[1]

[cor1]Corresponding author

# Charm and beauty isolation from heavy flavor decay electrons in p+p and
Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV at LHC

Dongsheng Li  Fan Si  Yidan Zhao  Pengyu Zhou  Yifei Zhang<EMAIL_ADDRESS>Xiujun Li  Chengxi Yang  State Key Laboratory of Particle Detection and
Electronics, University of Science and Technology of China, Hefei 230026,
China

###### Abstract

We present an analysis on the heavy flavor hadron decay electrons with charm
and beauty contributions decomposed via a data driven method in p+p and Pb+Pb
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV at LHC. The transverse
momentum $p_{\mathrm{T}}$ spectra, nuclear modification factor
$R_{\mathrm{AA}}$ and azimuthal anisotropic flow $v_{2}$ distributions of
electrons from charm and beauty decays are obtained. We find that the electron
$R_{\mathrm{AA}}$ from charm ($R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$) and
beauty ($R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$) decays are suppressed at
$p_{\mathrm{T}}$ $>$ 2.0 GeV/$c$ in Pb+Pb collisions, which indicates that
charm and beauty interact with and lose their energy in the hot-dense medium.
A less suppression of $R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ than
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ at 2.0 $<$ $p_{\mathrm{T}}$ $<$
8.0 GeV/$c$ is observed, which is consistent with the mass-dependent partonic
energy loss scenario. A non-zero electron $v_{2}$ from beauty decays
($v_{2}^{\mathrm{b\rightarrow e}}$) is observed and in good agreement with
ALICE measurement. At low $p_{\mathrm{T}}$ from 1.0 to 3.0 GeV/$c$, a
discrepancy between RHIC and LHC results is observed with an 85% confidence
level, which might be a possible hint of an energy-dependent behavior of
beauty quarks reacting with the medium created in heavy-ion collisions. At 3.0
$<$ $p_{\mathrm{T}}$ $<$ 7.0 GeV/$c$, $v_{2}^{\mathrm{b\rightarrow e}}$
deviates from a number-of-constituent-quark (NCQ) scaling hypothesis, which
favors that beauty quark is unlikely thermalized in heavy-ion collisions at
LHC energy.

###### keywords:

Quark-Gluon Plasma charm beauty semileptonic decay nuclear modification factor
elliptic flow

##  1 Introduction

Strong interactions among quarks and gluons by exchanging color charges are
described by the theory of Quantum Chromodynamics (QCD). Quarks and gluons are
confined in color-neutral hadrons and can be liberated under an extremely hot
and dense condition such as the stage of a few microseconds after the big bang
of the universe. Ultra-relativistic heavy-ion collisions provide such
environment to create the state of matter known as quark-gluon plasma (QGP)
that behaves almost as a perfect fluid [1, 2, 3]. Heavy quarks (charm and
beauty) are produced mainly via initial hard scatterings in heavy-ion
collisions due to their large masses. Their production yield can be evaluated
by perturbative QCD. Thus, heavy quarks are believed to experience the whole
medium dynamic evolution after their creation and maintain conserved quark
numbers. Therefore, heavy quarks are an ideal probe to study QGP matter and
sensitive to its thermal dynamic properties [4, 5, 6]. In the past ten years,
open heavy flavor hadrons have been measured in many experiments at
Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) [7, 8,
9, 10]. It was found that the nuclear modification factors ($R_{\mathrm{AA}}$)
of both electrons from heavy flavor decays and open charm mesons were strongly
suppressed at high transverse momentum ($p_{\mathrm{T}}$) in central heavy-ion
collisions. This indicates that charm quarks significantly lose their energy
when traversing the hot-dense medium. A significant elliptic flow ($v_{2}$) of
open charm mesons is also observed and comparable with light flavor hadrons,
which indicates that charm quark reaches thermalization in heavy-ion
collisions.

Beauty, with a factor of three larger mass than charm, is predicted to lose
less energy than charm in the medium created in heavy-ion collisions either
with a suppression of the gluon radiation angle [11, 12, 13] or the elastic
collisions [4, 14]. And its $v_{2}$ is probably different from those of charm
quark and light flavors if it is far from thermalization. However, measurement
of beauty is challenging due to its small production cross section and low
hadronic decay branching ratios. Thus, in an alternative way, the beauty was
commonly measured indirectly via its decay products, such as electrons, non-
prompt $\mathrm{D^{0}}$ and $\mathrm{J/\psi}$ mesons. It was observed that the
$R_{\mathrm{AA}}$ of its decay daughters are less suppressed compared to those
of prompt open charm mesons at $p_{\mathrm{T}}$ up to $\sim$ 20 GeV/$c$, which
is consistent with the mass hierarchy of the parton energy loss in the medium
[15, 16, 17, 18, 19, 20, 21, 22]. Recently, a sizable electron $v_{2}$ from
beauty decays was also measured by ALICE [23], which indicates strong
interactions between beauty and the medium. Since charm looks no significant
difference compared with light quarks in terms of $R_{\mathrm{AA}}$ and
$v_{2}$ in RHIC and LHC energies [5, 24], it is crucial to study whether
beauty is sensitive to the different media created at RHIC and LHC energies
and behaves differently under different temperatures.

In a previous study, the charm and beauty contributions are separated from the
inclusive heavy flavor electrons (HFE) in 200 GeV Au+Au collisions at RHIC
[25]. It was found that the electron $R_{\mathrm{AA}}$ from beauty decays is
less suppressed compared to that from charm decays, which is consistent with
the mass-dependent energy loss expectation. And a first zero electron $v_{2}$
from beauty decays, lower than a number-of-constituent-quark (NCQ) scaling
hypothesis, was observed at low $p_{\mathrm{T}}$, which indicates that beauty
quark is unlikely thermalized and too heavy to be moved following the
collective flow of lighter partons in heavy-ion collisions at RHIC energy.

In this paper, we perform an analysis in a similar way to separate charm and
beauty contributions from heavy flavor decay electrons in 5.02 TeV p+p and
Pb+Pb collisions. The $R_{\mathrm{AA}}$ and $v_{2}$ of electrons from charm
and beauty decays in 5.02 TeV Pb+Pb collisions at LHC are obtained and
compared with those in 200 GeV Au+Au collisions at RHIC.

