###  4.1 Hadronic effects on $D$-meson observables

In HICs heavy quarks are primarily generated in hard processes that can be
studied through pQCD calculations. Consequently, their initial momentum
distribution can be theoretically evaluated and verified by experiment. The
initial HQ spectra are generally of power-law type and thus harder than what
one would expect in a thermalized medium (with some caveat due to the
collective flow of the medium). The HQ interactions with the constituents of
the bulk matter drive their spectra toward local thermal equilibrium, but
since the thermalization time is delayed by a factor of order $m_{Q}/T$ they
generally do not reach full equilibrium (which is precisely why they are an
excellent gauge of the interaction strength with the medium). The standard
approaches to study the time evolution of the HQ spectra in the QGP are the
relativistic Langevin equation [133, 134, 135, 77, 12] and the Boltzmann
transport equation [136, 137, 138, 30]. The input to the Langevin approach are
the drag and diffusion coefficients discussed in the preceding sections, while
for the Boltzmann equation HQ cross sections with light quarks and gluons
serve as the inputs. Accurately simulating the dynamics of heavy quarks at the
collision energies of RHIC and LHC also relies on providing a realistic space-
time evolution of the expanding medium. Hydrodynamic or transport simulations
are usually employed to simulate the evolving QGP phase, which can describe
the measured collective properties of the system. Once the temperature of the
QGP phase reaches the one of the quark-hadron cross-over transition, the heavy
quarks will hadronize and continue to interact with the surrounding hadronic
phase consisting of pions, kaons, $\eta$ mesons and nucleons, as well as their
excited states. The QGP evolution and hadronization of HF particles can thus
be viewed as an initial condition to their hadronic diffusion.

In Sec. 4.1.1 we will first review basic features of the time evolution of the
HQ spectra and their elliptic flow that are pertinent to the subsequent
hadronic evolution of $D$ mesons. We then analyze the results of various model
implementations, specifically the Langevin calculations of the Texas A&M
University group (TAMU) in Sec. 4.1.2, the parton-hadron string dynamics
calculations (PHSD) in Sec. 4.1.3, the results of the Nantes group who combine
Boltzmann transport in the QGP with Langevin simulations in the hadronic phase
in Sec. 4.1.4, and the Duke University model which utilizes Langevin transport
in the QGP and Boltzmann transport within the UrQMD model for the hadronic
phase in Sec. 4.1.5. In Sec. 4.1.6 we briefly discuss rather recent
developments utilizing hadronic 2-body correlation functions to assess final-
state interactions, specifically for the case of $D\pi$ and $DK$ channels.

####  4.1.1 Evolution of HQ spectra and elliptic flow in the QGP

Figure 29: The time evolution of the nuclear modification factor (left panel)
and the elliptic flow (right panel) of charm quarks at RHIC, as computed
within the TAMU transport approach. Figures adapted from Ref. [139].

To characterize HQ transport in the QGP phase, a quantitative assessment of
heavy-hadron transport in the hadronic phase is required. To understand the
possible impact of the hadronic phase on various experimental observables, the
nuclear modification factor, $R_{\mathrm{AA}}$, and elliptic flow, $v_{2}$, it
is instructive to understand the generic time evolution of both observables.
This is shown in Fig. 29 for the resonance+pQCD model in the fireball
evolution [12], taken from the review in Ref. [139]. One finds that the
$R_{\mathrm{AA}}$ of charm quarks is predominantly built up during the early
stages of medium expansion, characterized by relatively high temperatures and
energy densities, and tends to saturated within $t$=4-5 fm/$c$. This indicates
that the $R_{\mathrm{AA}}$ may not be very sensitive to further changes in the
hadronic phase following the QGP phase. However, the $v_{2}$ develops in the
later stages of the evolution. Early on, the bulk medium does not carry
significant $v_{2}$, as it requires time to develop it, with a time typical
scale comparable to the medium size characterizing the initial spatial
anisotropy. Subsequently, the bulk-matter $v_{2}$ in momentum space will be
transferred to the heavy quarks due to its interaction with the thermal bath.
This suggests that the time evolution of the HF $v_{2}$ will still be
significant in the hadronic phase, even if the transport coefficients are not
very large since the surrounding bulk-matter $v_{2}$ is substantial. Hence,
the time evolution of the HF $R_{\mathrm{AA}}$ and $v_{2}$ is responsive to
rather different stages of the fireball expansion, contrary to the strong
correlation implied by the experimental results. This will have important
consequences for the interpretation of the experimental data to disentangle
different energy loss models [11, 140] and to understand the impact of the
hadronic phase on $R_{\mathrm{AA}}$ and $v_{2}$.

Figure 30: Nuclear modification factor (left panel) and elliptic flow (right
panel) of charm quarks at LHC, as computed within the Catania transport
approach using pQCD*5 interaction; red and green lines correspond to the
results at $T_{pc}=155$ MeV and $T_{pc}=170$ MeV, respectively. Figures
adapted from Ref. [1].

To further motivate the possible impact of hadronic phase, Fig. 30 depicts the
impact of varying the pseudo-critical temperature, $T_{\rm pc}$ on the final
spectra and $v_{2}$ of charm quarks as a function of $p_{\rm T}$. This study
utilizes the pQCD*5 interaction [1] within the Catania Langevin transport
framework [141], examining two settings with varying critical temperatures:
$T_{\rm pc}$=155 MeV (red lines) and $T_{\rm pc}$=170 MeV (green lines). As
shown in Fig. 30, the $R_{\mathrm{AA}}$ remains relatively unchanged, while
the $v_{2}$ experiences a noticeable increase of up to 20$\%$ when the
evolution is extended to lower temperatures. This finding aligns with prior
studies [142, 140], which identify the suppression seen in the
$R_{\mathrm{AA}}$ as primarily driven by density, with its greatest impact
observed during the initial stages of the fireball. Conversely, the transfer
of $v_{2}$ from the medium to heavy quarks demonstrates its highest
effectiveness when the fireball’s $v_{2}$ is large, typically occurring in the
later phases of evolution closer to $T_{\rm pc}$. This effect becomes more
pronounced when the coupling strength of the medium reaches a maximum near
$T_{\rm pc}$. The pQCD*5 interaction model employed in this study does not
feature an enhanced strength near $T_{\rm pc}$. Consequently, the anticipated
increase in HQ $v_{2}$ near $T_{\rm pc}$ is expected to be even more
noticeable in situations involving nonperturbative interactions [143]. Again,
this is suggestive for a further development of the HF $v_{2}$ in the hadronic
phase, which in the following four section will be investigated quantitatively
in various model implementations.

A similar study has been carried out in Ref. [144] at FAIR/SPS energies, in
Au-Au collisions with a beam energy of 25 GeV (corresponding to $\sqrt{s_{\rm
NN}}\simeq 7$ GeV). A coarse-graining procedure of the UrQMD transport model
has been carried out to obtain local temperatures and baryochemical potentials
(alternatively, ideal hydrodynamic simulations initialized by UrQMD have also
been considered). These results have then been employed as a background medium
for Langevin simulations of $c$-quarks in a QGP (using the nonperturbative
resonance model of Ref. [77]) and $D$ mesons in hadronic matter (using the
coefficients from the EFT calculations of Ref. [106]), both including the
effects of finite baryo-chemical potentials, and connected with a
coalescence/fragmentation model for hadronization. It was found that the
$D$-meson $v_{2}$ right after hadronization varies by up to a factor of
$\sim$2-3 at $p_{\rm T}\simeq 1.5$ GeV when varying the hadronization
temperature from $T$=160 MeV down to 145 or 130 MeV. On the other hand, the
$R_{\mathrm{AA}}$ was much less affected, although it showed an overall
interesting effect of becoming very large at toward the kinematic limit in pp
collisions at this collision energy, $p_{\rm T}^{\rm pp,max}\simeq 1.6-2$ GeV.

####  4.1.2 TAMU model

Figure 31: Nuclear modification factor (left panel) and elliptic flow (right
panel) of charm quarks (green dashed line), $D$-mesons (purple dashed-dotted
line) and $D_{s}$-mesons (red line) at RHIC, as computed within the TAMU
transport approach; the blue dashed double-dotted line represents the
equilibrium limit for $D_{s}$ mesons in the hydrodynamic medium. The red
uncertainty band in the nuclear modification factor is due to the inclusion or
omission of a Cronin effect in the initial charm spectra. The red uncertainty
band in the elliptic flow is due to the inclusion or neglect of diffusion
effects in the hadronization window. Figures adapted from Ref. [145].

