colorless hadrons is generally described as a non-
perturbative fragmentation of the quark into lower-momentum partons that, in
the final state, are converted into colorless hadrons. At high $p_{\rm T}$,
this process produces a spray of particles traveling in a similar direction,
called “jet”. A thorough characterization of the in-vacuum heavy quark
fragmentation process can be obtained by measuring the angular correlation
distribution between heavy-flavor “trigger” particles and other “associated”
charged particles produced in the same event, in pp collisions. Under the
assumption of a leading-order production process of the quark–anti quark pair,
two peaks can be observed in the correlation distribution for
$\Delta\varphi\approx 0$ (“near-side” peak) and $\Delta\varphi\approx\pi$
(“away-side” peak). These peaks are produced by particles in the jets emerging
from the heavy quark pair fragmentation, which are approximately collinear
with the quark’s directions. While the presence of next-to-leading-order heavy
quark production processes—that are quite significant at LHC energies—breaks
this picture for the away-side peak, the features of the near-side peak
retains a clear connection to the original parton shower features. In
particular, studying the peak shape, its particle content and composition, and
the $p_{\rm T}$ distribution of its constituents for different kinematic
regimes allows us to retrieve information about the heavy quark fragmentation
process. By comparing the features of the near-side peak with predictions from
theoretical models or Monte Carlo simulations that implement different
techniques to model the heavy quark fragmentation (and, in general, for the
description of processes involving heavy quarks), it becomes possible to
discriminate models and to validate those that yield the most accurate
description of the data. In general, these comparisons allow us to determine
constraints on the model configuration, parameters, and tuning.

In the presence of a deconfined medium, like the quark–gluon plasma produced
in ultra-relativistic heavy-ion collisions, the hadronization process can be
modified with respect to in-vacuum fragmentation. In particular, an additional
hadronization mechanism, the coalescence, is expected to play a prominent
role, where neighboring quarks in phase space recombine into higher-momentum
bound hadrons Fries et al. (2003); Greco et al. (2003); Ravagli and Rapp
(2007). The role of coalescence, as a competing mechanism to the quark
fragmentation, is already hinted from studies of charm/hadron production
ratios in heavy-ion collisions Adam et al. (2021); Acharya et al. (2023,
2022). The modified hadronization should result in a significant modification
of the final-state jet produced by the heavy quark. Such a modification can be
evidenced and quantified by comparing the properties of the near-side peak of
angular correlations between heavy-flavor particles and other particles, in
Pb–Pb collisions to the reference system of pp collisions.

In this context, the ALICE Collaboration measured the azimuthal correlation
distribution between D mesons and charged particles in pp collisions at
$\sqrt{s}=13$ TeV Acharya et al. (2022). A weighted average of the correlation
distributions of ${\rm D^{0}}$, ${\rm D^{+}}$, and ${\rm D^{*+}}$ mesons was
considered, at central rapidity ($|y|<0.5$) in the transverse momentum range
$3<p_{\rm T}^{\rm D}<36$ GeV/$c$, while the associated particles were
reconstructed in $\eta<0.8$ for $p_{\rm T}^{\rm assoc}>0.3$ GeV/$c$. Only
pairs with $|\Delta\eta|$ < 1 were considered. The correlation distribution
was fitted with a function composed of a generalized Gaussian, describing the
near-side peak, a Gaussian describing the away-side peak, and a constant,
accounting for the physically uncorrelated pairs, assumed to be flat along
$\Delta\varphi$. This model allowed for obtaining a quantitative description
of the properties of the peaks in terms of their integral (peak yield) and
width for the different kinematic ranges studied. The correlation
distributions and the near- and away-side peak properties were found to be
consistent with the results obtained for lower center-of-mass energies
($\sqrt{s}=5.02$ TeV Acharya et al. (2020) and $\sqrt{s}=7$ TeV Adam et al.
(2017)). The similarity of the near-side features implies that the charm quark
fragmentation process is independent of the collision energy, at least for the
energy ranges studied at the LHC.

Focusing further on the near-side correlation peak properties, a significant
increase in its yield with increasing values of $p_{\rm T}^{\rm D}$ was
observed. This can be explained by the corresponding increase in charm quark
$p_{\rm T}$, on average, which implies that a larger amount of energy is
available for the production of associated particles during its fragmentation.
At the same time, a narrowing of the peak width can be observed when probing
larger D meson $p_{\rm T}$. Such an effect is related to the increased Lorentz
boost of the charm quark, leading to a more collimated spray of particles
produced by the fragmentation in the laboratory frame.

A comparison of the ALICE results with several model predictions, including
PYTHIA8 Sjostrand et al. (2008) with 4C tune, POWHEG+PYTHIA8 Nason (2004);
Frixione et al. (2007) using hard parton scattering matrix elements at LO or
at NLO accuracy, HERWIG 7 Bellm et al. (2016), and EPOS 3.117 Drescher et al.
(2001); Werner et al. (2010), is shown in Figure 8. In particular, the near-
side yields (widths) are shown in the first (third) row, and their model-to-
data ratios are reported in the second (fourth) row. Although all the models
are able to reproduce the increase in peak yields for increasing D meson
$p_{\rm T}$, the strength of such a dependence, and the absolute values of the
yields differ substantially among the various models. In particular, an
ordering is found for the predicted near-side yields, with the lowest values
observed for HERWIG (which tends to underestimate the data for $p_{\rm T}^{\rm
D}<16$ GeV/$c$), followed by PYTHIA, POWHEG+PYTHIA, and EPOS, which
overestimates the yield values in most of the $p_{\rm T}$ intervals. Among the
tested predictions, POWHEG+PYTHIA8 and PYTHIA generators are those that better
reproduce the measured data and are thus more suited to quantify the number of
particles emerging from charm quark fragmentation in association with the D
meson. The various model predictions for the near-side widths show instead
similar values in all the studied kinematic ranges, and overall, all models
are consistent with the ALICE measurements within the uncertainties.

Figure 8: First  (third) row: near-side associated yields (widths) of
azimuthal correlation distributions between D mesons and charged particles in
pp collisions at $\sqrt{s}=13$ TeV, measured by the ALICE Collaboration and
predicted by several models, as a function of $p_{\rm T}^{\rm D}$, for
different $p_{\rm T}^{\rm assoc}$ intervals. Second (fourth) row: model-to-
data ratios of near-side associated yields (widths) of D meson and charged
particle correlation distributions Acharya et al. (2022).