##  2 Analysis Technique and Results

###  2.1 Transverse Momentum Spectra and $R_{\mathrm{AA}}$

We adopt the data-driven method developed in Ref. [25] to separate charm and
beauty decay contributions from the inclusive HFE spectra [26] in 0–10% and
30–50% central Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV and p+p
collisions at $\sqrt{s}$ = 5.02 TeV at LHC by taking advantage of the most
recent charmed hadron measurements. Different from the previous study at RHIC
energy, the HFE measurement [26] from the ALICE experiment was not subtracted
by the $\mathrm{J/\psi}$ contribution (about 5% at 2 < $p_{\mathrm{T}}$ < 3
GeV/$c$ and much lower in other $p_{\mathrm{T}}$ regions), and thus, the
$\mathrm{J/\psi\rightarrow e^{+}e^{-}}$ contribution is also studied in this
work. In Pb+Pb collisions, the ALICE experiment has precisely measured the
$p_{\mathrm{T}}$ spectra of prompt $\mathrm{D^{0}}$, $\mathrm{D^{\pm}}$ and
$\mathrm{D_{s}}$ mesons (0 $<$ $p_{\mathrm{T}}$ $<$ 50 GeV/$c$) and
$\mathrm{\Lambda_{c}}$ baryons (0 $<$ $p_{\mathrm{T}}$ $<$ 24 GeV/$c$)
including their antiparticles at mid-rapidity ($|y|$ $<$ 0.5) at 0–10% and
30–50% centrality classes [27, 28, 29]. The $\mathrm{J/\psi}$ spectra ($|y|$
$<$ 0.9) in both centrality classes are provided by the TAMU model predictions
[30, 31, 32] combined with the measurement by the CMS experiment [20] at 6.5
$<$ $p_{\mathrm{T}}$ $<$ 50 GeV/$c$, which is scaled from 0–100% centrality
with the average number of binary nucleon-nucleon collisions ($\left\langle
N_{\mathrm{coll}}\right\rangle$) [33]. All the spectra are then parameterized
with a Levy [34] function, and the uncertainties are given by a quadratic sum
of two components: (a) 1-$\sigma$ band from the parameterization with
statistical uncertainties and (b) systematic uncertainties from the
parameterized spectra scaled to upper and lower limits. The parameterization
at 30–50% centrality is shown as an example in Fig. 1, which works well
compared with the experimental data (black points) and the TAMU prediction
(black band). In p+p collisions, the prmopt D-mesons ($|y|$ $<$ 0.5) [35],
$\mathrm{\Lambda_{c}}$ ($|y|$ $<$ 0.5) [36] and $\mathrm{J/\psi}$ ($|y|$ $<$
0.9) [37] spectra measured by ALICE are parameterized with the same steps as
those in Pb+Pb collisions. To extend the $p_{\mathrm{T}}$ ranges, the
measurements of $\mathrm{J/\psi}$ by the ATLAS experiment [38] at 12 $<$
$p_{\mathrm{T}}$ $<$ 40 GeV/$c$ and $\mathrm{\Lambda_{c}}$ by CMS [39] at 10
$<$ $p_{\mathrm{T}}$ $<$ 20 GeV/$c$ are combined with the ALICE measurements
for parameterization.

Figure 1: Parameterized charmed hadron ($\mathrm{D^{0}}$, $\mathrm{D^{\pm}}$,
$\mathrm{D_{s}}$, $\mathrm{\Lambda_{c}}$, $\mathrm{J/\psi}$) spectra at mid-
rapidity ($|y|$ $<$ 0.9 for $\mathrm{J/\psi}$ and 0.5 for others) in 30–50%
central Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. Uncertainties
are shown as shaded color bands. The experimental measurements and the TAMU
prediction are shown as black points and black bands, respectively. The
$\mathrm{J/\psi}$ measurement in 20–40% centrality class by ALICE [40] scaled
to 30–50% by $\left\langle N_{\mathrm{coll}}\right\rangle$ [41] is also shown
for comparison (open black stars) but not included in the analysis. The
measurement on prompt $\mathrm{J/\psi}$ in 0–100% centrality class by CMS [20]
scaled to 30–50% by $\left\langle N_{\mathrm{coll}}\right\rangle$ [33] (solid
black stars) is combined with the TAMU prediction on inclusive
$\mathrm{J/\psi}$ production in the parameterization. The spectra of charmed
hadrons are scaled with different factors to separate the merged points.

The charmed hadrons described above are then simulated to decay to electrons
via semileptonic decay ($\mathrm{e^{+}e^{-}}$ for $\mathrm{J/\psi}$) channels
with their parameterized $p_{\mathrm{T}}$ spectra at 0 $<$ $p_{\mathrm{T}}$
$<$ 20 GeV/$c$ and Gaussian rapidity distributions ($\mu$ = 0 and $\sigma$ =
1.7 for open charm hadrons, $\mu$ = 0 and $\sigma$ = 1.9 for
$\mathrm{J/\psi}$) as inputs. These rapidity distributions have been checked
by the PYTHIA [42] event generator. By varying the rapidity distributions with
a sigma of 1.7 $\pm$ 0.3, the obtained electron spectra changes $<$ 1% in 0
$<$ $p_{\mathrm{T}}$ $<$ 20 GeV/$c$. The open charm hadron decays are
conducted with the help of decay formfactors in the hadron rest frame which
are sampled from the measured distribution by the CLEO experiment [43], while
the $\mathrm{J/\psi}$ decays are simulated with PYTHIA. Note that a non-zero
$\mathrm{J/\psi}$ polarization could have impact on the decay kinematics.
Recently the ALICE experiment measured $\mathrm{J/\psi}$ polarization which
shows some hint of non-zero polarization but with large uncertainties [44].
Thus here the effect of potential $\mathrm{J/\psi}$ polarization on the decay
electron spectra is not taken into account. Figure 2 shows the electron
spectra from $\mathrm{D^{0}}$ (blue dashed curve), $\mathrm{D^{\pm}}$ (brown
dot-dot-dashed curve), $\mathrm{D_{s}}$ (green dot-dashed curve),
$\mathrm{\Lambda_{c}}$ (cyan long-dot-dashed curve) and $\mathrm{J/\psi}$
(violet dot-dot-dot-dashed curve) decays and the summed charm contributions
($\mathrm{c\rightarrow e}$, black solid curve) at mid-rapidity ($|y|$ $<$ 0.8
at $p_{\mathrm{T}}$ $<$ 3 GeV/$c$ and $|y|$ $<$ 0.6 at $p_{\mathrm{T}}$ $>$ 3
GeV/$c$) in 5.02 TeV 0–10% and 30–50% central Pb+Pb collisions and p+p
collisions. All components of the total $\mathrm{c\rightarrow e}$ production
from individual charmed hadron decays are summarized in Table 1. The maximum
$p_{\mathrm{T}}$ of charmed hadron measurements and the contributions of
unsimulated regions of charmed hadron decays
($p_{\mathrm{T}}^{\mathrm{charm}}$ $>$ 20 GeV/$c$) to the electrons in 0–8
GeV/$c$ are summarized in Table 2. The contributions are found to be less than
0.04% to insure that most electrons in the region of interest come from the
simulated region of charmed hadrons. The uncertainties of the charmed hadron
$p_{\mathrm{T}}$ inputs are propagated into the decay electron spectra, and
the uncertainties of branching ratios [45] are also taken into account. The
total uncertainties of electron spectra from individual charmed hadron decays
are shown as shaded bands in Fig. 2. All contributions of the total
uncertainties of $\mathrm{c\rightarrow e}$ within 0 $<$ $p_{\mathrm{T}}$ $<$ 8
GeV/$c$ are summarized in Table 3. The total uncertainties of
$\mathrm{c\rightarrow e}$, quadratically summed from all uncorrelated
components, are 8.0%–40.6% and 6.4%–17.9% at 0–10% and 30–50% centrality
classes in Pb+Pb collisions, and 4.2%–8.2% in p+p collisions, respectively.
The electron spectra from beauty hadron decays ($\mathrm{b\rightarrow e}$, red
solid circles), are then calculated by subtracting the $\mathrm{c\rightarrow
e}$ contribution from the inclusive HFE spectra (black open squares) [26].
Figure 3 presents the $\mathrm{b\rightarrow e}$ spectra in 0–10% (black solid
squares) and 30–50% (red solid circles) central Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV and in p+p collisions at $\sqrt{s}$ = 5.02
TeV with the previously obtained result (blue open squares) [25] in minimum
bias Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV and the FONLL
prediction (green open crosses) [46] at $\sqrt{s}$ = 5.02 TeV shown for
comparison. The p+p result obtained in this work is consistent with FONLL
upper limit. And note that the $p_{\mathrm{T}}$-differential yield spectra in
p+p collisions is obtained by dividing the $p_{\mathrm{T}}$-differential cross
section spectra in Fig. 3 by the minimum bias trigger cross section (50.9
$\pm$ 0.9 mb) at $\sqrt{s}$ = 5.02 TeV. In all of the following uncertainty
calculations, statistical and systematic uncertainties are quadratically
summed into total uncertainties, unless otherwise specified.