One of the first attempts to characterize the impact of the hadronic phase was
reported in Ref. [145] (see also Ref. [146] for an early more schematic
study). In accordance with the widely accepted and empirically supported idea,
the authors argued that multi-strange hadrons decouple not only chemically but
also kinetically [147, 148] in the proximity of the hadronization transition;
the main argument for this is that these hadrons largely lack the resonant
cross sections (like $\pi N\to\Delta\to\pi N$ or $\pi K\to K^{*}\to\pi K$)
that are abundantly operative in the light sector; a similar decoupling is
then expected for $D_{s}$ meson. Therefore, by calculating the impact of the
hadronic diffusion on the $D$ meson, a quantitative assessment of the hadronic
transport coefficient can be achieved through a comparison of observables with
the $D_{s}$. In this study, the dynamics of charm quarks was studied at RHIC
energy using relativistic Langevin simulations. During the QGP phase, they
utilized nonperturbative $T$-matrix calculations [143, 149] for the HQ
transport coefficients, with input potentials approximated by HQ internal
energies computed in thermal LQCD. The resulting $T$-matrices reveal resonant
states near the threshold in mesonic ($D$ and $D^{*}$) and color-antitriplet
diquark channels close to the pseudo-critical temperature. These resonances
substantially augment the interaction strength, by a factor of around $\sim$5
compared to the pQCD Born amplitudes. The space-time evolution of the medium
was approximated through boost-invariant ideal hydrodynamics. After undergoing
diffusion through the QGP, the charm-quark distributions are converted into
charmed hadrons by utilizing the resonance recombination model (RRM) [150],
wherein thermal light and strange quarks combine into $D$ and $D_{s}$ mesons
on the hydrodynamic hypersurface at a critical temperature of $T_{\rm pc}$=170
MeV) [151]. The strength of the interaction of the $D$ meson with light mesons
and baryons has been assessed using existing microscopic models for $D$-hadron
scattering [56]. These models are constrained by considerations of chiral
symmetry and vacuum spectroscopy, as described in Sec. 2 for $D_{s}$ mesons,
and the pertinent transport coefficients are displayed in Fig. 17.

Fig. 31 (left panel) displays the variation of $R_{\mathrm{AA}}$ of the $D$
and $D_{s}$ mesons in semicentral Au+Au collisions at RHIC; the results for
the $D$-meson $R_{\mathrm{AA}}$ are consistent with STAR data in the 0-80%
centrality class [152]. A significant enhancement in the $D_{s}$-meson
$R_{\mathrm{AA}}$ was found, which, however, is associated with a chemistry
effect in the hadronization process due to charm-quark recombination with the
enhanced strangeness abundance in Au+Au collisions, relative to pp. To provide
further illustration of this phenomenon, the results are compared to the
$R_{\mathrm{AA}}$ for charm quarks at $T_{\rm pc}$, representing the spectra
of $D$ and $D_{s}$ mesons if coalescence were not present and only
$\delta$-function fragmentation was applied. The impact of coalescence is
evident, notably diminishing beyond $p_{\rm T}$ values of approximately 5 GeV
where fragmentation takes precedence, leading to the convergence of the
$R_{\mathrm{AA}}$ for $D$, $D_{s}$, and charm quarks. It was furthermore found
that hadronic diffusion has a minimal impact on the $R_{\mathrm{AA}}$ of $D$
mesons (and was absent by construction for the $D_{s}$), attributed to a
compensatory effect arising from the simultaneous decrease in temperature and
increase in the flow of the medium. Fig. 31 (right panel) displays the $p_{\rm
T}$ dependence of the $v_{2}$ of the $D$ and $D_{s}$ meson in semicentral
Au+Au collisions at RHIC. While the charm-quark diffusion in the QGP imparts a
sizable $v_{2}$ reaching up to approximately 4.5%, coalescence with thermal
light and strange quarks enhances this value by about 50 $\%$ for both $D$ and
$D_{s}$ mesons. However, even as the $D_{s}$ spectra undergo freeze-out after
hadronization, the coupling of $D$ meson to the hadronic medium persists [56].
This coupling enables the $D$-meson to pick up more of the elliptic flow of
the bulk medium generated during the QGP phase, leading to an additional
increase in $v_{2}$ by approximately 30 $\%$ due to its interaction in the
hadronic phase (dash-dotted line compared to the upper end of the red band for
$D_{s}$ mesons in Fig. 31). These findings support the expectations based on
Fig. 29, demonstrating that $D$-meson $R_{\mathrm{AA}}$ remains largely
unaffected by interactions in the hadronic phase, while, in contrast, the
impact on $v_{2}$ is notable, being sensitive to the later stages of the
evolution with a significant enhancement generated by interactions in the
hadronic phase. More quantitatively, the $v_{2}$ splitting between $D$ and
$D_{s}$, inspired by the early freeze-out of multistrange hadrons in the
underlying hydro-evolution, has been suggested as a measure of the $v_{2}$
generated in the hadronic evolution, remaining essentially independent of the
modeling of QGP diffusion and hadronization. Even in cases where the $D_{s}$
undergoes re-interactions in the hadronic phase, the $v_{2}$ splitting between
$D$ and $D_{s}$ continues to serve as a lower limit for the transport
coefficient in the hadronic medium.

Figure 32: Nuclear modification factor (left panel) and elliptic flow (right
panel) of charm quarks (blue line) and $D$-mesons (red and green lines) at
LHC, as computed within the TAMU transport approach; the red and green lines
correspond to the $D$-meson results with and without the contribution of
hadronic diffusion, respectively. In the nuclear modification factor, the
bands indicate uncertainties due to the shadowing of charm production. In the
elliptic flow, the bands indicate uncertainties due to the total charm-quark
coalescence probability. Figures adapted from Ref. [153].

A similar study has been conducted at LHC energies within the same model. In
Fig. 32 (left panel), the $R_{\mathrm{AA}}$ as a function of $p_{\rm T}$ is
shown for the charm quarks, $D$ mesons just after hadronization and $D$ mesons
after the kinetic freeze-out in central Pb+Pb collisions at $\sqrt{s_{\rm
NN}}$=2.76 TeV. The HQ diffusion in the QGP is calculated by via Langevin
dynamics with transport coefficients from a thermodynamic $T$-matrix approach
[149], and hadronization is realized in the RRM, and the hadronic diffusion
utilizes the hadronic coefficients from Fig. 17 [56]. Again, a relatively
minor impact on the $R_{\mathrm{AA}}$, the larger radial flow at the LHC leads
to a slight enhancement and shift to higher $p_{\rm T}$ of the maximum
structure (“flow bump”) at low $p_{\rm T}$. From the right panel of Fig. 32,
which shows the $p_{\rm T}$ dependence of the elliptic flow coefficient, one
finds that the $D$-meson $v_{2}$ just after hadronization is enhanced due to
the hadronic diffusion until kinetic freeze-out by a maximum of up to
$\sim$25%. The observed increase in elliptic flow due to diffusion in the
hadronic phase appears to play an important factor in the simultaneous
description of both observables, the $R_{\mathrm{AA}}$ and $v_{2}$.

####  4.1.3 PHSD model

Figure 33:  Nuclear modification factor of $D$-mesons at RHIC, as computed
within the PHSD transport approach; the red solid line and blue dashed line
corresponds to results with contribution of hadronic diffusion. The red dot-
dashed line and blue dotted line represents the results without contribution
of hadronic diffusion. Figures adapted from Ref. [17].

The production of charm quarks and hadrons in ultra-relativistic heavy-ion
collisions has been investigated through the Parton-Hadron-String Dynamics
(PHSD) transport approach [17]. The PHSD model [154, 155] is based on a
microscopic off-shell transport methodology designed for elucidating the
behavior of strongly interacting hadronic and partonic matter, both in and
out-of equilibrium. It relies on solving the Kadanoff–Baym equations within a
first-order gradient expansion in phase space. This framework enables the
depiction of the time evolution of nonperturbative interacting systems.

The transport of HF particles in HICs considers the evolution in both the
partonic and hadronic phases. Charm quarks generated in primordial NN
collisions undergo scattering in the QGP with off-shell partons whose masses
and widths are determined by the Dynamical Quasi-Particle Model (DQPM) [156,
157] via fits to reproduce the LQCD equation-of-state in thermal equilibrium.
The interactions of charm quarks with the partons in the QGP have been
obtained using DQPM propagators and couplings [158]. In PHSD, charm quarks
undergo hadronization into $D$ mesons through coalescence and fragmentation
processes near the critical energy density of the phase transition.
Subsequently, the hadronized $D$ mesons interact with various hadrons in the
hadronic phase, with cross sections calculated using a unitarized approach
effective theory approach that incorporates heavy-quark spin symmetry [58,
66], cf. Sec. 2.2.2.1.

In Fig. 33, the $D$-meson $R_{\mathrm{AA}}$ is displayed as a function of
$p_{\rm T}$ with and without hadronic scattering in AU-Au collisions at
different centrality classes for the highest RHIC energy obtained within PHSD
transport approach. The results for charm-quark hadronization through
coalescence are presented for two different radii. It is found that hadronic
rescattering plays a significant role in the $R_{\mathrm{AA}}$ at low $p_{\rm
T}$, by enhancing the flow bump by up to 20% and shifting it to higher $p_{\rm
T}$ by up to $\sim$ 1 GeV. The medium effects and thus the overall impact of
hadronic scattering is milder in peripheral compared to central collisions.

Figure 34: The elliptic flow of of $D$-mesons at RHIC, as computed within the
PHSD transport approach; the red solid line and blue dashed line corresponds
to results with contribution of hadronic diffusion. The red dot-dashed line
and blue dotted line represents the results without contribution of hadronic
diffusion. Figures adapted from Ref. [17].

In Fig. 34, the $D$-meson $v_{2}$ is displayed as a function of $p_{\rm T}$
with and without hadronic scattering for 0-80% Au-Au collisions at the highest
RHIC energy, as obtained from PHSD. The hadronic rescattering has a very
sizable effect, producing an enhancement of up to a maximum of near 40$\%$.

####  4.1.4 Nantes model

Figure 35: Nuclear modification factor (left panel) and elliptic flow (right
panel) of $D$-mesons (top) and single non-photonic electrons (bottom)
originating from the decays of HF mesons at RHIC, as computed within Nantes
transport approach; green dot-dashed and blue dotted lines corresponds to
results with contribution of hadronic diffusion. The red solid line represents
the results without contribution of hadronic diffusion. Figures adapted from
Ref. [75].