The in-vacuum behavior of heavy quarks, including their parton shower and
hadronization, can be altered in the presence of a nuclear medium.
Measurements in p–Pb collisions are sensitive to the influence of cold-
nuclear-matter effects on the heavy quarks, and can thus act as a reference to
help disentangle and understand those modifications that are instead induced
by the quark–gluon plasma environment in heavy-ion collisions. To probe
whether cold-nuclear-matter effects play a role in the charm quark
fragmentation process, the ALICE Collaboration has measured the azimuthal
correlation distribution of D mesons and charged particles at mid-rapidity in
multiplicity-integrated p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV. The
same analysis technique used in the pp collision results discussed above
Acharya et al. (2022) was exploited, with the same kinematic coverage. The
shape of the correlation distribution and its evolution with transverse
momenta of D mesons and associated charged particles were compared, and found
to be fully consistent, with those obtained in pp collisions at the same
center-of-mass energy, as shown in Figure 9. Specifically, the comparison of
the near-side peak yields and widths in the two collision systems is shown in
Figure 10. No modification of the near-side yield values, and the same
increase in D meson $p_{\rm T}$ measured in pp collisions were observed. For
the near-side widths, the tendency for a collimation of the peak at larger
$p_{\rm T}^{\rm D}$ is possibly less pronounced, but pp and p–Pb results are
similar within uncertainties. From these results, no indications for a
modified fragmentation process of charm quarks due to cold-nuclear-matter
effects, or for any alteration of the charm hadronization mechanism, are
observed.

Figure 9: Comparison of the azimuthal correlation distributions between D
mesons and charged particles in pp and p–Pb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV, for the kinematic ranges studied by the ALICE Collaboration
Acharya et al. (2020).

In recent years, the high-energy nuclear physics community’s interest in the
dynamics of partons produced in small collision systems at very high
multiplicities has grown. Although no clear modifications of high-$p_{\rm T}$
particle production yields (beyond what is expected from nuclear modification
of the parton distribution functions) have been measured, several indications
of collective-like effects have been observed at the LHC in the recent past,
including measurements in the heavy-flavor sector Acharya et al. (2019); Aad
et al. (2020); Sirunyan et al. (2018, 2021) as discussed in more detail in
Section 5. In general, the evaluation of flow coefficients in small collision
systems is based on two- or multi-particle correlation techniques, and relies
on the assumption that the contribution of jet peaks to the correlation
distribution has negligible dependence on the event multiplicity, and can be
removed from the high-multiplicity correlation distribution by measuring it in
low-multiplicity collisions, where no collective effects are present. Such an
assumption can be tested by studying the jet fragmentation properties at
different event multiplicities in small collision systems. In this regard, the
ALICE experiment has studied the dependence of the azimuthal correlation
distribution of D mesons and charged particles, and of its near-side peak
features on the event multiplicity in pp collisions at $\sqrt{s}=13$ TeV
Acharya et al. (2022) and p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV
Acharya et al. (2020). The results show that, within the experimental
uncertainties, the near-side peak yields and widths are consistent for all the
multiplicity ranges studied, suggesting a similar fragmentation of charm quark
into final-state D mesons and other associated particles that is independent
of the surrounding event activity.

Figure 10: Comparison of near-side associated yields (first row) and widths
(second row) of D meson and charged-particle azimuthal correlation
distributions in pp and p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV,
measured by the ALICE Collaboration, as a function of $p_{\rm T}^{\rm D}$, for
different $p_{\rm T}^{\rm assoc}$ intervals Acharya et al. (2020).

As an alternate approach for investigating the in-vacuum heavy quark
fragmentation and possible impact of cold-nuclear-matter effects, the ALICE
Collaboration has measured angular correlations between electrons produced
from decays of heavy-flavor hadrons (called herein “heavy-flavor decay
electrons” for simplicity) and charged particles in pp and p–Pb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV Acharya et al. (2023). Compared to correlations
with D mesons as heavy-flavor trigger particles, heavy-flavor decay electrons
have a looser connection with the direction and momentum of the original heavy
quark because of the hadron decay kinematics. On the other hand, such an
analysis profits from a larger statistical sample, which allows for the
transverse momentum range of the associated charged particles to be
significantly extended (up to 7 GeV/$c$). Additionally, at high $p_{\rm T}$,
the sample of heavy-flavor decay electrons is dominated by those from beauty
quarks, which enables the study of beauty quark fragmentation when focusing in
the $p_{\rm T}$ region above 7 GeV/$c$. The study is performed at mid-
rapidity, in the electron $p_{\rm T}$ range $4<p_{\rm T}<16$ GeV/$c$,
considering pairs with pseudorapidity displacement $|\Delta\eta|<1$. Also, in
this case, a quantitative assessment of the quark-into-jet fragmentation is
performed by fitting the correlation distribution with a function, composed as
the sum of two von Mises functions, to model the near- and away-side peaks,
plus a constant term.

In Figure 11, the near-side peak yields and widths measured by ALICE in pp
collisions are compared with predictions from the PYTHIA8 event generator
Sjostrand et al. (2008) with the Monash tune and from the EPOS3 event
generator Drescher et al. (2001); Werner et al. (2010). Two transverse
momentum ranges are considered for the electrons, i.e., $4<p_{\rm T}^{\rm
e}<7$ GeV/$c$, with a balanced contribution between charm and beauty origins,
and $7<p_{\rm T}^{\rm e}<16$ GeV/$c$, where the large majority of electrons
are produced by beauty hadron decays. For both $p_{\rm T}^{\rm e}$ ranges, the
largest contribution of charged particles produced in the fragmentation is
present below 2 GeV/$c$, pointing to a dominance of soft particle production
from the quark fragmentation. The fraction of high-$p_{\rm T}$-associated
particles significantly increases when probing the high-$p_{\rm T}^{\rm e}$
range, despite remaining subdominant. In addition, the absolute value of the
yields is substantially larger in the $7<p_{\rm T}^{\rm e}<16$ GeV/$c$ range
compared to the $4<p_{\rm T}^{\rm e}<7$ GeV/$c$ interval. This is similar to
what was observed for the D meson correlation with charged particles, and can
be ascribed to the larger average energy of heavy quarks producing
higher-$p_{\rm T}$ electrons, which generally leads to an increased
multiplicity of fragmenting particles. The values of the near-side widths are
fully consistent between the two $p_{\rm T}^{\rm e}$ ranges, and point towards
an emission of harder particles more collinear with the electron, while softer
particles are emitted at larger angles. Both PYTHIA8 and EPOS3 generators can
successfully describe the near-side yield values, with EPOS3 predicting larger
values for high $p_{\rm T}$ of the associated tracks. While PYTHIA8 also
correctly reproduces the near-side widths, EPOS3 tends to overestimate them at
high $p_{\rm T}^{\rm assoc}$, predicting a flatter trend than what is observed
in the data. The away-side peak yields and widths are also shown in the same
figure. The away-side peak has a connection to the fragmentation of the heavy
quark that did not lead to the production of the trigger particle, though such
a connection is less direct than that of the near-side peak. In contrast, the
away-side peak features are sensitive to the production mechanisms of the
heavy quark pairs, which induce different angular topologies, as discussed in
Section 2. Very similar considerations as for the near-side can be drawn for
the away-side peak yield values. The away-side peak widths are about a factor
2 larger than those measured for the near-side, with significantly larger
uncertainties. This is mainly due to the NLO production processes of heavy
quarks, which break the back-to-back topology of the quark pairs, and to the
additional smearing with respect to the original quark–anti-quark correlation
distribution induced by the hadronic decays.

Figure 11: Near- (left) and away-side (right) associated peak yields (top) and
widths (bottom) of correlation distributions between heavy-flavor hadron decay
electrons and charged particles in pp collisions at $\sqrt{s_{\rm NN}}=5.02$,
measured by the ALICE Collaboration and predicted by PYTHIA8 and EPOS3. The
results are reported for two $p_{\rm T}$ ranges, and the insets show the
ratios of the observable distributions in higher to lower $p_{\rm T}$ ranges
considered Acharya et al. (2023).