Figure 2: Electron spectra from charmed hadrons ($\mathrm{D^{0}}$,
$\mathrm{D^{\pm}}$, $\mathrm{D_{s}}$, $\mathrm{\Lambda_{c}}$,
$\mathrm{J/\psi}$) and the sum of them ($\mathrm{c\rightarrow e}$) and the
inclusive HFE spectra at mid-rapidity ($|y|$ $<$ 0.8 at $p_{\mathrm{T}}$ $<$ 3
GeV/$c$ and $|y|$ $<$ 0.6 at $p_{\mathrm{T}}$ $>$ 3 GeV/$c$) in 0–10% and
30–50% central Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV and p+p
collisions($|y|$ $<$ 0.8) at $\sqrt{s}$ = 5.02 TeV. The spectra of beauty
hadron decay electrons ($\mathrm{b\rightarrow e}$) are obtained by subtracting
the $\mathrm{c\rightarrow e}$ contributions from the HFE data from ALICE [26].
Uncertainties are shown as shaded bands. Figure 3: Electron spectra from
beauty hadron decays in 0–10% and 30–50% central Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV, p+p collisions at $\sqrt{s}$ = 5.02 TeV
compared with the FONLL prediction [46] and the result in minimum bias Au+Au
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV [25]. Table 1: Production
components of the $\mathrm{c\rightarrow e}$ spectra (0–8 GeV/$c$) from
individual charmed hadron decays.

centrality | 0–10% | 30–50% | p+p
---|---|---|---
$\mathrm{D^{0}\ \rightarrow e}$ | 25.7%-34.3% | 24.0%–33.6% | 36.8%–39.0%
$\mathrm{D^{\pm}\ \rightarrow e}$ | 35.7%–58.3% | 39.2%–56.4% | 39.7%–43.0%
$\mathrm{D_{s}\ \rightarrow e}$ | 6.7%–12.9% | 7.2%–10.5% | 6.3%–9.0%
$\mathrm{\Lambda_{c}\ \rightarrow e}$ | 3.5%–10.1% | 4.3%–9.4% | 2.4%–14.2%
$\mathrm{J/\psi\rightarrow e}$ |  $<$ 17.0% |  $<$ 15.7% |  $<$ 12.1%

Table 2: Contributions of unsimulated regions of charmed hadron decays
($p_{\mathrm{T}}^{\mathrm{charm}}$ $>$ 20 GeV/$c$) to the electrons in 0–8
GeV/$c$.

| measured $p_{\mathrm{T}}^{\mathrm{charm}}$ max (GeV/$c$) | contribution
---|---|---
centrality | 0–10% | 30–50% | p+p | 0–10% | 30–50% | p+p
$\mathrm{D^{0}\rightarrow e}$ | 50 | 35 | 35 | 0.005% | 0.016% | 0.026%
$\mathrm{D^{\pm}\rightarrow e}$ | 50 | 35 | 35 | 0.004% | 0.012% | 0.024%
$\mathrm{D_{s}\rightarrow e}$ | 50 | 50 | 25 | 0.037% | 0.010% | 0.037%
$\mathrm{\Lambda_{c}\rightarrow e}$ | 24 | 24 | 20 |  $<$ $10^{-4}$% | 0.004% | 0.002%
$\mathrm{J/\psi\rightarrow e}$ | 50 | 50 | 40 |  $<$ $10^{-4}$% | 0.006% | 0.011%

Table 3: Uncertainty components of the $\mathrm{c\rightarrow e}$ spectra (0–8
GeV/$c$) from electron spectra from individual charmed hadron decays.

| from input | from branching ratio | from centrality
---|---|---|---
centrality | 0–10% | 30–50% | p+p | 0–10% | 30–50% | p+p | 0–10% | 30–50% | p+p
$\delta\left(\mathrm{D^{0}\ \rightarrow e}\right)$ | 3.5%–7.6% | 2.6%–6.1% | 1.8%–4.3% | 0.4%–0.6% | 0.4%–0.6% | 0.1%–0.3% | 0.5%–0.7% | 0.4%–0.7% | $\times$
$\delta\left(\mathrm{D^{\pm}\ \rightarrow e}\right)$ | 5.2%–39.8% | 4.0%–15.7% | 1.7%–4.2% | 0.6%–1.1% | 0.7%–1.1% | 0.7%–0.9% | 0.7%–1.2% | 0.7%–1.2% | $\times$
$\delta\left(\mathrm{D_{s}\ \rightarrow e}\right)$ | 1.1%–4.8% | 1.0%–3.1% | 0.6%–1.9% | 0.4%–0.8% | 0.4%–0.7% | 0.3%–0.6% | 0.1%–0.3% | 0.1%–0.3% | $\times$
$\delta\left(\mathrm{\Lambda_{c}\ \rightarrow e}\right)$ | 0.8%–2.6% | 1.0%–2.8% | 0.3%–2.1% | 1.3%–3.8% | 1.6%–3.6% | 0.9%–5.4% | $\times$ | $\times$ | $\times$
$\delta\left(\mathrm{J/\psi\rightarrow e}\right)$ |  $<$ 2.4% |  $<$ 1.8% |  $<$ 0.9% |  $<$ 0.1% |  $<$ 0.1% |  $<$ 0.1% | $\times$ | $\times$ | $\times$