The effects of hadronic phase rescattering on the propagation of $D$ mesons
within the hadronic medium, along with its implications for heavy-flavor
observables, have further been studied by the Nantes group [75]. In their
investigation, the authors employ Monte Carlo propagation of heavy quarks,
denoted as MC@sHQ, within a 2+1D expanding ideal hydro medium [136, 159]. The
plasma evolution starts from smooth initial conditions and an equation of
state featuring a first-order phase transition. The interaction between heavy
quarks and plasma partons occurs through elastic and radiative collisions. The
transport of heavy quarks is carried out with the Boltzmann equation, solved
using the test particle method, employing Monte Carlo techniques. The
hadronization of heavy quarks is performed when the energy density of a fluid
cell falls below a critical value, taken as $\epsilon_{c}$ = 0.45 GeV/fm3.
This process involves a combination of coalescence and fragmentation
mechanisms [160]. The propagation of $D$ mesons in the hadronic medium is
treated using the Fokker-Planck (FP) equation [133] using elastic scattering
of $D$ mesons. The cross sections, needed to compute the transport
coefficients, for $D\pi\rightarrow D\pi$, $D\eta\rightarrow D\eta$, and
$DK(\bar{K})\rightarrow DK(\bar{K})$, as well as for $DN(\bar{N})\rightarrow
DN(\bar{N})$ and $D\Delta(\bar{\Delta})\rightarrow D\Delta(\bar{\Delta})$, are
taken from Ref. [66], recall Sec. 2.2.2.1. In addition, elastic scattering
processes involving $D$ mesons with higher excited mesons $m$ (not listed
above), $Dm\rightarrow Dm$, are simulated with a constant cross section of
$\sigma=10~{}{\rm mb}$, motivated by the work of He et al. [56]. The authors
then conducted calculations in two different scenarios for the hadronic
diffusion [75]. In scenario-I, they implemented the drag and diffusion
coefficients from their calculations without imposing the Einstein relation.
Consequently, the asymptotic solution of the FP equation does in general not
agree with the equilibrium distribution function. In scenario-II, they
utilized the drag coefficient obtained from their calculations and imposed the
Einstein relation for the momentum- diffusion coefficients. The Einstein
relation ensures that the asymptotic distribution function conforms to the
Boltzmann distribution function. In Fig. 35, the variation of the final
$D$-meson $R_{\mathrm{AA}}$ and $v_{2}$ for $D$ mesons is illustrated as a
function of $p_{\rm T}$, along with the corresponding quantities for single
non-photonic electrons (which also include bottom decay contributions), and
compared with experimental data from the PHENIX and STAR collaboration in AU-
Au collisions at maximum RHIC energy. Based on these results it has been
concluded that the rescattering of $D$ mesons in the hadronic gas has a
negligible impact on the $R_{\mathrm{AA}}$ but leads to an increase of up to
$\sim$20% of the elliptic flow for both $D$ mesons and their decay electrons.

####  4.1.5 Duke model

Figure 36: Nuclear modification factor (left flow) and elliptic flow (right
panel) of $D$ mesons at LHC, as computed within the Duke transport approach;
blue dashed and red lines correspond to the nuclear modification factor with
and without the contribution of hadronic diffusion, respectively. The green
dashed line represents the elliptic flow without the contribution of hadronic
diffusion, while the red and blue lines represent the elliptic flow with the
contribution of hadronic diffusion under two different initial conditions. The
error bands characterize the uncertainties introduced in the choice of the
cutoff parameter in the hadron form factors when calculating the heavy meson
scattering cross sections. Figures adapted from Ref. [161].

In Ref. [161] the influence of hadronic interactions on the $R_{\mathrm{AA}}$
and $v_{2}$ of $D$ mesons is addressed at both RHIC and LHC energies. Here a
modified Langevin equation [162] is employed to study the momentum evolution
of the heavy quarks as they traverse the QGP medium: in addition to the drag
and diffusion terms, an extra term is incorporated to account for the recoil
force acting on heavy quarks when they undergo medium-induced gluon radiation,
with a lower gluon energy cutoff of $\pi T$ to mitigate the violation of
detailed balance [162, 161]. In order to attain the best description of HF
data from RHIC and the LHC, a spatial diffusion coefficient of ${\cal
D}_{s}(2\pi T)=6$ is employed. For the radiated gluons, a distribution
function from higher-twist calculations is adopted [163]. The modified
Langevin setup is implemented into an expanding QGP medium, which is simulated
using a (2+1)D viscous hydrodynamic model [164]. Close to the critical
temperature, both the bulk matter of the QGP and heavy quarks undergo
hadronization, forming color-neutral bound states. To characterize the
hadronization process of the bulk matter, they utilized the numerical tool ISS
[165], which is based on the Cooper-Frye formula [166]. This tool is employed
to obtain soft hadrons from the hydrodynamic medium. For heavy quarks, they
employed a hybrid model combining fragmentation and coalescence to describe
their hadronization.

After hadronization, the $D$ mesons undergo rescattering with the hadrons of
the bulk medium, which is modeled through the UrQMD transport model [167,
168]. The latter utilizes microscopic cross-sections of hadronic scatterings
are the basic inputs, taken from data as available and assuming constituent
quark-scaling otherwise. The rescattering of $D$ mesons within a hadron gas is
incorporated to the UrQMD framework using scattering cross sections with $\pi$
and $\rho$ mesons taken from Refs. [76, 169] which are based on a hadronic
Lagrangian generated from local flavor SU(4) gauge symmetry. In this
calculation, there is appreciable uncertainty related to the choice of the
cutoff parameter in the hadron form factors, which is varied to obtain an
estimate of the uncertainty in the subsequent calculations of heavy-meson
observables. In Fig. 36, the $D$-meson $R_{\mathrm{AA}}$ (left panel) is
depicted as a function of $p_{\rm T}$ in Pb-Pb collisions at LHC energy,
$\sqrt{s_{\rm NN}}=2.76$ TeV. It is observed that the additional interaction
experienced by $D$ mesons within the hadron gas leads to a mild additional
suppression of their $R_{\mathrm{AA}}$ at large transverse momentum, while at
the low momentum, the impact of hadronic rescattering is essentially
unnoticeable. The impact of hadronic interactions on the elliptic flow of $D$
mesons at the LHC is illustrated in the right panel of Fig. 36. Owing to
additional scatterings of $D$ mesons in the hadron phase, their $v_{2}$ is
further enhanced by up to approximately 20$\%$, most notably in the peak
region around $p_{\rm T}\simeq 3$ GeV. The authors also showcase the effect of
different initial conditions for staring the hydrodynamic simulations. The KLN
model yields larger eccentricity in the initial entropy density profiles
compared to the Glauber model, entailing an additional 20$\%$ difference in
the $v_{2}$ of heavy mesons following their evolution through the QGP,
hadronization and the hadron gas.

Figure 37: Same as Fig. 36, but for collisions at the RHIC. Figures adapted
from Ref. [161].

Fig. 37 illustrates the variation of the $D$-meson $R_{\mathrm{AA}}$ (upper
panel) and $v_{2}$ at the top RHIC energy of 0.2 GeV. Similar to case at the
LHC, the inclusion of hadronic interactions using the UrQMD model leads to a
slight suppression of $D$ meson $R_{\mathrm{AA}}$ at large $p_{\rm T}$ and
significantly enhanced elliptic flow of up to almost $\sim$40%, not unlike the
results in the PHSD model.

In summary of the sections on hadronic $D$-meson diffusion in HICs, Secs.
4.1.2-4.1.3, it seems fair to say that all of the phenomenological studies of
the $D$-meson $R_{\mathrm{AA}}$ and $v_{2}$ that have considered rescattering
in the hadronic medium (in addition to the QGP and hadronization effects)
indicate that its impact on the nuclear modification factor is small. This is
mainly due to the fact that the softening (thermalization) of the additional
scattering in the hadronic phase is largely compensated by the additional
radial flow as they are dragged along with the collective expansion. However,
at the same time all calculations observe a very significant impact on the
elliptic flow, that typically amounts to an increase of up to 30-40% at RHIC
energies and around 20$\%$ at LHC energies, most notably in the region of
$p_{\rm T}\simeq 3$ GeV, where the effect of thermalization can be expected to
be largest. This is in large part due to the fact that the elliptic flow of
the expanding medium is already near its maximum once the hadronic phase
commences.

####  4.1.6 Femtoscopy studies

Figure 38: Femtoscopic correlation functions of $D^{+}\pi^{+}$ \+ c.c. (top,
left), $D^{+}\pi^{-}$ \+ c.c. (top, right), $D^{+}K^{+}$ \+ c.c. (bottom,
left) and $D^{+}K^{-}$ \+ c.c. (bottom, right), measured by the ALICE
collaboration in high-multiplicity pp($\sqrt{s}=13$ TeV) collisions.
Predictions of several theoretical models using low-energy scattering data are
also displayed (B.L.Huang et al. [170], L.Liu et al. [67], Z.H.Guo et al.
[70], X.Y.Guo et al. [171] and J.M.Torres-Rincon et al. [172]). Figures taken
from Ref. [131].

Two-particle correlation measurements in nuclear collisions have a long
history. Originally, the idea was to utilize them as a way to characterize
spatial properties of the particle emission source [173, 174, 175], by using
the most abundant, weakly-interacting particles, i.e., pions. In general, at
small relative momentum, however, they can carry information about the nature
of the final-state interactions, such as scattering lengths [176, 177, 178,
179], which has been utilized to study a diverse set of two-hadron systems.