In the same publication, the near- and away-side peak properties of the
azimuthal correlation between heavy-flavor decay electrons and charged
particles in pp and p–Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV are
compared for $4<p_{\rm T}^{\rm e}<12$ GeV/$c$ and various charged-particle
transverse momentum intervals. From the comparison, shown in Figure 12, fully
compatible peak yields and widths are found in the two collision systems. This
observation holds also for the high $p_{\rm T}^{\rm assoc}$ intervals not
covered by previous D-hadron and charged particle correlation measurements
Acharya et al. (2020). These results thus complement the findings observed for
that analysis, and confirm that the charm quark fragmentation is unaffected by
the presence of cold-nuclear-matter effects. No strong conclusions can be
drawn for the beauty, given the lack of a specific comparison in a
high-$p_{\rm T}^{\rm e}$ interval.

Figure 12: Comparison of near- (left) and away-side (right)-associated peak
yields (top) and widths (bottom) from the azimuthal correlation distributions
of heavy-flavor hadron decay electrons and charged particles in pp and p–Pb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, measured by the ALICE
Collaboration. The insets show the ratios of the observable distribution in pp
to p–Pb collisions Acharya et al. (2023).

##  4 Heavy Quark Energy Loss and Redistribution

In the presence of the quark–gluon plasma, produced in ultra-relativistic
heavy-ion collisions, high-$p_{\rm{T}}$ partons produced in hard scatterings
with high $Q^{2}$ lose energy via medium-induced gluon radiation and
collisions with the medium constituents Acharya et al. (2022, 2022); Gyulassy
and Plumer (1990); Baier et al. (1997); Thoma and Gyulassy (1991); Braaten and
Thoma (1991). The processes by which quarks and gluons lose energy in the QGP
can modify the internal structure and energy of the jet they produce, a
phenomenon known as jet quenching. Through the passage of a jet, the QGP can
itself be modified, due to the injection of energy and momentum lost by the
jet into the plasma. Because of momentum conservation, a “wake” is induced in
the medium as the jet loses energy and momentum, giving the medium a net
momentum in the jet direction, yielding a correlation between the bulk
dynamics of the medium and the jet direction Casalderrey-Solana et al. (2021).

Measurements of heavy-flavor jets and particle distributions within jets can
be used to constrain parton energy loss mechanisms and to probe how the “lost”
energy is redistributed to other partons and the subsequent particles emerging
from the collision. In general, these measurements and analyses may lead to a
better understanding of heavy quark propagation inside the medium Wang et al.
(2019); Nahrgang et al. (2014); Cao et al. (2015); Hambrock and Horowitz
(2017). Such measurements can provide complementary information to the
measurements of inclusive heavy-flavor mesons Dong et al. (2019), such as the
nuclear modification factor Adam et al. (2016, 2015); Sirunyan et al. (2018,
2017); Khachatryan et al. (2017) and azimuthal anisotropy Khachatryan et al.
(2017); Abelev et al. (2014); Sirunyan et al. (2018); Adamczyk et al. (2017a);
Acharya et al. (2018). Experiments at RHIC and at the LHC have been performed
to investigate the angular correlations of particles associated with heavy-
flavor jets. In this section, a brief overview of these measurements is
presented.

The PHENIX and STAR Collaborations at RHIC performed studies of angular
correlations of electrons from heavy-flavor hadron decays with charged hadrons
Adare et al. (2011), and angular correlations of D mesons with charged hadrons
Adam et al. (2020), respectively. In pp collisions, such a correlation
distribution is characterized by a jet peak at small $\Delta\varphi$ due to
particle pairs from the same fragmentating jet, and a jet peak at
$\Delta\varphi\sim\pi$ due to particle pairs from the fragmentating partons in
back-to-back di-jet. Angular correlation measurements in pp collisions are
discussed in much detail in the previous Section 3. In nucleus–nucleus
collisions, these correlations can provide information about the pattern of
energy loss for the back-to-back di-jet system as well as interaction between
the fast partons and the medium. The STAR Collaboration performed a study of
the centrality dependence of 2D angular correlations
($\Delta\eta,\Delta\varphi$) of $\rm{D}^{0}$ mesons
($2<p_{\rm{T}}^{\rm{D}^{0}}<10$ GeV$/c$), produced by charm quark
hadronization after it traverses the medium, and associated charged hadrons
($p_{\rm{T}}$ integrated) in Au–Au collisions at $\sqrt{s_{\rm{NN}}}=200$ GeV
Adam et al. (2020). The main focus of this analysis was the near-side
correlation distribution, within $\Delta\varphi\leq\pi/2$, measuring the 2D
widths of the jet-like peak, and the number of associated charged hadrons
associated with the triggered $\rm{D}^{0}$ meson. The near-side yield and peak
widths as a function of the collision centrality are shown in Figure 13. The
expectations from PYTHIA Monte Carlo simulations Sjostrand and van Zijl
(1987); Shi et al. (2015), as a proxy for pp collisions, are also included.
The yields and widths in 50–80% central Au–Au collisions are consistent with
the PYTHIA predictions within the measured uncertainties. The near-side yields
and the widths are observed to increase towards more central Au–Au collisions,
similar to what was measured for unidentified di-hadron correlations
Agakishiev et al. (2012). The increase in the near-side yields and widths in
most central collisions is observed for the same $p_{\rm{T}}$ range, where a
strong suppression in $\rm{D}^{0}$ meson yield is observed Adam et al. (2020),
thus bringing complementary information about charm quark propagation in the
QGP medium. The measurement could indicate that the energy lost by the charm
quark results in the production of new particles accompanying the D meson. The
PHENIX Collaboration measured angular correlations of electrons from heavy-
flavor hadron decays and charged particles in Au–Au collisions at
$\sqrt{s_{\rm{NN}}}=200$ GeV in 0–60% centrality, for two trigger electron
$p_{\rm{T}}$ intervals, $2<p_{\rm{T}}^{\rm{e}}<3$ GeV$/c$ and
$3<p_{\rm{T}}^{\rm{e}}<4$ GeV$/c$, and for different associated charged-
particle $p_{\rm{T}}$ intervals Adare et al. (2011). To investigate the
possible modification of the jet produced by the opposite heavy quark with
respect to one producing the trigger electron, the away-side
($1.25<\Delta\varphi<\pi$ rad) yield was obtained. The ratio of the away-side
yields in Au–Au collisions to pp collisions is shown in Figure 14. The
$I_{\rm{AA}}$ is the largest and above unity for low associated-particle
$p_{\rm{T}}$, and decreases with increasing associated particle $p_{\rm{T}}$.
The $I_{\rm{AA}}$ obtained for correlations of electrons from heavy-flavor
hadron decays is compared with the one obtained for unidentified di-hadron
correlations with similar average triggered hadron $p_{\rm{T}}$. The
$I_{\rm{AA}}$ for heavy-flavor trigger particles is consistent with that of
unidentified charged particles Adare et al. (2008), though within large
uncertainties, which could indicate similar modifications of charged particles
inside light-flavor and heavy-flavor jets due to interaction with the QGP
medium.