The beauty contribution fraction in Pb+Pb and in p+p collisions
($f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ and
$f_{\mathrm{pp}}^{\mathrm{b\rightarrow e}}$), shown as solid squares (0–10%),
solid circles (30–50%) and open circles (p+p) in Fig. 4, is calculated by
taking the ratio of the $\mathrm{b\rightarrow e}$ and inclusive HFE spectra,
and the $\mathrm{J/\psi}$ contribution is not removed from the denominator
(i.e., the inclusive HFE spectra). The relative uncertainty ($\delta$) of
$f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ (5.0%–55.7% for 0–10% and
6.2%–51.4% for 30–50%) and $f_{\mathrm{pp}}^{\mathrm{b\rightarrow e}}$
(5.6%–11.9%) is propagated from those of the $\mathrm{c\rightarrow e}$ and the
inclusive HFE spectra with

$\delta_{f^{\mathrm{b\rightarrow
e}}}^{2}=f_{\mathrm{cb}}^{2}\left(\delta_{\mathrm{c\rightarrow
e}}^{2}+\delta_{\mathrm{HFE}}^{2}\right),$ (1)

where $f_{\mathrm{cb}}=\left.\left(1-f^{\mathrm{b\rightarrow
e}}\right)\middle/f^{\mathrm{b\rightarrow e}}\right.$ in Pb+Pb or p+p
collisions. Here we compare the results in 0–10% and 30–50% central Pb+Pb
collisions, p+p collisions, the previous work in minimum bias Au+Au collisions
[25] and the FONLL prediction [46]. The FONLL prediction of the beauty
contribution fraction roughly match the p+p results of this work at $\sqrt{s}$
= 5.02 TeV. The beauty contributions at both centrality classes in Pb+Pb
collisions monotonically increase with $p_{\mathrm{T}}$ and are significantly
higher than that in p+p collisions at $p_{\mathrm{T}}$ $>$ 4 GeV/$c$. At
$p_{\mathrm{T}}$ $<$ 4 GeV/$c$, however, similar conclusion is still tenable
only in 0–10% Pb+Pb collisions, since the beauty contribution fractions in p+p
collisions are comparable with those in 30–50% Pb+Pb collisions. At first, for
$p_{\mathrm{T}}$ $<$ 2 GeV/$c$, the beauty contributions are smaller than
charm, then at $p_{\mathrm{T}}$ $\gtrsim$ 3 GeV/$c$, they overwhelm the charm
and start to be flattened, and finally at $p_{\mathrm{T}}$ $\sim$ 7 GeV/$c$,
the beauty contributions reach a level of 70%. It is also observed that the
result at 0–10% centrality are slightly higher than that at 30–50%, and that
the results at Pb+Pb 5.02 TeV show slightly lower than that at Au+Au 200 GeV
at $p_{\mathrm{T}}$ $>$ 4 GeV/$c$. This could be due to more energy loss of
beauty quark at higher collision energy. However, considering the large
uncertainties, the difference is not significant.

Figure 4: Beauty hadron decay electron fractions ($f^{\mathrm{b\rightarrow
e}}$) at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV in 0–10% and 30–50% central Pb+Pb
collisions ($\left|\eta\right|$ $<$ 0.8 at $p_{\mathrm{T}}$ $<$ 3 GeV/$c$ and
$\left|\eta\right|$ $<$ 0.6 at $p_{\mathrm{T}}$ $>$ 3 GeV/$c$) and at
$\sqrt{s}$ = 5.02 TeV in p+p collisions ($\left|\eta\right|$ $<$ 0.8) compared
with those by FONLL calculations [46] and in minimum bias Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV ($\left|\eta\right|$ $<$ 0.7) [25].

The $R_{\mathrm{AA}}$ of electrons from individual charm and beauty hadron
decays ($R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ and
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$) are extracted with the formulae
[25]

$\displaystyle R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}=$
$\displaystyle\frac{1-f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}}{1-f_{\mathrm{pp}}^{\mathrm{b\rightarrow
e}}}R_{\mathrm{AA}}^{\mathrm{ince}},$ (2) $\displaystyle
R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}=$
$\displaystyle\frac{f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}}{f_{\mathrm{pp}}^{\mathrm{b\rightarrow
e}}}R_{\mathrm{AA}}^{\mathrm{ince}},$ (3)

where the $R_{\mathrm{AA}}^{\mathrm{ince}}$ represents the $R_{\mathrm{AA}}$
of inclusive electrons from heavy flavor hadron decays ($\left|\eta\right|$
$<$ 0.8 at $p_{\mathrm{T}}$ $<$ 3 GeV/$c$ and $\left|\eta\right|$ $<$ 0.6 at
$p_{\mathrm{T}}$ $>$ 3 GeV/$c$) measured by ALICE [26]. Figure 5 shows the
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ and
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ at 0–10% (blue solid squares and
black solid circles) and 30–50% (green open squares and violet open circles)
centrality classes as functions of $p_{\mathrm{T}}$ extracted from Eqs. (2)
and (3). The results of $R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ are
consistent with the ALICE preliminary measurements [47] using a different
impact parameter method shown as red and orange solid stars at both centrality
classes. At each centrality class, clear suppression is observed for both
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ and
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ at $p_{\mathrm{T}}$ $>$ 2.0
GeV/$c$, which indicates that charm and beauty quarks strongly interact with
the hot-dense medium and lose energy. Meanwhile,
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ is less suppressed compared with
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ at $p_{\mathrm{T}}$ $>$ 2.0
GeV/$c$, which is consistent with the mass-dependent energy loss prediction.
The $R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ at 30–50% centrality is
observed to be higher than that at 0–10% at $p_{\mathrm{T}}$ $>$ 2.0 GeV/$c$
and a similar centrality dependence can be observed for
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ at $p_{\mathrm{T}}$ $>$ 4.5
GeV/$c$, which suggests that heavy quarks lose more energy in the interaction
with the medium created in more central collisions, as seen in Fig. 5.
Theoretical model predictions with the TAMU model [48, 49, 50] and Parton-
Hadron-String Dynamics (PHSD) [51] are also shown for comparison. TAMU model
calculations agree with the $R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$
results but seem to overestimate $R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$.
PHSD model calculations are consistent with
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ better but underestimate
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ for both centrality classes.