In Ref. [119], the ALICE collaboration has presented results for correlations
functions of $D^{(*)+}\pi^{+},D^{(*)+}\pi^{-},D^{(*)+}K^{+}$ and
$D^{(*)+}K^{-}$ in high-multiplicity pp collisions at $\sqrt{s}=13$ TeV,
together with several theoretical predictions including several of the EFT
calculations discussed in Sec. 2. In most of the models, only the information
at low energy (scattering length and scattering range) is accounted for, along
with the long-range Coulomb interaction via the Koonin-Pratt formula [173,
178, 175]. As can be seen in Fig. 38—where we reproduce the results from Ref.
[119]—the theoretical approaches show a common trend and a reasonable
agreement with ALICE data (within the rather large errors bars) in most the
channels [119, 172]. The most striking discrepancy between data and theory
occurs in the $D^{+}\pi^{-}$ channel, which seems to favor an absence of
strong interaction. In terms of low-energy scattering parameters, the theory
calculations predict a scattering length in the $I=1/2$ and $I=3/2$ channels
of around $a_{0}^{D\pi}(I=1/2)\simeq 0.5$ fm and
$a_{0}^{D\pi}(I=3/2)\simeq-0.1$ fm, while the experimental data point towards
values compatible with zero (or even a small positive value for
$a_{0}^{D\pi}(I=3/2)$)—in clear contradiction with all EFT predictions. This
difference is yet to be understood. A caveat in this puzzle is the possible
existence of broad states decaying into the two-particle final state, as
manifest in the calculation of [172], which generates the $D_{0}^{*}(2300)$
state, decaying into $D^{+}\pi^{-}$. In the top, right panel of Fig. 38 the
effect of this resonance can be traced to a shallow minima below $C(k^{*})=1$
around $k^{*}=250$ MeV in the theoretical prediction of [172] (light blue line
in the figure). Should this resonance have any visible effect on the $D\pi$
final state, the source function would need to be revisited to account for
this extra emission contribution. In the meson-baryon sector we should mention
that preliminary femtoscopy results for $DN$ interactions are reported in Ref.
[118] by the ALICE collaboration also in high-multiplicity pp collisions at
$\sqrt{s}=13$ TeV.

###  4.2 $B$ mesons and heavy baryons in the hadronic phase

The majority of the analysis regarding the impact of hadronic medium
scattering on heavy-hadron observables is centered around the $D$ meson, as
elaborated in the preceding section. We now turn to reviewing and analyzing
the information on available from the literature on hadronic HF diffusion in
HICs of hadrons other than the $D$ meson, specifically $B$ mesons in Sec.
4.2.1 and HF baryons ($\Lambda_{c}$ and $\Lambda_{b}$) in Sec. 4.2.2.

####  4.2.1 $B$-meson $R_{\mathrm{AA}}$ and $v_{2}$

The rescattering of $B$ mesons in the hadronic medium of HICs has been
evaluated in the Catania transport model [180] in Au-Au ($\sqrt{s_{NN}}=200$
GeV) collisions at RHIC. The dynamics governing the time evolution of heavy
quarks in the QGP and $B$ mesons in the hadronic phase were simulated using
Langevin dynamics. The drag and diffusion coefficients for heavy quarks in the
QGP are determined through calculations based on the quasi-particle model
(QPM) [140, 141]. The quasi-particle model incorporates non-perturbative
dynamics by introducing temperature-dependent quasi-particle masses for both
light quarks and gluons. The quasi-particle model can reproduces LQCD
thermodynamics [181] by appropriately fitting the strong-coupling parameter,
$g(T)$. The initial distribution of heavy quarks in momentum space is
determined based on the bottom-quark distribution in pp collisions, based on
the Fixed-Order-Next-to-Leading-Log (FONLL) framework [182, 183]. In
coordinate space, the distribution is assumed to follow the profile of binary
nucleon-nucleon collisions ($N_{\rm coll}$) derived from the Glauber model.
The bulk medium expansion to determine cooling collective flow of the bulk
matter is simulated with a 3+1D relativistic transport code. This model allows
for the description of fluid evolution with a constant $\eta/s$ in a manner
analogous to viscous-hydrodynamics simulation. In this way, one can simulate
the dynamical evolution of a fluid with a specified $\eta/s$ ratio using the
Boltzmann equation, see also Refs. [184, 185, 186]. For the hadronization
process, where bottom quarks transform into $B$-mesons, the Peterson
fragmentation function is employed. Subsequently, the Langevin dynamics is
employed to simulate the propagation of $B$-mesons within the hadronic bath
which includes $\pi$, $K$, $\bar{K}$, and $\eta$ particles. The interaction
between the $B$-mesons and the bath is evaluated using unitarized interactions
derived from effective field theories that uphold both chiral and heavy-quark
symmetries. Specifically, the interactions of $B$ mesons with $\pi$, $K$,
$\bar{K}$, and $\eta$ mesons are obtained through a unitarized coupled-channel
model that incorporates heavy-quark spin symmetry [93, 187, 109, 110, 188,
86]. We remind that in the absence of a net baryon density, the drag and
diffusion coefficients of $\bar{B}$ mesons are the same as those for $B$
mesons.

Figure 39: Results of the Catania transport model for $B$-meson observables in
in Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV. The nuclear modification
factor $R_{\mathrm{AA}}$ (left panel) and elliptic flow (right panel) are
shown considering either both QGP and hadronic phase interactions (black solid
lines) or only QGP phase interactions (black dashed lines). Figures adapted
from Ref. [180].

Figure 39 depicts the $p_{\rm T}$ dependence of the $R_{\mathrm{AA}}$ (left
panel) and $v_{2}$ (right panel) of the $B$ meson in minimum-bias Au-Au
collisions at RHIC with and without hadronic medium rescattering, as obtained
in Ref. [180]. In qualitative analogy to the $D$-meson case, the impact of
hadronic phase rescattering on the nuclear modification factor is negligible,
while the influence on the elliptic flow ($v_{2}$) is quite significant,
amounting to a maximum of approximately 20% at $p_{\rm T}\simeq m_{B}$.

####  4.2.2 Heavy-flavor baryon $R_{\mathrm{AA}}$ and $v_{2}$

Heavy-flavor baryons ($\Lambda_{c}$ and $\Lambda_{b}$) play a key role in
probing HF dynamics in QCD matter beyond the realm of heavy mesons ($D$ and
$B$). Most notably, this concerns the hadro-chemistry in the hadronization
process (which will be discussed in Sec. 4.3 below) but is also of interest
for re-interactions in the hadronic phase. On the one hand, the baryons
provide an additional mass scale which affects the kinetic diffusion
properties of HF particles, but on the other hand one may also ask in how far
the presence of two light quarks (rather than one in mesons) affects the
interactions and thus the dynamics of the hadronic re-interactions of HF
baryons.

The nuclear modification factor of the $\Lambda_{c}$ has been recently
measured in Pb-Pb collisions at the LHC [129]. The increased presence of heavy
baryons [189, 190, 111], possibly triggered by quark coalescence processes,
can impact the nuclear modification factor of the heavy mesons in terms of a
reduction in their yields. Additionally, the baryon-to-meson ratio [191],
$\Lambda_{c}$/$D$ and $\Lambda_{b}$/$B$, when measured as a function of
$p_{\rm T}$, can provide insights into in-medium hadronization, in particular
on the additional collective flow (both radial and elliptic) that is imparted
on the HF baryons through the presence of an extra light quark in the
coalescence process. Recent experimental measurements of the charm-baryon-to-
meson ratio [129, 19, 20] have provided intriguing data that may become a
valuable tool for disentangling various hadronization mechanisms [180, 192,
193, 18, 23]. Again, this renders compelling motivation, if not a mandatory
task, to quantify the impact of hadronic rescattering of $\Lambda_{c}$ and
$\Lambda_{b}$ on the pertinent experimental observables. Transport
coefficients of the heavy baryons, $\Lambda_{c}$ and $\Lambda_{b}$, in the
hadronic medium have been reported in Refs. [111, 33] (see Sec. 3.4).

Let us now turn to the available phenomenological studies. In Ref. [180] the
influence of rescattering within the hadronic medium on the $R_{\mathrm{AA}}$
and $v_{2}$ of the $\Lambda_{c}$ has been investigated. The dynamics governing
the time evolution of heavy quarks in the QGP and heavy baryons in the
hadronic phase were modeled using Langevin dynamics. Hadronization was
performed using Peterson fragmentation functions [194] whose functional form
given by

$f(z)\propto\frac{1}{z\left[1-\frac{1}{z}-\frac{\epsilon_{c}}{1-z}\right]^{2}}\
,$ (22)

where $\epsilon_{c}$ serves as a free parameter used to calibrate the shape of
the fragmentation function for individual charm hadrons to be consistent with
experimental data obtained in pp collisions. Specifically, in the absence of
pp collision data, the shape of the $\Lambda_{c}$ fragmentation was determined
by utilizing electron-positron annihilation data. In electron-positron
annihilation, the value $\epsilon_{c}$=0.2 for the $\Lambda_{c}$ baryon turns
out to be approximately twice as large as that for the $D$ meson. This is in
line with the common expectation that the fragmentation function for heavier
hadrons is softer compared to that of lighter one, or, from a different angle,
requires the production of an extra quark necessitating an additional
splitting.