Figure 13: Near-side jet-like peak properties of $\rm{D}^{0}$ meson and hadron
correlation distribution in Au–Au collisions at $\sqrt{s_{\rm{NN}}}=200$ GeV,
measured by the STAR Collaboration. The near-side peak width along
$\Delta\eta$ (top left), width along $\Delta\varphi$ (top right), and
correlated hadron yield per $\rm{D}^{0}$ trigger (bottom) are shown. PYTHIA
predictions and di-hadron results Agakishiev et al. (2012) are also included.
Vertical bars show the statistical errors, and cross-bars show the systematic
uncertainties Adam et al. (2020). Figure 14: $I_{\rm AA}$ determined from the
per-trigger yield of away-side $\Delta\varphi$ distribution of electrons from
heavy-flavor hadron decays and charged particles in Au–Au collisions at
$\sqrt{s_{\rm NN}}=200$ GeV, measured by the PHENIX Collaboration. The
$\Delta\varphi$ range used is $1.25<\Delta\varphi<\pi$ rad. For comparison,
the di-hadron $I_{\rm AA}$ values Adare et al. (2008) are also shown for
trigger $p_{\rm T}$ selections where the parent heavy meson has similar
$p_{\rm T}$ as the trigger light hadron Adare et al. (2011).

Measurements of angular correlations between heavy-flavor mesons and jets can
be used to constrain parton energy loss mechanisms and to better understand
the heavy quark diffusion (i.e., propagation) inside the QGP medium Wang et
al. (2019). The charm quark diffusion with respect to the jet axis was
measured by the CMS Collaboration Sirunyan et al. (2020) in pp and Pb–Pb
collisions, for two $\rm{D}^{0}$ meson $p_{\rm{T}}$ intervals, a lower
$p_{\rm{T}}$ interval of $4<p_{\rm{T}}^{\rm{D}^{0}}<20$ GeV$/c$ and high
$p_{\rm{T}}>20$ GeV$/c$, for $p_{\rm{T}}^{\rm{Jet}}>60$ GeV$/c$. The measured
observable is the radial distribution of the $\rm{D}^{0}$ mesons with respect
to the jet axis, $r=\sqrt{\Delta\varphi^{2}+\Delta\eta^{2}}$, defined as the
quadratic sum of the differences in pseudorapidity and azimuth between the
$\rm{D}^{0}$ meson and the jet axis direction, shown in Figure 15 for
$4<p_{\rm{T}}^{\rm{D}^{0}}<20$ GeV$/c$. The average value of $r$ for
low-$p_{\rm{T}}$ $\rm{D}^{0}$ mesons was measured to be $0.198\pm
0.015~{}(\rm{stat})\pm 0.005~{}(\rm{sys})$ and $0.160\pm
0.007~{}(\rm{stat})\pm 0.009~{}(\rm{sys})$ for Pb–Pb and pp collisions,
respectively. This indicates that $\rm{D}^{0}$ mesons at low $p_{\rm{T}}$ are
farther away from the jet axis in Pb–Pb compared to pp collisions. At higher
$\rm{p}_{T}$, the radial distribution of $\rm{D}^{0}$ mesons is more similar
in Pb–Pb and pp collisions. The pp results are compared with PYTHIA Sjöstrand
et al. (2015) and SHERPA Gleisberg et al. (2009) event generators, which
capture the data trend well within the measured uncertainties. The Pb–Pb
distribution is compared to an energy loss model, CCNU Wang et al. (2019),
which includes in-medium elastic (collisional) and inelastic (radiative)
interactions for both the heavy and the light quarks. The model predicts a
small depletion of the $\rm{D}^{0}$ meson yield at small $r$ and an
enhancement of yield at larger $r$ compared to pp collisions, which is
consistent with the trend seen in the data, as can be seen in the ratio plot
of Figure 15. This measurement hints at a modification of the $\rm{D}^{0}$
meson radial profile in Pb–Pb collisions at low $p_{\rm{T}}$, possibly induced
by interactions of the charm quark with the medium constituents, which alter
the original quark direction. This modification vanishes at higher
$p_{\rm{T}}$.

Figure 15: Distribution of $\rm{D}^{0}$ mesons ($4<p_{\rm T}^{D^{0}}<20$
GeV$/c$) in jets, as a function of the distance from the jet axis ($r$) for
$p_{\rm T}^{\rm{jet}}>60$ GeV$/c$ and $|\eta^{\rm{jet}}|<1.6$ in pp and Pb–Pb
collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV from the CMS Collaboration. The
Pb–Pb spectra are compared to the CCNU energy loss model, while the pp spectra
are compared with predictions from the PYTHIA and SHERPA pp MC event
generators. The ratios of the $\rm{D}^{0}$ meson radial distributions in Pb–Pb
to the pp data are shown in the middle panel. In the bottom panel, the ratios
of the $\rm{D}^{0}$ meson radial distributions predicted by the two MC event
generators to the CMS results in pp collisions are presented Sirunyan et al.
(2020).

Similarly, correlation measurements can be used to study the modification of
jet shapes in heavy-ion collisions, using charged hadron constituents as a
function of their radial distance from the jet axis, as performed by the CMS
experiment Tumasyan et al. (2023). These measurements can give insight into
details of jet quenching and medium response to the evolving jet. The CMS
experiment used jets initiated by beauty quarks (b-jets) to provide unique
experimental means to investigate the mass dependence of quenching effects and
parton shower evolution. The transverse momentum profile $P(\Delta r)$ of
charged particles in the jets, defined as $P(\Delta r)=\frac{1}{\Delta
r_{\rm{b}}-\Delta
r_{\rm{a}}}\frac{1}{N_{\rm{jet}}}\sum_{\rm{jets}}$$\sum_{\rm{trk}}$${}_{\in(\Delta
r_{\rm{a}},\Delta r_{\rm{b}})}p_{\rm{T}}^{\rm{trk}}$, was measured, where
$\Delta r=\sqrt{\Delta\varphi^{2}+\Delta\eta^{2}}$ is the radial distance
between a track and the jet axis defined in pseudorapidity and azimuthal
angle, $\Delta r_{\rm{a}}$ and $\Delta r_{\rm{b}}$ are the edges of rings in
$\Delta r$, and $p_{\rm{T}}^{\rm{trk}}$ is the charged particle’s transverse
momentum. The $P(\Delta r)$ distribution is normalized to unity within $\Delta
r<1$ to produce the jet shape distribution, $\rho(\Delta r)$, that indicates
how the momentum of charged particles is distributed with respect to the jet
axis. The shapes of b-jets and inclusive jets of $p_{\rm{T}}>120$ GeV$/c$ for
charged particles with $p_{\rm{T}}>1$ GeV$/c$ was measured for Pb–Pb and pp
collisions, as shown in Figure 16. The b-jets are found to be broader than
inclusive jets. The ratio of $\rho(\Delta r)$ distribution in Pb–Pb to pp
collisions ($2^{\rm{nd}}$ row panels) shows a depletion of particles for low
$\Delta r$ and a strong enhancement at large $\Delta r$, indicating
redistribution of $p_{\rm{T}}$ of jet constituents from small to large
distances from the jet axis. The large $\Delta r$ enhancement in Pb–Pb
collisions is centrality-dependent and is most significant in central
collisions, indicating a modification of energy flow around the jet axis in
the presence of the QGP medium. The large $\Delta r$ enhancement in Pb–Pb
collisions is more pronounced for b-jets than inclusive jets, showing mass
dependent interactions in the QGP. The difference in the transverse momentum
profile between Pb–Pb and pp collisions characterizes the magnitude of the
measured excess momentum as shown in the third row of the figure. A more
significant transverse momentum excess in Pb–Pb collisions at intermediate and
high $\Delta r$ is found for b-jets than for inclusive jets. A comparison of
b-jet shapes to those of inclusive jets is shown in the bottom panel for pp
and Pb–Pb collisions. In pp collisions, b-jets show a depletion for $\Delta
r<0.05$, with respect to inclusive jets, that could be interpreted as being
due to the dead-cone effect Dokshitzer et al. (1991) (suppression of collinear
parton radiation from a massive emitter such as a heavy quark). In Pb–Pb
collisions, the depletion at small $\Delta r$ is similar to pp collisions. For
higher $\Delta r$, b-jet shapes are broader than inclusive jet shapes in pp
and Pb–Pb collisions, with a significant enhancement in the most central Pb–Pb
collisions. This measurement provides new constraints for theoretical
calculations of parton flavor dependence of energy loss and jet–medium
interactions in the quark–gluon plasma.