Figure 5: The nuclear modification factors ($R_{\mathrm{AA}}$) of
$\mathrm{c\rightarrow e}$ and $\mathrm{b\rightarrow e}$ at mid-rapidity
($\left|\eta\right|$ $<$ 0.8 at $p_{\mathrm{T}}$ $<$ 3 GeV/$c$ and
$\left|\eta\right|$ $<$ 0.6 at $p_{\mathrm{T}}$ $>$ 3 GeV/$c$) in Pb+Pb
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. The solid bars represent
the total uncertainties in Pb+Pb collisions and brackets denote those in p+p
collisions. The ALICE preliminary measurements ($\left|\eta\right|$ $<$ 0.8)
[47] together with theoretical results by the TAMU model [48, 49, 50] and by
the PHSD model [51] are shown for comparison with those of this work at 0–10%
(a) and 30–50% (b) centrality classes. The same results are drawn in panel (c)
for the centrality comparison.

From Eqs. (2) and (3), the relative uncertainties of
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ and
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$ can be calculated by

$\displaystyle\delta_{R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}}^{2}=$
$\displaystyle
f_{\mathrm{bc,Pb}}^{2}\delta_{f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}}^{2}+f_{\mathrm{bc,pp}}^{2}\delta_{f_{\mathrm{pp}}^{\mathrm{b\rightarrow
e}}}^{2}+\delta_{R_{\mathrm{AA}}^{\mathrm{ince}}}^{2},$ (4)
$\displaystyle\delta_{R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}}^{2}=$
$\displaystyle\delta_{f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}}^{2}+\delta_{f_{\mathrm{pp}}^{\mathrm{b\rightarrow
e}}}^{2}+\delta_{R_{\mathrm{AA}}^{\mathrm{ince}}}^{2},$ (5)

respectively, where $f_{\mathrm{bc}}=\left.f^{\mathrm{b\rightarrow
e}}\middle/\left(1-f^{\mathrm{b\rightarrow e}}\right)\right.$ in Pb+Pb or p+p
collisions. Table 4 summarizes the uncertainty components of
$R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ and
$R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$.

Table 4: Uncertainty components of $R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$
and $R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$

| $f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ | $f_{\mathrm{pp}}^{\mathrm{b\rightarrow e}}$ | $R_{\mathrm{AA}}^{\mathrm{ince}}$ | total
---|---|---|---|---
centrality | 0–10% | 30–50% | 0–10% | 30–50% | 0–10% | 30–50% | 0–10% | 30–50%
$\delta\left(R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}\right)$ | 13.0%–17.5% | 13.1%–14.9% | 5.0%$-$6.7% | 5.0%$-$6.7% | 10.5%$-$17.4% | 11.1%$-$17.4% | 17.5%–24.0% | 18.3%–23.7%
$\delta\left(R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}\right)$ | 5.0%–13.4% | 6.4%–13.7% | 4.0%–7.6% | 4.0%–7.6% | 10.5%–17.4% | 11.1%–17.4% | 12.8%–20.3% | 14.9%–19.2%

###  2.2 Elliptic Flow $v_{2}$

Similar procedures to Ref. [25] are applied to extract the $v_{2}$ of
$\mathrm{c\rightarrow e}$ and $\mathrm{b\rightarrow e}$ and their
uncertainties:

The latest measurement of prompt D-meson $v_{2}$ in 30–50% central Pb+Pb
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV by ALICE [52] is
parameterized from $p_{\mathrm{T}}$ = 0 to 24.0 GeV/$c$ with

$v_{2}=\frac{p_{0}n}{1+{\mathrm{exp}}\left(\frac{p_{1}-\left.\left(m_{\mathrm{T}}-m_{0}\right)\middle/n\right.}{p_{2}}\right)}-\frac{p_{0}n}{1+{\mathrm{exp}}\left(\frac{p_{1}}{p_{2}}\right)}+p_{3}\left(m_{\mathrm{T}}-m_{0}\right),$
(6)

where $m_{\mathrm{T}}=\sqrt{p_{\mathrm{T}}^{2}+m_{0}^{2}}$ and $m_{0}$ denote
the transverse and rest masses of the particle, respectively, $n$ is the
number of constituent quarks and $p_{i}$ ($i$ = 0, 1, 2, 3) are free
parameters. Meanwhile, the $v_{2}$ of $\mathrm{\Lambda_{c}}$ ($n$ = 3) is also
obtained from Eq. (6) by assuming that it follows the NCQ scaling as D-mesons
($n$ = 2).

The azimuthal angle ($\phi$) distributions of D-mesons and
$\mathrm{\Lambda_{c}}$ in each $p_{\mathrm{T}}$ bin is sampled according to
the input $v_{2}$ with

$\frac{{\mathrm{d}}N}{{\mathrm{d}}\phi}=1+2v_{2}{\mathrm{cos}}(2\phi).$ (7)

Then, semileptonic decay simulations are performed with the $p_{\mathrm{T}}$
spectra and $\phi$ distributions of D-mesons and $\mathrm{\Lambda_{c}}$ as
inputs. The output $\phi$ distributions of decay electrons are fitted by Eq.
(7) in each $p_{\mathrm{T}}$ bin, and the $v_{2}$ of $\mathrm{D\rightarrow e}$
($v_{2}^{\mathrm{D\rightarrow e}}$) and $\mathrm{\Lambda_{c}\rightarrow e}$
($v_{2}^{\mathrm{\Lambda_{c}\rightarrow e}}$) can be obtained. However,
different from the previous work, $\mathrm{J/\psi}$ also exhibits a finite
$v_{2}$ at LHC energy, so the $\mathrm{J/\psi}$ contributions (more than 10%
at 3.0 $<$ $p_{\mathrm{T}}$ $<$ 4.6 GeV/$c$ according to the following
calculation) should also be considered. The latest measurement on
$\mathrm{J/\psi}$ $v_{2}$ only covers the 20–40% centrality at
$\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV [53] and gives poor precision at mid-
rapidity ($\left|y\right|$ $<$ 0.9), so the theoretical prediction mentioned
in Ref. [53] is applied instead as an estimation [32, 54]. Then the $v_{2}$ of
$\mathrm{J/\psi\rightarrow e}$ ($v_{2}^{\mathrm{J/\psi\rightarrow e}}$) is
obtained in the same way mentioned above.