For the $\Lambda_{b}$, an even smaller value of $\epsilon_{c}$=0.006 has been
used. This approach has been applied to study $\Lambda_{c}$ and $\Lambda_{b}$
production at RHIC and LHC energies Using Langevin simulations for both QGP
and hadronic matter. The drag and diffusion coefficients for heavy quarks in
the QGP are determined through calculations based on the quasi-particle model
(QPM) [140, 141] and for $\Lambda_{b,c}$ baryons in hadronic matter within a
unitarized meson-baryon coupled-channel model that incorporates heavy-quark
spin symmetry [93, 187, 109, 110, 188, 86] for interactions with $\pi$, $K$,
$\bar{K}$, and $\eta$ particles, as discussed in the preceding section on
$B$-meson observables (which also contains the information on the initial
conditions and the Boltzmann background medium).

Figure 40: Nuclear modification factor (left panel) and elliptic flow (right
panel) of $\Lambda_{c}$ baryons in minimum-bias Au+Au collisions at RHIC, as
computed within the Catania transport approach; solid and dashed lines
correspond to the results with and without the contribution of hadronic
diffusion, respectively. Figures adapted from Ref. [180].

Fig. 40 depicts the $R_{\mathrm{AA}}$ (left panel) and $v_{2}$ (right panel)
for direct $\Lambda_{c}$ production (i.e., without feed-down contributions) as
a functions of $p_{\rm T}$ following from $c$-quark diffusion in the QGP plus
fragmentation, and for additional hadronic phase diffusion at top RHIC energy.
The influence of the hadronic phase on the $R_{\mathrm{AA}}$ is almost
imperceptible, while the $v_{2}$ is enhanced due to the presence of the
hadronic phase, with an increase of up to approximately 20%.

Figure 41: Same as Fig. 40, but for 0-20% central Pb-Pb (5.5 TeV) collisions
at the LHC. Figures adapted from Ref. [180].

Within the same setup, the results for $\Lambda_{c}$ in central Pb-Pb
collisions at the LHC are shown in Fig. 41. Again, the impact of the hadronic
phase on the $R_{\mathrm{AA}}$ is minimal, while $v_{2}$ exhibits a
noticeable. The absolute magnitude of the effects is slightly diminished
compared to the results at RHIC energy, although it is somewhat surprising
that the absolute values of the $v_{2}$ (with and without hadronic diffusion)
are smaller than at RHIC. However, the trend that the hadronic phase is more
important at RHIC than at the LHC persists.

Figure 42: Nuclear modification factor (left panel) and elliptic flow (right
panel) of $\Lambda_{b}$ baryons in central Au+Au collisions at RHIC, as
computed within the Catania transport approach; solid and dashed lines
correspond to the results with and without the contribution of hadronic
diffusion. Figures adapted from Ref. [180] )

The Catania group has conducted the same type of study, within the same setup
as for the $\Lambda_{c}$, for the $\Lambda_{b}$, with appropriately updated
transport coefficients for QGP and hadronic phase using the same framework for
the interactions, as well as accordingly adapted fragmentation functions. In
Figs. 42 and 43 the $R_{\mathrm{AA}}$ and $v_{2}$ results are compiled for Au-
Au at RHIC and Pb-Pb at the LHC. By and large, the same findings have been
reported: negligible effects from the hadronic phase on the $R_{\mathrm{AA}}$
(left panels) but significant ones on the $v_{2}$ (right panels), more
pronounced art RHIC, although overall quantitatively slightly smaller. There
is, however, a small hint of an enhancement of the $R_{\mathrm{AA}}$ at the
lowest $p_{\rm T}\lesssim 1$ GeV.

Figure 43: Same as Fig. 42, but for 0-20% central Pb-Pb (5.5 TeV) collisions
at the LHC. Figures adapted from Ref. [180].

###  4.3 Heavy-flavor hadro-chemistry

Figure 44: The ratio $\Lambda_{c}/D$ as a function of $p_{\rm T}$ is shown for
minimum bias Au+Au collisions at RHIC (left panel) and for 0-20% central Pb-Pb
(5.5 TeV) collisions at the LHC (right panel), as computed within the Catania
transport approach using coalescence plus fragmentation and fragmentation-
only. The dashed and solid lines correspond to the results with and without
the contribution of hadronic diffusion, respectively. Figures adapted from
Ref. [180].

The presence of a deconfined medium in HICs is expected to affect the
hadronization process of heavy quarks into open heavy-flavor hadrons, relative
to elementary reactions with small hadron multiplicities in the final state.
In the light-flavor sector, the enhancement observed in HICs for baryon-over-
meson ratios (such as $p/\pi$ or $\Lambda/K$) at intermediate $p_{\rm T}$ has
been interpreted in terms of coalescence processes [195], although their
manifestation is not always straightforwardly disentangled from the effects of
radial flow (which leads to a stronger “blue-shift” for more massive
particles); in addition, the $p_{\rm T}$-integrated ratios, which are the more
relevant quantity for the hadro-chemistry, did not exhibit significant changes
from pp to AA collisions at the same collision energy. The classic example for
a chemistry effect is the strangeness enhancement, or more precisely, the
hadro-chemical equilibration of strange particles in AA collisions relative to
an under-saturation in pp collisions. This occurs already at relatively low
collision energies, such as at the fixed-target energies of the SPS
($\sqrt{s_{\rm NN}}$=17.3 GeV). The heavy-baryon-to-meson ratios,
$\Lambda_{c}$/$D$ and $\Lambda_{b}$/$B$, are providing a fresh perspective on
these questions, and arguably a better controlled one, as it is not the total
$c\bar{c}$ number that is expected to equilibrate (since secondary pair
production or annihilation is strongly suppressed at the typical medium
temperatures), but rather the relative abundance of charm and bottom hadrons.

Recent experimental measurements, reported in Refs. [129, 19, 196], have
highlighted the heavy baryon-to-meson ratio as a valuable probe for
elucidating different hadronization mechanisms. The observed enhancement in
the $\Lambda_{c}^{+}/D^{0}$ yield ratio at midrapidity in pp and pPb
collisions at the LHC, as reported by ALICE [197, 196, 198] and CMS
experiments [21], is substantial compared to baseline value of $\sim$0.1 in
$e^{-}e^{+}$ and $e^{-}p$ collisions [199, 200, 201]. This enhancement can
reach up to a factor of 2–5 for $p_{\rm T}$ below 8 GeV. While the
$\Lambda_{c}^{+}/D^{0}$ ratio in Pb-Pb collisions, as measured by ALICE [129,
20], was observed to reach a maximum value of nearly one at $p_{\rm T}\simeq
5$ GeV, it is important to note that the integrated ratio maybe be very
similar to the one in pp and pPb collisions. These findings can be described
by theoretical calculations incorporating hadronization through both
coalescence and fragmentation mechanisms [192, 18], although based on
different micro-physics. While Ref. [192] largely relies on controlling the
production abundances through hadron wave function properties (in particular,
different hadron radii), the approach in Ref. [18] essentially invokes large
decay feed-down contributions from excited states (in the HF sector, many of
these excited states have not been discovered yet, but are expected to exist
based on the analogous states in the strange and light sectors and have been
predicted by, e.g., relativistic quark models [202]). The population of these
states has been originally put forward to explain the large
$\Lambda_{c}^{+}/D^{0}$ in pp collisions at the LHC at mid-rapidity (and is
also consistent with pPb measurements) [22]; the idea (and assumption) is that
these states can be effectively populated once the particle multiplicity is
large enough to justify a statistical treatment; once the multiplicity becomes
too small, the ratio should decrease, which is indeed observed in both charm
[203, 204] and bottom sectors [205], and in agreement with model predictions
[206] through a canonical-ensemble induced suppression (due to enforcing exact
quantum-number conservation).

In practice, the influence of the hadronic phase on the $\Lambda_{c}/D$ ratio
has been studied in Ref. [180], within the same setup (Langevin simulations in
QGP and hadronic phase) as described in the context of Figs. 40-43 of the
preceding section. However, for hadronization, a scenario with additional
recombination with light partons at $T_{\rm pc}$ has been considered, based on
the instantaneous coalescence model of Ref. [207]. It is worth mentioning that
the contribution from resonance decays impacts the heavy-baryon-to-meson
ratios, as it involves the ratio of two distinct hadron species with varying
contributions from resonance decays [190]. The influence of resonance decay on
the $R_{\mathrm{AA}}$ is usually less pronounced (especially if the chemistry
does not change significantly), as it pertains to ratios involving the same
hadrons. Fig. 44 illustrates the variation of $\Lambda_{c}/D$ as a function of
$p_{\rm T}$ at top RHIC (left panel) and LHC energy (right panel) for both the
QGP and QGP plus hadronic phase. The effects of coalescence are mostly
relevant in the low and intermediate-$p_{\rm T}$ regions, where they lead to a
significant increase in the ratio (basically since the charm baryon pick up
two light quarks from the thermal medium vs. one for the charm meson).
However, the ratio remains essentially unaffected by the rescattering in the
hadronic phase. This observation reiterates the previously discussed findings,
i.e., that the $R_{\mathrm{AA}}$ (and hence, the momentum spectra figuring in
tit numerator) are affected very little, if any, by rescattering in the
hadronic phase. This implies that the heavy-baryon-to-meson ratios are a
rather clean probe of the hadronization dynamics (recall that hadronization
into different species is based on the same underlying HQ distribution as
resulting from diffusion through the QGP).