At momenta comparable to or smaller than the quark mass, heavy quarks are
thought to undergo Brownian-like motion in the quark–gluon plasma, with their
transport characterized by a diffusion coefficient Casalderrey-Solana and
Teaney (2006). The process of losing energy via gluon radiation is referred to
as radiative energy loss Djordjevic and Gyulassy (2004). Heavy quarks, when
produced at LO, will have a back-to-back correlation in azimuthal angle
between the quark and anti-quark, due to momentum conservation. As a
consequence of the multiple interactions with the medium, this initial
correlation can broaden around $\Delta\varphi=\pi$. Energy loss via a
radiative mechanism may dampen this broadening Zakharov (2021). Angular
correlations of heavy quarks can thus be sensitive to the relative
contribution of collisional and radiative energy loss processes Nahrgang et
al. (2014); Cao et al. (2015). The ATLAS Collaboration performed measurements
of angular correlations between muon pairs in pp and Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}=5.02$ TeV Aad et al. (2023). Muon pairs from beauty hadron
decays were selected by considering muons with the same charge with both muons
having $p_{\rm{T}}>4$ GeV$/c$. The $\Delta\varphi$ distribution of muon pairs
shows a clear peak on the away-side, consistent with the back-to-back
configuration of beauty quark pair production from hard scattering processes.
The width of the away-side peaks was characterized by fitting the
$\Delta\varphi$ distribution with a Cauchy–Lorentz function. The centrality
dependence of the away-side width was measured, as shown in Figure 17. The
widths obtained for different centralities in Pb–Pb collisions are consistent
with the value measured in pp collisions, with a slightly reduced value
observed for the 0–10% centrality interval. To further investigate the
mechanisms by which a heavy quark interacts with the QGP constituents, this
measurement can be extended to different $p_{\rm{T}}$ regions of muon pairs.
Comparison with model calculations including collisional and/or radiative
energy loss would help with the interpretation of the current observation. The
measurement will provide important constraints on theoretical descriptions of
the dynamics of heavy quarks inside the quark–gluon plasma.

Figure 16: First row: jet shape distributions $\rho(\Delta r)$ for inclusive
and b-jets with $p_{\rm{T}}>120$ GeV$/c$ in three centrality bins (left to
right) for Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV, and in pp
collisions, both measured by the CMS Collaboration. Second row: ratio of Pb–Pb
to pp jet shape results for inclusive jets (red) and b-jets (blue). Third row:
difference between the charged-particle transverse momentum profile between
Pb–Pb and pp collisions for inclusive and b-jets. Fourth row: ratio of b- to
inclusive jet shapes for several Pb–Pb centrality bins (green), as well as pp
collisions (identical in all three panels). In all panels, the vertical bars
and shaded boxes represent the statistical and systematic uncertainties,
respectively Tumasyan et al. (2023). Figure 17: The measured widths of the
away-side peak of the angular correlations between same-sign muon pairs in pp
and Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV from the ATLAS
Collaboration. The vertical bars and shaded bands on the data points represent
the statistical uncertainties, and systematic uncertainties, respectively. The
horizontal lines indicate the nominal pp values plotted across the full
centrality range Aad et al. (2023).

##  5 Small-System Collective-Like Effects for Heavy Quarks

As discussed in the previous section, Section 4, the properties of the QGP are
studied by analyzing high-energy heavy-ion collisions Adcox et al. (2005);
Adams et al. (2005); Back et al. (2005); Arsene et al. (2005); Aamodt et al.
(2011); ALI (2022). One of the key signatures of the formation of the QGP in
these collision systems is the azimuthal anisotropy of the produced particles
Voloshin et al. (2010), due to the onset of collective motion of the system
that is driven by the specific geometrical structure of the overlap region of
the two colliding nuclei. During the medium expansion, the initial-state
spatial anisotropy is translated into a momentum anisotropy of the particles
emerging from the medium Qin et al. (2010). The magnitude of the azimuthal
anisotropies is quantified via a Fourier decomposition of the particle
azimuthal distribution Voloshin and Zhang (1996), where the Fourier
coefficients $v_{n}$ characterize the strength of the anisotropy. For non-
central A–A collisions, where the overlap region typically has an almond
shape, the largest contribution to the azimuthal anisotropy is provided by the
second-order Fourier coefficient $v_{2}$, referred to as the elliptic flow
coefficient. Its value is used to characterize the strength of the collective
motion of the system. In two-particle angular correlation distributions
measured in non-central A–A collisions, the effect of the elliptic flow can be
seen as pronounced structures on the near and away sides along
$\Delta\varphi$, extending over a large $\Delta\eta$ region, which are
commonly referred to as “ridges” Abelev et al. (2009). The measurements are
well described by models invoking a hydrodynamic expansion of the hot and
dense asymmetrical medium produced in the collision. Surprisingly, similar
long-range ridge structures and a positive $v_{2}$ coefficient were also
observed for light-flavor particles in high-multiplicity pp and p–Pb
collisions at the LHC Abelev et al. (2013); Aaboud et al. (2017); Chatrchyan
et al. (2013); Abelev et al. (2013); Khachatryan et al. (2015, 2010), and in
high-multiplicity $d-$Au and 3He–Au collisions at RHIC Adare et al. (2013);
Adamczyk et al. (2015). The interpretation of the positive $v_{2}$ in these
small systems is currently highly debated Loizides (2016). It has raised the
question of whether a fluid-like QGP medium with a size smaller than that
produced in A–A collisions is created Werner et al. (2011); Deng et al.
(2012). Alternate explanations foresee mechanisms involving initial-state
effects, such as gluon saturation within the color glass condensate effective
field theory Dusling and Venugopalan (2013); Bzdak et al. (2013), or final-
state color–charge exchanges Dumitru et al. (2014); Wong (2011).