The $v_{2}$ of $\mathrm{c\rightarrow e}$ ($v_{2}^{\mathrm{c\rightarrow e}}$)
and $\mathrm{b\rightarrow e}$ ($v_{2}^{\mathrm{b\rightarrow e}}$) are
calculated with

$\displaystyle v_{2}^{\mathrm{c\rightarrow e}}$
$\displaystyle=w_{\mathrm{D\rightarrow e}}v_{2}^{\mathrm{D\rightarrow
e}}+w_{\mathrm{\Lambda_{c}\rightarrow e}}v_{2}^{\mathrm{\Lambda_{c}\rightarrow
e}}+w_{\mathrm{J/\psi\rightarrow e}}v_{2}^{\mathrm{J/\psi\rightarrow e}},$ (8)
$\displaystyle v_{2}^{\mathrm{b\rightarrow e}}$
$\displaystyle=\frac{v_{2}^{\mathrm{ince}}-\left(1-f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}\right)v_{2}^{\mathrm{c\rightarrow
e}}}{f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}},$ (9)

where $w_{\mathrm{D\rightarrow e}}$, $w_{\mathrm{\Lambda_{c}\rightarrow e}}$
and $w_{\mathrm{J/\psi\rightarrow e}}$ denote the weights by the relative
yields of D-mesons, $\mathrm{\Lambda_{c}}$ and $\mathrm{J/\psi}$ in
$\mathrm{c\rightarrow e}$ calculated in Section 2.1, and
$v_{2}^{\mathrm{ince}}$ denotes the $v_{2}$ of inclusive HFE from the
parameterized average of the ALICE preliminary measurements
($\left|\eta\right|$ $<$ 0.7) in 30–50% 5.02 TeV and 20–40% 2.76 TeV Pb+Pb
collisions [55]. There is no large difference between the
$v_{2}^{\mathrm{ince}}$ measurements at both centrality classes and collision
energies, so they are combined together to help perform a better fit and
extrapolation.

The statistical and systematic uncertainties of measured D-meson $v_{2}$ are
quadratically summed and propagated into the uncertainties of
$v_{2}^{\mathrm{D\rightarrow e}}$ and $v_{2}^{\mathrm{\Lambda_{c}\rightarrow
e}}$ through the decay simulations, and they are correlated as a result. The
absolute uncertainties ($\sigma$) of $v_{2}^{\mathrm{c\rightarrow e}}$ and
$v_{2}^{\mathrm{b\rightarrow e}}$ are calculated with

$\begin{split}&\sigma_{v_{2}^{\mathrm{c\rightarrow
e}}}=\left(w_{\mathrm{D\rightarrow e}}\sigma_{v_{2}^{\mathrm{D\rightarrow
e}}}+w_{\mathrm{\Lambda_{c}\rightarrow
e}}\sigma_{v_{2}^{\mathrm{\Lambda_{c}\rightarrow
e}}}\right)^{2}+w_{\mathrm{J/\psi\rightarrow
e}}^{2}\sigma_{v_{2}^{\mathrm{J/\psi\rightarrow e}}}^{2}\\\
&+\left(\frac{\Delta v_{2}\mathrm{[D,\Lambda_{c}]}}{\mathrm{c\rightarrow
e}}\right)^{2}\left(w_{\mathrm{\Lambda_{c}\rightarrow
e}}^{2}\sigma_{\mathrm{D\rightarrow e}}^{2}+w_{\mathrm{D\rightarrow
e}}^{2}\sigma_{\mathrm{\Lambda_{c}\rightarrow e}}^{2}\right)\\\
&+\left(\frac{\Delta v_{2}\mathrm{[D,J/\psi]}}{\mathrm{c\rightarrow
e}}\right)^{2}\left(w_{\mathrm{J/\psi\rightarrow
e}}^{2}\sigma_{\mathrm{D\rightarrow e}}^{2}+w_{\mathrm{D\rightarrow
e}}^{2}\sigma_{\mathrm{J/\psi\rightarrow e}}^{2}\right)\\\
&+\left(\frac{\Delta v_{2}\mathrm{[J/\psi,\Lambda_{c}]}}{\mathrm{c\rightarrow
e}}\right)^{2}\left(w_{\mathrm{\Lambda_{c}\rightarrow
e}}^{2}\sigma_{\mathrm{J/\psi\rightarrow e}}^{2}+w_{\mathrm{J/\psi\rightarrow
e}}^{2}\sigma_{\mathrm{\Lambda_{c}\rightarrow e}}^{2}\right),\end{split}$ (10)

where $\Delta v_{2}\mathrm{[A,B]}=v_{2}^{\mathrm{A\rightarrow
e}}-v_{2}^{\mathrm{B\rightarrow e}}$, and

$\begin{split}\sigma_{v_{2}^{\mathrm{\mathrm{b\rightarrow
e}}}}=&{f_{\mathrm{1b}}^{2}\sigma_{v_{2}^{\mathrm{ince}}}^{2}+f_{\mathrm{cb}}^{2}\sigma_{v_{2}^{\mathrm{\mathrm{c\rightarrow
e}}}}^{2}}+f_{\mathrm{1b}}^{4}\left(v_{2}^{\mathrm{\mathrm{c\rightarrow
e}}}-v_{2}^{\mathrm{ince}}\right)^{2}\sigma_{f_{\mathrm{PbPb}}^{\mathrm{\mathrm{b\rightarrow
e}}}}^{2},\end{split}$ (11)

where $f_{\mathrm{1b}}=1/f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ and
$f_{\mathrm{cb}}=\left(1-f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow
e}}\right)/f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$. The low values of
$f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ at low $p_{\mathrm{T}}$ result
in large uncertainties of $v_{2}^{\mathrm{b\rightarrow e}}$.

Figure 6: The elliptic flows ($v_{2}$) of $\mathrm{c\rightarrow e}$ and
$\mathrm{b\rightarrow e}$ at mid-rapidity ($\left|\eta\right|$ $<$ 0.8) in
30–50% central Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV. The
major contributions to the uncertainty of $v_{2}^{\mathrm{b\rightarrow e}}$
are from $v_{2}^{\mathrm{ince}}$ (bars), $v_{2}^{\mathrm{c\rightarrow e}}$
(brackets) and $f_{\mathrm{PbPb}}^{\mathrm{b\rightarrow e}}$ (grey bands). In
panel(a), the $v_{2}^{\mathrm{b\rightarrow e}}$ measured in the same collision
system by ALICE [23] and in minimum bias Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 200 GeV [25] by RHIC are shown as orange and green
squares, respectively. In panel(b), the $v_{2}^{\mathrm{b\rightarrow e}}$ with
B-meson $v_{2}$ NCQ scaling assumption are shown as the red dashed curve.
Theoretical predictions of $v_{2}^{\mathrm{c\rightarrow e}}$ and
$v_{2}^{\mathrm{b\rightarrow e}}$ from TAMU [48, 49, 50] and PHSD [51] models
are shown as well.