###  4.4 Heavy-flavor diffusion: QGP vs. hadronic phase

Figure 45: Spatial diffusion coefficient as a function of temperature for both
the QGP (only from LQCD computations) and hadronic phase obtained within
different models. The $D$-meson and $B$-meson results are take from Ref. [66]
and Ref. [105], respectively. The $\Lambda_{c}$ and $\Lambda_{b}$ results are
taken from Ref. [33]. The in-medium results for $D$-mesons and $B$-mesons are
taken from Ref. [84] and Ref. [96], respectively. The LQCD results on charm,
bottom, and $m_{H}\rightarrow\infty$ are taken from Ref. [208]. The LQCD
results from Banerjee et al. are taken from Ref. [209]. The LQCD results from
Altenkort et al. are taken from Ref. [210]. The ALICE Collaboration’s model
fits to their data are taken from Ref. [13].

A primary objective of all phenomenological investigations concerning open
heavy-flavor observables is to extract the heavy-flavor spatial diffusion
coefficient, ${\cal D}_{s}$ [145]. As discussed in Appendix A, this quantity
can be defined through the drag coefficient in the zero-momentum limit,

${\cal D}_{s}=\frac{T}{A(p\rightarrow 0)m_{H}},$ (23)

where $m_{H}$ is the mass of the heavy quark in the QGP phase or the mass of
the heavy hadron in the hadronic phase. The diffusion coefficient serves to
quantify the interaction between heavy quarks and the surrounding bulk medium.
For example, in pQCD it is of the parametric form ${\cal D}_{s}\propto
1/(\alpha_{s}^{2}T)$, i.e., does not, to lowest order, depend on the mass of
the HF particle. This illustrates why the HF diffusion coefficient contains
fundamental information about the QCD medium. As a result, it is thought to be
closely connected to other transport coefficients, such as shear viscosity or
electric conductivity. To facilitate comparison and interpretation as
indicators of the interaction strength within the medium, these coefficients
are commonly scaled to dimensionless quantities. For instance, the HF
diffusion coefficient is scaled by the inverse “thermal wavelength” of the
medium, $1/(2\pi T)$, resulting in $2\pi T{\cal D}_{s}$. Similarly, the shear
viscosity is scaled by the entropy density ($s$) to obtain the dimensionless
ratio $\eta/s$, while the electric conductivity is scaled by temperature
($T$), yielding $\sigma_{\rm el}/T$. In pQCD, these quantities are all
proportional to the square of the inverse coupling constant,
$1/\alpha_{s}^{2}$.

Transport models [145, 141] have proven to be highly effective in reproducing
experimentally measured observables such as the nuclear modification factor
and elliptic flow of $D$-mesons. These models rely on inputs such as drag and
momentum diffusion coefficients, which encapsulate microscopic details
regarding the interactions of heavy quarks with the QGP medium. In contrast,
the Boltzmann equation relies on inputs such as cross-sections. After
comparing the results obtained within the transport model with experimental
data, different groups estimate the spatial diffusion coefficient, ${\cal
D}_{s}$, from the transport coefficients that allow them to describe the
experimental data effectively using Eq. (23). Experimental data on $D$-meson
production, obtained from both RHIC and the LHC, have reached a level of
precision that permits quantitative constraints on the spatial diffusion
coefficient. These constraints can be established by comparing the data with
theoretical models. This has been carried out, e.g., by the ALICE
collaboration using their recent measurements of the $R_{\mathrm{AA}}$ and
$v_{2}$ of $D$ mesons [13]. By conducting $\chi^{2}$ analysis of the available
model calculations, the ones with a reasonably good confidence were selected
and used to extract the underlying values of the diffusion constant as 1.5
$\leq(2\pi T){\cal D}_{s}\leq\,4.5$. Several of these models did not account
for hadronic diffusion.

The spatial diffusion coefficient of heavy quarks has also been computed using
LQCD [209, 211, 212]. For $N_{f}$=2+1-flavor QCD with light dynamical quarks
corresponding to a pion mass of around 320 MeV recent computations have been
carried out in Ref. [210]. It has been found that the $c$-quark diffusion
coefficient is quite close to the conjectured lower bound at temperatures just
above $T_{\rm pc}$, $(2\pi T){\cal D}_{s}\simeq 1.2\pm 0.25$ and rises only
moderately with temperature, to about 3$\pm$1 at 2$T_{\rm pc}$; these results
are not inconsistent with the phenomenological extraction from heavy-ion data.
As a further advancement, its mass dependence has been reported, which indeed
turned out to be weak, with a 20-30% reduction when going from $c$-quarks to
the static limit ($m_{H}\to\infty$) [208].

This once again underscores the significance of open HF hadrons as probes of
the transport properties of hot QCD matter, providing direct connections
between LQCD and experiments via the phenomenology of transport models.

The spatial diffusion coefficient ${\cal D}_{s}$ is directly related to the HF
thermalization time. One can estimate the thermalization time of heavy quarks
as $\tau_{\rm{th}}$, defined as the inverse of the drag coefficient,
$\tau_{\rm{th}}=A^{-1}$. Hence,

$\tau_{\rm{th}}=\frac{m_{H}{\cal D}_{s}}{T}=\frac{m_{H}}{2\pi T^{2}}\ (2\pi
T{\cal D}_{s})\ .$ (24)

In Fig. 45, we compile the temperature dependence ${\cal D}_{s}$ as a function
of temperature in both the QGP (where we limit our selection to LQCD results)
and the hadronic phase, as thoroughly discussed in this review. The hadronic
model calculations have a clear tendency to drop to rather small values when
extrapolated to temperatures in the vicinity of $T_{\rm pc}$, suggestive for a
continuous transition from the confined to the deconfined medium through a
minimum near the quark-hadron transition. This indicates the strength of the
interaction is maximum near the quark-hadron transition temperature. As
discussed above, in the QGP phase there is again a tendency for $D_{s}$ to
increase but the values remain overall very small, far from a weakly coupled
gas of quarks and gluons.

If the thermalization time remains constant, it implies that the drag
coefficient will be independent of temperature, meaning $2\pi T{\cal
D}_{s}\propto T^{2}$. Therefore, a constant value of $2\pi T{\cal D}_{s}$
corresponds to a thermalization time that is proportional to $1/T^{2}$. The
LQCD results for the QGP indicate a (mild) linear increase in $2\pi T{\cal
D}_{s}$ with temperature, implying that the interaction strength decreases.
One might wonder whether the marked drop of ${\cal D}_{s}(2\pi T)$ with $T$ in
the hadronic phase indicates a strong increase in the interaction strength;
this is probably not the case, but rather due to the fact that the total
hadronic density increases with a rather large power in $T$, due to numerous
and rather massive excited states that are increasingly populated toward
higher temperatures. As a result, the friction coefficient has a stronger
temperature scaling; for example, if $A(0)\propto T^{4}$ (i.e., two additional
powers compared to pQCD), one would obtain a parametric dependence of the
diffusion coefficient as ${\cal D}_{s}(2\pi T)\propto 1/T^{2}$).

In phenomenological applications (and extractions of ${\cal D}_{s}$), one has
to keep in mind that the 3-momentum dependence of the friction coefficient
plays an important role. This is due to the fact that thermal motion and the
collective flow of the expanding medium will always results in a smearing of
the 3-momentum dependence of the transport coefficient relative to the
observed $p_{\rm T}$ in the lab-frame of experiment. In particular, it has
been shown through model comparisons that there is a direct correlation
between a larger value of ${\cal D}_{s}$ along with a harder 3-momentum
dependence of the corresponding friction coefficient, $A(p)$, to yield a
comparable description of $D$-meson data, see also Ref. [4] for additional
details.

##  5 Summary and Outlook

The theoretical investigation of the transport properties of open heavy-flavor
particles in hot hadronic matter essentially commenced a little more than a
decade ago, originally driven by the phenomenological need to study the impact
on their transverse-momentum spectra in heavy-ion collisions. While this is
essential for a quantitative assessment of the transport properties of heavy
quarks in the strongly coupled quark-gluon plasma and its hadronization, it is
also of interest in its own right in a comprehensive understanding of hot QCD
matter and its underlying quantum many-body physics. In addition, the
construction and applications of effective hadronic interactions have been
mutually beneficial to other sub-fields, such as the properties of HF hadrons
in the cold nuclear medium or spectroscopic applications in particular to as
exotic states.

In this article, we have attempted to review the progress made in the
evaluation of open HF diffusion in hot hadronic matter, specifically pertinent
drag and diffusion coefficients, and to illuminate its quantitative role in
transport model simulations of the fireball in heavy-ion collisions. Most of
this research to date, in both microscopic calculations and applications to
heavy-ion collisions, has focused on the transport coefficients of $D$ mesons.
While initial approaches have focused on LO approximations in heavy-meson
effective theory, it quickly became clear that for the temperatures
encountered in experiment, resumed interactions—in particular those involving
resonance excitations—are required for realistic descriptions of $D$-meson
interactions with light mesons in a heat bath. This led to the development of
systematic approaches where symmetry constraints from both the heavy-quark and
the light-quark sectors ($\chi$PT) have been combined, and pertinent diagrams
utilized via unitarization techniques which enable to extend the calculations
to higher temperatures. We have found the different approaches to be in fair
agreement when compared under the same assumptions and/or inputs, lending
robustness to the results for the transport coefficients. A bigger uncertainty
originates from the proliferation of states in the hadron resonance gas when
approaching temperatures near the pseudo-critical one. Here, constituent-quark
scaling arguments for effective cross sections have been put forward, but it
is not known how accurate they are especially if higher excited resonances are
involved that are not necessarily accessible from interactions based on
symmetry arguments (but are rather rooted in the quark-level spectroscopy).
These arguments become even more pertinent if one considers the diffusion of
HF baryons.