Heavy quarks, produced during the early stages of hadronic collisions Cacciari
et al. (1998, 2001); Kniehl et al. (2006, 2008), can be used to probe both
initial- and final-state effects of the collision dynamics Bierlich et al.
(2018); Dusling et al. (2016); Zhang et al. (2019, 2020); Dong et al. (2019).
In A–A collisions, strong elliptic flow signals were observed for leptons from
the decay of heavy-flavor hadrons and open-charm D mesons Adam et al. (2016a,
b); Acharya et al. (2017); Abelev et al. (2014); Sirunyan et al. (2018); ALI
(2022), suggesting that charm quarks develop significant collective behavior
via their strong interactions with the bulk of the QGP medium. Measurements of
elliptic flow of hidden-charm J$/\psi$ mesons provide further evidence for
strong rescatterings of charm quarks Acharya et al. (2017). Recent
measurements of positive $v_{2}$ for non-prompt D mesons (i.e., D mesons
produced from the decays of beauty hadrons), though with smaller values than
prompt D meson $v_{2}$, were released by the CMS CMS (2022) and ALICE
Collaborations Acharya et al. (2023), suggesting that beauty quarks could also
participate in the medium collective motion, though with lesser extent than
charm quarks.

In small colliding systems, the study of heavy-flavor hadron collectivity has
the potential to disentangle possible contributions from both initial- and
final-state effects Zhang et al. (2019); Li et al. (2019); Zhang et al.
(2020). In particular, heavy-flavor hadrons may be more sensitive to possible
initial-state gluon saturation effects. The Collaboration at the LHC performed
several measurements to evaluate the $v_{2}$ of charm and beauty hadrons in pp
and p–Pb collisions, using two-particle angular correlation techniques Acharya
et al. (2018, 2019); Sirunyan et al. (2019); Aad et al. (2020); Sirunyan et
al. (2021); Acharya et al. (2023). The general procedure performed was to
obtain the angular correlations of heavy-flavor trigger particles with charged
particles Acharya et al. (2019) in events with high multiplicity. The
correlation distribution in these events contains contributions related to
collectivity and jet fragmentation, with the latter being referred to as non-
flow effects. These non-flow contributions can be suppressed by requiring a
pseudorapidity separation ($\Delta\eta$) between heavy-flavor particles and
charged particles. The azimuthal correlation distribution normalized to the
number of trigger particles is obtained. While a large $\Delta\eta$ separation
largely reduces non-flow contributions, especially for the near-side region, a
significant contribution from recoil jet fragmentation still remains on the
away-side of the $\Delta\varphi$ distribution. This can be subtracted by using
the azimuthal correlation distributions measured in low-multiplicity events.
The subtraction method relies on the assumptions that the jet correlations on
the away side remain unmodified as a function of the event multiplicity and
that there are no significant correlations due to collective effects in low-
multiplicity collisions. The non-flow subtracted $\Delta\varphi$ distribution
is fit with a Fourier decomposition, $a[1+2V_{1\Delta}^{\rm{HF-
ch}}\cos(\Delta\varphi)+2V_{2\Delta}^{\rm{HF-ch}}\cos(2\Delta\varphi)]$. The
second-order coefficient $2V_{2\Delta}^{\rm{HF-ch}}$, which is the dominant
term, is obtained. Using the assumption that $V_{2\Delta}^{\rm{HF-ch}}$ can be
factorized as a product of single-particle $v_{2}$ coefficients, the elliptic
flow coefficient of the heavy-flavor particle, $v_{2}^{\rm{HF}}$, is extracted
from the equation $v_{2}^{\rm{HF}}=V_{2\Delta}^{\rm{HF-ch}}/v_{2}^{\rm{ch-
ch}}$. The $v_{2}$ values of several heavy-flavor particle species in small
systems were measured by the ALICE Collaboration using leptons from heavy-
flavor hadron decays Acharya et al. (2019, 2023) and J$/\psi$ Acharya et al.
(2018), by the ATLAS Collaboration using muons from charm and beauty hadron
decays Aad et al. (2020), and by the CMS Collaboration using prompt
$\rm{D}^{0}$ Sirunyan et al. (2018), non-prompt $\rm{D}^{0}$ mesons Sirunyan
et al. (2021), and J$/\psi$ Sirunyan et al. (2019) mesons.

The $v_{2}$ of prompt $\rm{D}^{0}$ mesons Sirunyan et al. (2018), non-prompt
$\rm{D}^{0}$ mesons Sirunyan et al. (2021), and prompt J$/\psi$ Sirunyan et
al. (2019) as a function of $p_{\rm{T}}$ for p–Pb collisions at
$\sqrt{s_{\rm{NN}}}=8.16$ TeV, measured by the CMS Collaboration, is shown in
Figure 18 (left panel). The $v_{2}$ of strange hadrons Sirunyan et al. (2018)
is also shown. A positive $v_{2}$ value for prompt $\rm{D}^{0}$ and J$/\psi$
is observed, with a rising and then a decreasing trend with $p_{\rm{T}}$,
peaking at about 3–4 GeV/$c$. A clear ordering in the $v_{2}$ values is
observed for the low-$p_{\rm{T}}$ region ($p_{\rm{T}}<3$ GeV$/c$), where
heavier particles have smaller $v_{2}$ at a given $p_{\rm{T}}$ value. A
similar mass ordering for the $v_{2}$ of $\rm{D}^{0}$ mesons and strange
hadrons in semi-central Pb–Pb collisions is observed, although the
multiplicity range in Pb–Pb collisions is much larger Sirunyan et al. (2018).
In Pb–Pb collisions, this behavior is understood to be due to particle
emission from a collectively expanding source with a common velocity field.
This might indicate a significant collective behavior of charm quarks in high-
multiplicity p–Pb systems at LHC energies. For non-prompt $\rm{D}^{0}$ mesons,
the $v_{2}$ values are consistent with zero at low $p_{\rm{T}}$, while at high
$p_{\rm{T}}$, a hint of a positive $v_{2}$ value is observed, but it is not
significant within statistical and systematic uncertainties. The non-prompt
$\rm{D}^{0}$ $v_{2}$ is observed to be smaller than that of prompt
$\rm{D}^{0}$ mesons with a significance of 2.7 standard deviations. This also
indicates a mass hierarchy of the original quark participating in the
collective-like dynamics. In Pb–Pb collisions, a similar mass ordering for
muon $v_{2}$ from charm and beauty decay at low $p_{\rm{T}}$ was observed Aad
et al. (2020), and is understood as being due to final-state scattering
mechanisms Nahrgang et al. (2014); Ke et al. (2019); Katz et al. (2020). The
values of $v_{2}$ from a calculation within the color glass condensate (CGC)
framework Dusling et al. (2016); Zhang et al. (2019, 2020) for prompt J$/\psi$
mesons, prompt and non-prompt $\rm{D}^{0}$ mesons in p–Pb collisions are
compared with the data in Figure 18 (left panel). Within the CGC framework,
correlations in the initial stage of the collision between partons originating
from projectile protons and dense gluons in the lead nucleus can generate a
sizable elliptic flow. The model qualitatively describes the data, suggesting
that initial-state effects may play an important role in the generation of
collectivity for these particles in p–Pb collisions. The CGC framework also
predicts a mass hierarchy between prompt and non-prompt $\rm{D}^{0}$ mesons
for $p_{\rm{T}}\sim$ 2–5 GeV$/c$, consistent with the data within
uncertainties.