Figure 6 shows the results of $v_{2}^{\mathrm{c\rightarrow e}}$ and
$v_{2}^{\mathrm{b\rightarrow e}}$ as the blue solid curve with an uncertainty
band and the red solid circles in 30–50% central Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV , respectively. In Fig. 6(a), different
measurements on $v_{2}^{\mathrm{b\rightarrow e}}$ are shown for comparison.
Measurements by ALICE [23] are shown as the orange squares with statistical
(bars) and systematic (brackets) uncertainties, which are consistent with the
results obtained in this work. Results obtained in minimum bias Au+Au
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 200 GeV [25] are shown as green
squares and are consistent with zero at 1 $<$ $p_{\mathrm{T}}$ $<$ 3 GeV/$c$.
While in this work, the $v_{2}^{\mathrm{b\rightarrow e}}$ results in 5.02 TeV
Pb+Pb collisions are observed with a 2.94-$\sigma$ significance
($\chi^{2}/ndf$ = 17.77/5) deviating from zero and with a confidence level of
85% ($\chi^{2}/ndf$ = 6.78/4) to be higher than that in 200 GeV Au+Au
collisions, which might be a possible hint of an energy-dependent behavior of
beauty quarks reacting with the medium created in heavy-ion collisions. At
$p_{\mathrm{T}}$ $>$ 3 GeV/$c$, there is no significant energy dependence seen
with current precision.

In Fig. 6(b), we compare the results of $v_{2}^{\mathrm{c\rightarrow e}}$ and
$v_{2}^{\mathrm{b\rightarrow e}}$ in this work with those by theoretical
predictions. The red dashed curve with an uncertainty band represents the
$v_{2}^{\mathrm{b\rightarrow e}}$ assuming that B-meson $v_{2}$ follows the
NCQ scaling with the same technique as the $v_{2}^{\mathrm{D\rightarrow e}}$
calculation and the decay formfactor in the B-meson rest frame sampled by CLEO
[56] applied. The uncertainty band is calculated from the D-meson $v_{2}$
uncertainties assuming that with removing the mass effect by
$m_{\mathrm{T}}-m_{0}$, the B-meson $v_{2}$ and D-meson $v_{2}$ are the same.
The $v_{2}^{\mathrm{b\rightarrow e}}$ of this work deviates from the red
dashed curve at 3 $<$ $p_{\mathrm{T}}$ $<$ 7 GeV/$c$ with a 4.50-$\sigma$
significance ($\chi^{2}/ndf$ = 31.73/5), and this may indicate that the beauty
quark elliptic flow is smaller than that of light and charm quarks which
follow the NCQ scaling because the beauty quarks may be too heavy to
thermalize and to move with the collective flow of light quarks. The TAMU
model predictions [48, 49, 50] shown as pink dot-dashed curves
($v_{2}^{\mathrm{c\rightarrow e}}$) and violet dot-dot-dashed curves
($v_{2}^{\mathrm{b\rightarrow e}}$) are consistent with results of this work
at $p_{\mathrm{T}}$ $\gtrsim$ 4 GeV/$c$, while they, especially the
$v_{2}^{\mathrm{c\rightarrow e}}$, deviate from data at low $p_{\mathrm{T}}$.
Another theoretical predictions by PHSD model [51] are also shown as the black
long-dashed curve ($v_{2}^{\mathrm{c\rightarrow e}}$) and green dot-dot-dot-
dashed curve ($v_{2}^{\mathrm{b\rightarrow e}}$), which agree with data
better, but the prediction of $v_{2}^{\mathrm{b\rightarrow e}}$ still
underestimates the data. In a previous work, a compilation of calculations of
the dimensionless heavy quark diffusion coefficient, $2\pi TD_{s}$ that
describe the $v_{2}$ and $R_{\mathrm{AA}}$ of D-mesons, is constrained in the
range of 1.5–7 at the critical temperature $T_{c}$ [52]. In this work, since
we only separate the electron contributions from charm and beauty decays, the
charm diffusion coefficient extracted from the electron $R_{\mathrm{AA}}$ and
$v_{2}$ from charm decays should be consistent with that extracted from open
charm hadrons. The conclusion about the charm diffusion coefficient will not
change. However, there is very little discussion on the beauty diffusion
coefficient from theoretical calculations due to absence of the measurement on
open beauty hadrons in heavy-ion collisions. Usually the heavy flavor spatial
diffusion coefficients at zero-momentum reflect the long-wavelength structure
of the QCD medium. Therefore, the theoretical treatment of the spatial
diffusion coefficients for charm and bottom should be the same. However, there
maybe some difference practically between them because the large mass
approximation may not work the same way for charm and bottom. Testing the
universality of heavy quark spatial diffusion coefficients between charm and
bottom is an important next step to constrain the theoretical uncertainties
and provide a deep understanding on the QGP microscopic dynamics. Our results
could provide a new reference for further theoretical calculations. Especially
the electron $R_{\mathrm{AA}}$ and $v_{2}$ from beauty decays for both RHIC
and LHC energies may provide different results for theoretical calculations to
extract the QGP transport parameters with different temperatures.

##  3 Summary

This paper presents an isolation of charm and beauty contributions from heavy
flavor decay electrons in p+p and Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}$
= 5.02 TeV at LHC. The $p_{\mathrm{T}}$ spectra, $R_{\mathrm{AA}}$ at 0–10%
and 30–50% centrality classes and $v_{2}$ at 30–50% centrality of electrons
from charm and beauty decays at mid-rapidity in Pb+Pb collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV are reported.

The electron $R_{\mathrm{AA}}$ from charm
($R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$) and beauty
($R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$) decays are suppressed at
$p_{\mathrm{T}}$ $>$ 2.0 GeV/$c$ in Pb+Pb collisions, respectively, which
indicates that charm and beauty quarks interact with the hot-dense medium and
lose their energy. The $R_{\mathrm{AA}}^{\mathrm{c\rightarrow e}}$ at 2.0 $<$
$p_{\mathrm{T}}$ $<$ 8.0 GeV/$c$ and $R_{\mathrm{AA}}^{\mathrm{b\rightarrow
e}}$ at 4.0 $<$ $p_{\mathrm{T}}$ $<$ 8.0 GeV/$c$ are more suppressed at 0–10%
centrality than at 30–50% centrality, which suggests that heavy quarks lose
more energy in the medium created in more central collisions. However, at low
$p_{\mathrm{T}}$ $<$ 4.0 GeV/$c$, $R_{\mathrm{AA}}^{\mathrm{b\rightarrow e}}$
shows no significant centrality dependence. A less suppression of electron
$R_{\mathrm{AA}}$ from beauty decays compared with that from charm decays in
the whole $p_{\mathrm{T}}$ region from 2.0 to 8.0 GeV/$c$ is observed, which
is consistent with the mass-dependent partonic energy loss mechanism. A non-
zero electron $v_{2}$ from beauty decays ($v_{2}^{\mathrm{b\rightarrow e}}$)
is observed and consistent with the ALICE measurement in the overlapping
$p_{\mathrm{T}}$ region. At low $p_{\mathrm{T}}$ from 1.0 to 3.0 GeV/$c$, a
discrepancy between results in 200 GeV Au+Au collisions and in 5.02 TeV Pb+Pb
collisions is observed with an 85% confidence level, which might be a hint
that beauty quark reacts differently to the two different media created in the
heavy-ion collisions with two very different collision energies. At 3.0
GeV/$c$ $<$ $p_{\mathrm{T}}$ $<$ 7.0 GeV/$c$, $v_{2}^{\mathrm{b\rightarrow
e}}$ deviates from a NCQ scaling hypothesis, which favors that beauty quark is
unlikely thermalized even in such high energy heavy-ion collisions at LHC. The
$R_{\mathrm{AA}}$ and $v_{2}$ results of electrons with charm and beauty
contribution separation could provide a new reference for theoretical models
to extract the beauty diffusion coefficient with further theoretical
calculations.