Quantitatively, the theoretical estimates of the collisional widths of
$D$-mesons turn out to be at least $\sim$50 MeV from known interactions at
temperatures $T\simeq$ 150 MeV, and for corresponding drag coefficient of
about 0.05 fm-1. In reality, they are probably larger, e.g., due to further
excited states in the hadron resonance gas that are not taken into account.
Nevertheless, these widths are significant (and threshold and mass gaps in
$D$-meson spectra are often rather small, e.g., a $D$-$D^{*}$ mass splitting
which essentially amounts to the pion mass), quantum many-body treatments are
in order and have been carried out. They may lead to further enhanced effects
(e.g., by making bound-state accessible in $2\to 2$ scatterings), and require
additional investigations.

In principle, the above outlined approach directly translate into bottom
sector, although the empirical constraints on the interactions (and excited
states) are more scarce. On general grounds, one expects the thermalization
rate (or drag coefficient) for $B$-mesons to be smaller than that for $D$
mesons by a mass ratio $m_{D}/m_{B}$. In the definition of the spatial
diffusion coefficient the leading mass dependence is taken out, and therefore
they should be quite comparable (this has been confirmed in recent LQCD
computations for bottom and charm quarks in the quark-gluon plasma).

To assess the potential impact of the hadronic phase on various heavy hadron
observables, we have conducted a review of the results obtained within
different transport models. Again, almost all pertinent works have focused on
observables related to $D$ mesons. Basic arguments derived from the typical
time evolution of the nuclear modification factor and elliptic flow of heavy
quarks in the QGP already suggest that latter will be significantly stronger
affected in the hadronic phase than the former (since the $R_{\mathrm{AA}}$
saturates much earlier than the $v_{2}$). This has been corroborated by a
studies that terminate the HQ transport at different “critical temperatures”
found large (small) variations on the $v_{2}$ ($R_{\mathrm{AA}}$), which, of
course, implies an incomplete simulation.

Quantitative studies in various transport models have confirmed these
expectations. More concretely, the addition to the maximal $v_{2}$ value due
to the inclusion of hadronic diffusion can reach up to $\sim$30% in Au-Au
collisions at RHIC, and possibly up to $\sim$20% in Pb-Pb collisions at the
LHC. In either case, this is not a small effect, but it also suggests that the
better testing ground would be toward lower collision energies (as to be
expected from a relatively shorter QGP phase). Caveats include that the bulk-
medium $v_{2}$ tends to be lower at smaller collision energies (perhaps
because the mechanism of converting the initial spatial geometry into momentum
anisotropy is less effective, e.g., due to longer thermalization times), and,
on the other hand, the higher concentration of baryons which could enhance the
hadronic diffusion effects. Existing results from lower-energy runs at RHIC
tend to indicate a lower overall HF $v_{2}$, but its precision and
interpretation remains inconclusive to date. Precision charm-hadron
measurements at SPS energies ($\sqrt{s_{\rm NN}}\simeq$ 6-17 GeV) as planned
by the NA60+ experiment will be highly illuminating.

With the current best estimates of $D$-meson transport coefficients, values of
down to $\sim$5 for the (scaled) spatial diffusion coefficient, ${\cal
D}_{s}(2\pi T)$, at temperatures of $T\simeq$160 MeV seem attainable. This is
still significantly larger than the results of recent LQCD computations on the
QGP side, where values near 1-2 were found. Two possible reasons for this are
that (i) the minimum of ${\cal D}_{s}(2\pi T)$ as a function of temperature is
significantly above $T_{\rm pc}$, and (ii) current hadronic calculations are
still incomplete. Nevertheless, it is encouraging that the discrepancy is not
that large, and we are confident that future theoretical efforts will be able
to make significant progress on this, as well as on the quantitative
implementations of hadronic diffusion into heavy-ion phenomenology.

##  6 Acknowledgements

SKD acknowledges Jan-e Alam, Sourav Sarkar, Sabyasachi Ghosh, and Mohammad
Yousuf Jamal for their valuable discussions. The work of JMTR has been
supported by the project number CEX2019-000918-M (Unidad de Excelencia “María
de Maeztu”), PID2020-118758GB-I00, financed by the Spanish MCIN/
AEI/10.13039/501100011033/, the EU STRONG-2020 project, under the program
H2020- INFRAIA-2018-1 grant agreement no. 824093, and DFG project no.
315477589 - TRR 211 (Strong-interaction matter under extreme conditions). The
work of RR has been supported by the US National Science Foundation under
grant no. PHY-2209335 and the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics through the Topical Collaboration in Nuclear Theory
on Heavy-Flavor Theory for QCD Matter (HEFTY) under award no. DE-SC0023547.

##  Appendix A Appendix: Heavy-flavor transport coefficients

In this Appendix we recall the formal derivation of the Fokker-Planck equation
from the master (Boltzmann) equation, and the definition of the relevant
transport coefficients, as well as their connection to the Langevin
description of a relativistic Brownian particle. For further details we refer
we readers to the Refs. [213, 133, 214, 215, 29, 58].

We consider the elastic collisions of a generic $D$-meson (denoted by $H$, for
heavy hadrons in general) with a light particle (denoted by $l$) from the
thermal bath. The thermal bath might consist of hadrons ($\pi$, $K$,
$\bar{K}$, $\eta$, nucleons, $\Delta$, etc.), characterized by their thermal
phase space distributions.

The momentum evolution of the heavy hadron’s with momentum, ${\bm{p}}$,
$f_{H}(t,{\bm{p}})$, in the thermal bath can be studied using the Boltzmann
equation,

$\displaystyle\frac{\partial f_{H}(t,\bm{p})}{\partial t}=\left[\frac{\partial
f_{H}(t,\bm{p})}{\partial t}\right]_{\textrm{coll}}\ .$ (25)

The right-hand side of Eq. (25) represents the collision integral, wherein the
phase-space distribution function of the bulk medium appears as an integrated
quantity. This form of the Boltzmann equation assumes no external forces,
indicating that the evolution of $f_{H}(t,{\bm{p}})$ is solely governed by
collisions. Given the low density of heavy mesons, collisions between them can
be safely disregarded. Therefore our focus can be directed solely towards the
interaction of the heavy hadrons with the light particles in the thermal bath.
Heavy hadrons can enter and exit the momentum phase space element $d{\bm{p}}$
around ${\bm{p}}$ through collisions with the thermal particles. Consequently,
the collision term consists of two parts associated with a gain and a loss
term. If we define $w(p,k)$ as the rate of collisions that change the momentum
of $H$ from $p$ to $p-k$, then we can write

$\displaystyle\left[\frac{\partial f_{H}(t,{\bm{p}})}{\partial
t}\right]_{\textrm{coll}}=\int
d^{3}{\bm{k}}\left[f_{H}(t,{\bm{p}}+{\bm{k}})w({\bm{p}}+{\bm{k}},{\bm{k}})-f_{H}(t,{\bm{p}})w({\bm{p}},{\bm{k}})\right]\
.$ (26)

The first term on the right-hand side of Eq. (26) represents the gain of
probability through collisions, which cause $H$ to enter the volume element of
momentum space at ${\bm{p}}$. Conversely, the second term represents the loss
of probability out of that element; $w$ is the aggregate of contributions
arising from heavy-hadron scattering off all thermal particles. In essence,
the Boltzmann equation should be treated as a quantum Boltzmann-Uehling-
Uhlenbeck equation. This involves considering the Bose enhancement effect in
the final state, where factors like $1+f_{H}$ encode the increased probability
of a $D$ meson scattering into an already occupied state. As long as the
number of $D$-mesons is very small, we can approximate $1+f_{H}\sim 1$.
However, this approximation may not hold for the light thermal particles.
Expanding $f_{H}(t,{\bm{p}}+{\bm{k}})w({\bm{p}}+{\bm{k}},{\bm{k}})$ around
${\bm{p}}$,

$\displaystyle f_{H}(t,{\bm{p}}+{\bm{k}})w({\bm{p}}+{\bm{k}},{\bm{k}})\simeq
f_{H}(t,{\bm{p}})w({\bm{p}},{\bm{k}})+k_{i}\frac{\partial}{\partial
p_{i}}[f_{H}(t,{\bm{p}})w({\bm{p}}+{\bm{k}})]+\frac{1}{2}k_{i}k_{j}\frac{\partial^{2}}{\partial
p_{i}\partial p_{j}}[f_{H}(t,{\bm{p}})w({\bm{p}}+{\bm{k}})]+\cdots\ ,$ (27)

and substituting into this into Eq. (26), we obtain:

$\displaystyle\left[\frac{\partial f_{H}(t,{\bm{p}})}{\partial
t}\right]_{\textrm{coll}}=\frac{\partial}{\partial
p_{i}}\left\\{A_{i}({\bm{p}})f_{H}+\frac{\partial}{\partial
p_{j}}\left[B_{ij}({\bm{p}})f_{H}\right]\right\\}\ ,$ (28)

where the kernels are defined as

$\displaystyle A_{i}({\bm{p}})=\int d^{3}{\bm{k}}\ w({\bm{p}},{\bm{k}})k_{i}$
$\displaystyle B_{ij}({\bm{p}})=\int d^{3}{\bm{k}}\
w({\bm{p}},{\bm{k}})k_{i}k_{j}\ .$ (29)

The classical interpretation of these kernels is that they represent drag and
momentum-space diffusion coefficients affecting the kinematics of $H$. For a
collection of different notations used in the literature, cf. Table 1.