Figure 18: (Left) $v_{2}$ of prompt Sirunyan et al. (2018) and non-prompt
$\rm{D}^{0}$ mesons Sirunyan et al. (2021), $\rm{K}^{0}_{s}$ mesons and
$\Lambda$ baryons at mid-rapidity Sirunyan et al. (2018), and prompt J$/\psi$
mesons at forward rapidity Sirunyan et al. (2019), measured by the CMS
Collaboration as a function of $p_{\rm{T}}$ in high-multiplicity p–Pb
collisions at $\sqrt{s_{\rm{NN}}}=8.16$ TeV. (Right) The $p_{\rm{T}}$
differential $v_{2}^{\mu}$ of inclusive muons measured by the ALICE
Collaboration at forward rapidity in high-multiplicity p–Pb collisions at
$\sqrt{s_{\rm{NN}}}=8.16$ TeV Acharya et al. (2023), compared with predictions
of $v_{2}$ of muons from heavy-flavor hadron decays from CGC and AMPT models.
The vertical bars correspond to the statistical uncertainties, while the boxes
denote the systematic uncertainties.

The ALICE Collaboration measured the $v_{2}$ of muons at forward
($2.03<y_{\rm{cms}}<3.53$) and backward rapidity ($-4.46<y_{\rm{cms}}<-2.96$)
in high-multiplicity p–Pb collisions at $\sqrt{s_{\rm{NN}}}=8.16$ TeV Acharya
et al. (2023). For $p_{\rm{T}}>2$ GeV$/c$, a dominant contribution of muons is
produced from heavy-flavor hadron decays. A positive $v_{2}$ was measured at
both rapidities. To better understand the source of the observed azimuthal
anisotropies in small collision systems, the measurement was compared with a
multi-phase transport (AMPT) model Lin et al. (2005); Li et al. (2019); Lin
and Zheng (2021) and CGC Dusling et al. (2016); Zhang et al. (2019, 2020)
model calculations for muons from heavy-flavor hadron decays. The results for
muons measured at forward rapidity are shown in the right panel of Figure 18.
The AMPT model provides a microscopic evolution of parton interactions,
including a parton escape mechanism described via a parton cascade model Zhang
(1998). The AMPT model generates a positive $v_{2}$, mainly driven by the
anisotropic parton escape mechanism, where partons have a higher probability
to escape the interaction region along its shorter axis He et al. (2016). In
the CGC calculations, the correlations in the initial stage of the collision
between partons in the colliding proton and gluons in the dense Pb ion
generate a significant $v_{2}$ signal that persists till the final state and
is observed in the heavy-flavor hadron decay muon measurement. The CGC-based
calculations provide a larger $v_{2}$ compared to AMPT calculations at low
$p_{\rm{T}}$, up to 3 GeV$/c$, while the two models provide similar results
and describe the data at high $p_{\rm{T}}$, where heavy-flavor hadron decays
dominate the muon sample. This comparison indicates that both initial- and
final-state effects can explain the azimuthal anisotropies observed in small
collision systems.

The $v_{2}$ of J$/\psi$ at forward and backward rapidity in high-multiplicity
p–Pb collisions was compared with measurements in non-central Pb–Pb collisions
by the ALICE experiment Acharya et al. (2018). The $p_{\rm{T}}$-dependent
$v_{2}$ values in p–Pb collisions are consistent with those measured in Pb–Pb
collision within uncertainties. In Pb–Pb collisions, at low $p_{\rm{T}}$, the
$v_{2}$ coefficient is believed to originate from the recombination of charm
quarks thermalized in the medium, which is described by thermal models Du and
Rapp (2015). In p–Pb collisions, the amount of produced charm quarks is small
and, therefore, the contribution from recombination should be negligible. For
$p_{\rm{T}}>4$ GeV$/c$, the $v_{2}$ in Pb–Pb collisions is expected to come
from path length-dependent suppression inside the QGP medium. In p–Pb
collisions, the medium, if any, is expected to have a much smaller size, and
hence, very feeble path length-dependent effects are expected.

The $v_{2}$ of prompt $\rm{D}^{0}$ mesons as a function of $p_{\rm{T}}$ in pp
collisions at $\sqrt{s}=13$ TeV, measured by the CMS Collaboration Sirunyan et
al. (2021), is shown in Figure 19 (left panel). A positive $v_{2}$ signal for
prompt charm hadrons over a $p_{\rm{T}}$ range up to 6 GeV$/c$ is observed,
with a decreasing trend towards higher $p_{\rm{T}}$ values. The $v_{2}$ for
prompt $\rm{D}^{0}$ mesons is found to be comparable, within uncertainties,
with that of light-flavor hadron species, i.e., unidentified charged particles
(dominated by pions), $\rm{K}_{s}^{0}$ mesons and $\Lambda$ baryons
Khachatryan et al. (2017), that are also presented in the same figure. At
similar event multiplicities, the prompt $\rm{D}^{0}$ meson $v_{2}$ values are
found to be comparable within uncertainties in pp and p–Pb systems.

Figure 19: (Left) $v_{2}$ of prompt $\rm{D}^{0}$ mesons measured as a function
of $p_{\rm{T}}$ at mid-rapidity in high-multiplicity pp collisions at
$\sqrt{s}=13$ TeV Sirunyan et al. (2021), measured by the CMS Collaboration.
The prompt $\rm{D}^{0}$ meson measurement is compared with those of charged
particles, $\rm{K}^{0}_{s}$ mesons and $\Lambda$ baryons Khachatryan et al.
(2017). (Right) $v_{2}$ of muons from charm and beauty hadron decays as a
function of track multiplicity for muons with $4<p_{\rm{T}}<6$ GeV$/c$ in pp
collisions at $\sqrt{s}=13$ TeV Aad et al. (2020), measured by the ATLAS
Collaboration. The vertical bars correspond to the statistical uncertainties,
while the shaded areas denote the systematic uncertainties.

The elliptic flow of muons from the decay of charm and beauty hadrons in pp
collisions at $\sqrt{s}=13$ TeV was measured by the ATLAS Collaboration Aad et
al. (2020) for $4<p_{\rm{T}}<7$ GeV$/c$ and $|\eta|<2.4$. It is shown as a
function of reconstructed track multiplicity in Figure 19 (right panel). A
significant non-zero $v_{2}$ is observed for muons from charm hadron decays,
without significant dependence on multiplicity. The $v_{2}$ of muons from
charm hadron decays decreases with increasing $p_{\rm{T}}$, and is consistent
with zero for $p_{\rm{T}}\gtrsim 6$ GeV/$c$. The $v_{2}$ for muons from beauty
hadron decays was measured to be consistent with zero within uncertainties
through all the multiplicity ranges measured, and also shows no $p_{\rm{T}}$
dependence. These results also indicate a mass hierarchy of the $v_{2}$ signal
at the partonic level.