Drag Force
---
$A(p)$ |  [56],[57],[1],[84],[4]
$F(p)$ |  [58],[66],[33]
$\gamma(p)$ | [75]
$\eta_{D}$ | [55]
Momentum-space diffusion coefficients
$B_{0}(p),B_{1}(p)$ |  [57],[75],[1],[84],[4]
$\Gamma_{0},\Gamma_{1}$ |  [58],[66],[33]
$\kappa/2$ | [55]
Coordinate-space diffusion coefficient
${\cal D}_{s},D_{s}$ |  [1],[84],[4]
$D_{x}$ |  [57], [58] ,[66], [75],[33]
$D$ | [55]
Table 1: List of notations for the different transport coefficients used in
the works referenced in this review.

The function $w({\bm{p}},{\bm{k}})$ is given by

$\displaystyle
w({\bm{p}},{\bm{k}})=g_{l}\int\frac{d^{3}q}{(2\pi)^{3}}f_{l}(q)v_{q,p}\sigma_{p,q\rightarrow
p-k,q+k}\ ,$ (30)

where $f_{l}(q)$ represents the phase space distribution of the light
particles in the thermal bath, $g_{l}$ is their statistical degeneracy,
$v_{q,p}$ indicates the relative velocity between the collision partners and
$\sigma_{p,q\rightarrow p-k,q+k}$ stands for the interaction cross-section of
$H$ with the light particles in the heat bath.

With these approximations, the Boltzmann Eq. (25) reduces to the Fokker-Planck
equation,

$\displaystyle\frac{\partial f_{H}(t,{\bm{p}})}{\partial
t}=\frac{\partial}{\partial
p_{i}}\left\\{A_{i}({\bm{p}})f_{H}+\frac{\partial}{\partial
p_{j}}\left[B_{ij}({\bm{p}})f_{H}\right]\right\\}\ .$ (31)

In this context, $A_{i}({\bm{p}})$ acts like a friction term, indicating the
average momentum change of $H$, while $B_{ij}({\bm{p}})$ acts as a diffusion
coefficient in momentum space, responsible for the broadening of the average
momentum distribution of $H$. Since $A_{i}$ and $B_{ij}$ solely depend on
$p_{i}$, in a isotropic background medium they can be represented by three
scalar functions of $p$,

$\displaystyle A_{i}(p)$ $\displaystyle=A(p^{2})p_{i}$ $\displaystyle
B_{ij}(p)$
$\displaystyle=B_{0}(p^{2})\Delta_{ij}+B_{1}(p^{2})\frac{p_{i}p_{j}}{p^{2}}\
,$ (33)

where

$\displaystyle\Delta_{ij}\equiv\delta_{ij}-\frac{p_{i}p_{j}}{p^{2}}.$ (34)

The explicit expressions of these three scalar coefficients, as given in (A)
in terms of $w({\bm{p}},{\bm{k}})$, are as follows:

$\displaystyle A(p)$ $\displaystyle=$
$\displaystyle\frac{p^{i}A_{i}}{p^{2}}=\int d^{3}{\bm{k}}\
w({\bm{p}},{\bm{k}})\ \frac{k_{i}p^{i}}{p^{2}}\ ,$ (35) $\displaystyle
B_{0}(p)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Delta_{ij}B^{ij}=\frac{1}{4}\int d^{3}{\bm{k}}\
w({\bm{p}},{\bm{k}})\left[{\bm{k}}^{2}-\frac{(k_{i}p^{i})^{2}}{p^{2}}\right]\
,$ (36) $\displaystyle B_{1}(p)$ $\displaystyle=$
$\displaystyle\frac{p_{i}p_{j}}{p^{2}}B^{ij}=\frac{1}{2}\int d^{3}{\bm{k}}\
w({\bm{p}},{\bm{k}})\ \frac{(k_{i}p^{i})^{2}}{p^{2}}\ ,$ (37)

where pertinent scattering matrix elements, $|\mathcal{M}_{Hl}|$, dictate the
dynamics through the cross-section given in Eq. (30).

The three coefficients, $A(p)$, $B_{0}(p)$ and $B_{1}(p)$, in the Fokker-
Planck equation are not independent; instead, they are related through the
fluctuation-dissipation relation, which can be derived from the Fokker-Plank
equation (see, e.g., Ref.[77]. One should expect that in the long-time limit,
the heavy hadrons reach the same thermal distribution as the medium
constituents, $f_{H}={\rm e}^{-E_{H}/T}$. However, if the drag and diffusion
coefficients are directly taken from their calculations, this is generally not
the case. Since the drag and diffusion coefficients represent the first two
moments of the full collisional integral, not all momentum transfers can be
accurately captured solely by these two moments. Establishing a connection
between the drag and diffusion coefficients is necessary to ensure that the
correct equilibrium distribution $f_{eq}=e^{-E/T}$ is attained as the
asymptotic distribution. In equilibrium, the right-hand side of Eq. (31)
vanishes, and a connection between the drag and diffusion coefficients can be
established. At $p\rightarrow 0$, when $B_{0}=B_{1}=B$, Eq. (31) simplifies to
Einstein’s famous fluctuation-dissipation relation, $B=m_{H}AT$. Further
details can be found in Refs. [216, 29]. Employing the fluctuation-dissipation
relation reduces the problem to a single transport coefficient. This
highlights that the Fokker-Planck equation serves as a consistent
approximation to the Boltzmann equation, as it accurately captures the proper
equilibrium limit. It is essential to include both friction and diffusion
terms to maintain the principle of detailed balance. However, one should keep
in mind that if the momentum transfer is significant too large, i.e., does not
satisfy $k\ll m_{H}$, the Fokker-Planck equation may not serve as an
appropriate approximation of the Boltzmann equation, and it may deviate from
it [30].

The fluctuation-dissipation relation suggests that a pure diffusion equation,
$A=0$, is only valid in the limit of $T\rightarrow\infty$ for a particle with
finite mass. On the other hand, in phenomenological applications to heavy-ion
collisions, the evolution of high-$p_{\rm T}$ particles is often approximated
using an energy-loss treatment to compute the nuclear modification factor,
which involves neglecting the diffusion term, $B=0$. This implies that only
momentum or energy-degrading processes are considered, a behavior akin to the
$T\rightarrow 0$ limit in the fluctuation-dissipation relation. The absence of
momentum diffusion implies that both momentum randomization and energy gain
processes are neglected, thus preventing particles from achieving
equilibration.

The diffusion diffusion in momentum space or drag coefficient can be
translated into a spatial diffusion coefficient, which quantifies the variance
in the particle’s position as it evolves over time. The spatial diffusion
coefficient, represented as ${\cal D}_{s}$, can be expressed as follows:

$\langle({\bm{r}}-{\bm{r}}_{0})^{2}\rangle=6{\cal D}_{s}t\ .$ (38)

The spatial diffusion coefficient can be linked to the drag and momentum-
diffusion coefficient through the following relation,

${\cal D}_{s}=\frac{T}{m_{H}A}=\frac{T^{2}}{B}\ .$ (39)

The conventional method for studying the HQ momentum evolution within the QGP
phase and the momentum evolution of open HF hadrons within the hadronic medium
through the Fokker-Planck equation is a stochastic one using the Langevin
equations. The relativistic Langevin equations governing the temporal changes
in the position and momentum of heavy quarks or heavy hadrons can be expressed
as follows:

$\left\\{\begin{array}[]{rcl}dx_{i}&=&\frac{p_{i}}{E}dt\ ,\\\ \\\
dp_{i}&=&-A(p)p_{i}dt+C_{ij}(p)\rho_{j}\sqrt{dt}\ ,\end{array}\right.$

where for each discrete time step $dt$, the change in the coordinates and
momenta, denoted as $dx_{i}$ and $dp_{i}$ respectively, represent the shifts
in position and momentum. $A(p)$ represents the drag force, while $C_{ij}(p)$
denotes the so-called covariance matrix; $\rho$ represents a stochastic noise,
which adheres to the probability distribution of independent Gaussian-normal
distributed random variables, described by
$P(\rho)=(2\pi)^{-3/2}e^{-\rho^{2}/2}$. Additionally, it satisfies the
relations $\langle\rho_{i}\rho_{j}\rangle=\delta_{ij}$ and
$\langle\rho_{i}\rangle=0$. The covariance matrix is associated with the
transverse and longitudinal diffusion coefficients through the equation,

$\displaystyle C_{ij}(p)=\sqrt{2B_{0}(p)}\ \Delta_{ij}+\sqrt{2B_{1}(p)}\
\frac{p_{i}p_{j}}{p^{2}}\ ,$ (40)

where $\Delta_{ij}$ is given by Eq. (34). Under the assumption that
$B_{0}(p)=B_{1}(p)=B(p)$, Eq. (40) simplifies to $C_{ij}=\sqrt{2B(p)}\
\delta_{ij}$. This assumption, which is strictly valid for $p\rightarrow 0$,
is commonly employed for the heavy-quark/hadron momentum evolution at finite
momentum $p$. The Langevin equation can be effectively solved by incorporating
the drag and diffusion coefficients as inputs, as they encapsulate the
microscopic intricacies of the interactions between heavy hadrons and the
thermal bath. In addition, one needs to specify the initial distribution of
coordinates and momenta of the heavy hadrons (usually taken from the Glauber
model) and a realistic space-time of the bulk medium. Subsequently, the
results obtained from solving the Langevin equation can be utilized to compute
various observables that are measured in experiments, such as the nuclear
modification factor and flow coefficients. The Fokker-Planck and Langevin
equations are equivalent when considering the first two moments, which
correspond to the drag and diffusion coefficients in the Fokker-Planck
equation, see Eq. (31).

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