The presented measurements of heavy-flavor $v_{2}$ in high-multiplicity pp and
p–Pb collisions, the indication of mass dependence of $v_{2}$ in p–Pb
collisions, and the comparison with models capable of describing the
measurements, can provide insights into the origin of heavy-flavor quark
collectivity in small colliding systems. However, the identification of the
source of the observed collective-like effects remains a topic of debate.

##  6 Summary

This article reviews the most recent experimental measurements of correlated
heavy-flavor particle production as a function of the relative azimuthal angle
and/or rapidity. Angular correlation techniques are used to study the
production, fragmentation, and hadronization of heavy quarks in pp collisions
and to understand how these processes are modified in the presence of a
strongly interacting quark–gluon plasma in heavy-ion collisions. Azimuthal
anisotropy observations in high-multiplicity pp and p–A collisions and their
possible origin are also reviewed.

In pp collisions, angular correlations of heavy-flavor particle pairs allow
for testing pQCD calculations at different order in $\alpha_{\rm s}$. Such
correlation distributions were measured at RHIC and at the LHC through
different experiments, and were compared with predictions from different Monte
Carlo generators, such as PYTHIA, POWHEG, HERWIG, MADGRAPH, and SHERPA. It is
challenging for these models to provide an optimal description of the data.
These kinematic correlation observables provide sensitivity to the underlying
differences within the models. The shape of the correlation distribution is
also used to study the multiple production of heavy quarks in a single pp and
p–A collision. The angular separation between heavy-flavor particle pairs
shows distinct structural differences when produced in single or double parton
scattering processes.

A thorough characterization of the in-vacuum heavy quark fragmentation process
can be performed by measuring the angular correlation distribution between a
trigger heavy-flavor particle and associated charged particles, as performed
by the ALICE Collaboration. The angular correlation distribution and more
quantitative observables extracted from it, such as the near- and away-side
peak yields and widths measured in pp and p–Pb collisions, were compared with
different Monte Carlo generators such as PYTHIA, POWHEG+PYTHIA, HERWIG, and
EPOS. The first two models provide predictions that are closest to the data,
but further configuration and parameter tuning could be helpful for a better
reproduction of the properties of the correlation distribution. By comparing
the correlation distributions in pp and p–Pb collisions, effects from cold
nuclear matter on the heavy quark fragmentation process can be studied. No
significant impact was observed in the kinematic ranges measured in the
experiments. It is also important that the fragmentation studies can be
performed as a function of event multiplicity, as correlation techniques used
to evaluate flow coefficients in small systems rely on the assumption that jet
fragmentation is independent of event multiplicity. Results of correlation
measurements as a function of charged-particle multiplicity in pp and p–Pb
collisions indicate that, within the experimental uncertainties, the
correlation distributions and their properties are consistent for all
multiplicity ranges studied, implying a similar fragmentation of charm quarks
into final-state mesons.

In ultra-relativistic heavy-ion collisions, heavy-flavor jets and particle
distributions within jets are excellent tools for characterizing heavy quark
propagation and to constrain energy loss mechanisms affecting partons
traversing the quark–gluon plasma. The STAR and PHENIX experiments at RHIC
examined the angular correlations between a trigger heavy-flavor particle and
associated charged particles in Au–Au collisions and compared the results with
those obtained in pp collisions and with predictions from the PYTHIA event
generator. The trigger-particle $p_{\rm{T}}$-integrated near-side yields and
widths were observed to increase towards more central Au–Au collisions. The
measurement indicates that charm quarks lose energy in the QGP, and the lost
energy is converted into additional low-$p_{\rm{T}}$ particles accompanying
the charm meson. On the away side, a higher yield of
low-$p_{\rm{T}}$-associated particles was observed in Au–Au collision compared
to pp, which decreased and hinted towards a suppression for increasing the
associated particle $p_{\rm{T}}$. The charm quark diffusion inside the QGP
medium, studied at the LHC by the CMS Collaboration, implies that
low-$p_{\rm{T}}$ D mesons in Pb–Pb collisions tend to be further displaced
from the jet axis compared to pp collisions, due to interaction with the
medium constituents. Studies of the modification of jet shapes using
correlation techniques, by measuring the distribution of charged particles
inside a jet as a function of their radial distance from the jet axis, was
performed by the CMS Collaboration for jets initiated by beauty quarks
(b-jets). The measurement indicates a redistribution of the momentum of jet
constituents from smaller to larger radial distances from the jet axis in the
presence of the QGP medium. These measurements provide new constraints for
theoretical calculations of parton-flavor dependence of energy loss and
jet–medium interactions in the QGP.

To understand the long-range ridge structures and the positive $v_{2}$ values
observed for light-flavor particles in high-activity collisions of smaller
systems as pp, p–Pb, and d–Au, the experiments also searched for non-zero
$v_{2}$ signals in the heavy-flavor sector with the aim to provide insights
whether the observed collectivity originates from initial- or final-state
effects, or both. Experiments at the LHC performed several measurements to
evaluate the $v_{2}$ of charm and beauty hadrons in pp and p–Pb collisions,
using two-particle correlation techniques. A positive $v_{2}$ value for charm
hadrons was measured, showing a rising and then decreasing trend with
$p_{\rm{T}}$, and a mass ordering in the low-$p_{\rm{T}}$ region where heavier
particles have smaller $v_{2}$ at a given $p_{\rm{T}}$. These trends are
similar to those observed in Pb–Pb collisions, where they originate from
parton interactions with the medium constituents described by hydrodynamic
laws. The measurements in p–Pb collisions were compared with AMPT model
calculations that generate positive $v_{2}$ via an anisotropic parton escape
mechanism, and with CGC calculations that predict correlations in the initial
stage of the collision between partons originating from projectile protons and
gluons in the dense lead nucleus, resulting in a sizable $v_{2}$. Both models
qualitatively provide compatible results and are able to describe the data,
indicating that both initial- and final-state effects could explain the
azimuthal anisotropies observed in small systems, while leaving the question
of the exact origin of these effects still open.

The measurements presented in this review were performed by the STAR and
PHENIX Collaborations at RHIC, and by the ALICE, ATLAS, CMS, and LHCb
Collaborations at the LHC, using data collected in 2018. While the current
measurements have provided important information about various aspects of
heavy-flavor production in hadronic collisions as discussed above, the
uncertainties in several of these measurements are high, and extension to
different kinematic regions are required to achieve the goals intended with
these measurements. With the ongoing Run 3 data collection at the LHC, and the
new sPHENIX experiment at RHIC, we can expect additional correlation
measurements with improved precision, as well as access to further, more
differential observables. These ongoing and future efforts will help to
improve our knowledge of the abovementioned topics, and to shed further light
on some of the open questions in the field.

Conceptualization, F.C. and D.T.; investigation, F.C. and D.T.;
writing—original draft preparation, F.C. and D.T.; writing—review and editing,
F.C. and D.T. All authors have read and agreed to the published version of the
manuscript.

This work was supported by U.S. Department of Energy Office of Science under
contract number DE-SC0013391.

The authors declare no conflicts of interest.

References

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