Acknowledgments: This work was supported by National Natural Science
Foundation of China with Grant No. 11890712 and 12061141008, National Key
Research and Development Program of China with Grant No. 2018YFE0104700 and
2018YFE0205200, Strategic Priority Research Program of Chinese Academy of
Sciences with Grant No. XDB34000000, and Anhui Provincial Natural Science
Foundation with Grant No. 1808085J02.

Dongsheng Li and Fan Si contribute equally to this work as first co-authors.

## References

  * Gyulassy [2004] Gyulassy, M., 2004.  The QGP discovered at RHIC, in: NATO Advanced Study Institute: Structure and Dynamics of Elementary Matter, pp. 159–182.  arXiv:nucl-th/0403032.
  * Bass et al. [1999] Bass, S.A., Gyulassy, M., Stoecker, H., Greiner, W., 1999\.  Signatures of quark gluon plasma formation in high-energy heavy ion collisions: A Critical review.  J. Phys. G 25, R1--R57.  doi:10.1088/0954-3899/25/3/013, arXiv:hep-ph/9810281.
  * Karsch [1995] Karsch, F., 1995.  The Phase transition to the quark gluon plasma: Recent results from lattice calculations.  Nucl. Phys. A 590, 367C--382C.  doi:10.1016/0375-9474(95)00248-Y, arXiv:hep-lat/9503010.
  * Moore and Teaney [2005] Moore, G.D., Teaney, D., 2005\.  How much do heavy quarks thermalize in a heavy ion collision?  Phys. Rev. C 71, 064904\.  doi:10.1103/PhysRevC.71.064904, arXiv:hep-ph/0412346.
  * Tang et al. [2020] Tang, Z.B., Zha, W.M., Zhang, Y.F., 2020.  An experimental review of open heavy flavor and quarkonium production at RHIC.  Nucl. Sci. Tech. 31, 81\.  doi:10.1007/s41365-020-00785-8, arXiv:2105.11656.
  * Luo et al. [2020] Luo, X., Shi, S., Xu, N., Zhang, Y., 2020.  A Study of the Properties of the QCD Phase Diagram in High-Energy Nuclear Collisions.  Particles 3, 278--307.  doi:10.3390/particles3020022, arXiv:2004.00789.
  * Adams et al. [2005] Adams, J., et al. (STAR Collaboration), 2005.  Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions.  Nucl. Phys. A 757, 102--183.  doi:10.1016/j.nuclphysa.2005.03.085, arXiv:nucl-ex/0501009.
  * Adcox et al. [2005] Adcox, K., et al. (PHENIX Collaboration), 2005.  Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration.  Nucl. Phys. A 757, 184--283.  doi:10.1016/j.nuclphysa.2005.03.086, arXiv:nucl-ex/0410003.
  * Muller et al. [2012] Muller, B., Schukraft, J., Wyslouch, B., 2012.  First Results from Pb+Pb collisions at the LHC.  Ann. Rev. Nucl. Part. Sci. 62, 361--386.  doi:10.1146/annurev-nucl-102711-094910, arXiv:1202.3233.
  * Akiba et al. [2015] Akiba, Y., et al., 2015.  The Hot QCD White Paper: Exploring the Phases of QCD at RHIC and the LHC arXiv:1502.02730.
  * Dokshitzer and Kharzeev [2001] Dokshitzer, Y.L., Kharzeev, D.E., 2001\.  Heavy quark colorimetry of QCD matter.  Phys. Lett. B 519, 199--206.  doi:10.1016/S0370-2693(01)01130-3, arXiv:hep-ph/0106202.
  * Djordjevic et al. [2005] Djordjevic, M., Gyulassy, M., Wicks, S., 2005.  The Charm and beauty of RHIC and LHC.  Phys. Rev. Lett. 94, 112301\.  doi:10.1103/PhysRevLett.94.112301, arXiv:hep-ph/0410372.
  * Buzzatti and Gyulassy [2012] Buzzatti, A., Gyulassy, M., 2012\.  Jet Flavor Tomography of Quark Gluon Plasmas at RHIC and LHC.  Phys. Rev. Lett. 108, 022301\.  doi:10.1103/PhysRevLett.108.022301, arXiv:1106.3061.
  * Cao et al. [2018] Cao, S., Luo, T., Qin, G.Y., Wang, X.N., 2018.  Heavy and light flavor jet quenching at RHIC and LHC energies.  Phys. Lett. B 777, 255--259.  doi:10.1016/j.physletb.2017.12.023, arXiv:1703.00822.
  * Adamczyk et al. [2014] Adamczyk, L., et al. (STAR Collaboration), 2014.  Observation of $D^{0}$ Meson Nuclear Modifications in Au+Au Collisions at $\sqrt{s_{NN}}=200$ GeV.  Phys. Rev. Lett. 113, 142301\.  doi:10.1103/PhysRevLett.113.142301, arXiv:1404.6185. [Erratum: Phys.Rev.Lett. 121, 229901 (2018)].
  * Adam et al. [2019] Adam, J., et al. (STAR Collaboration), 2019.  Centrality and transverse momentum dependence of $D^{0}$-meson production at mid-rapidity in Au+Au collisions at ${\sqrt{s_{\rm NN}}=\rm{200\,GeV}}$.  Phys. Rev. C 99, 034908\.  doi:10.1103/PhysRevC.99.034908, arXiv:1812.10224.
  * Abelev et al. [2007] Abelev, B.I., et al. (STAR Collaboration), 2007.  Transverse momentum and centrality dependence of high-$p_{T}$ non-photonic electron suppression in Au+Au collisions at $\sqrt{s_{NN}}=200$\,GeV.  Phys. Rev. Lett. 98, 192301\.  doi:10.1103/PhysRevLett.98.192301, arXiv:nucl-ex/0607012. [Erratum: Phys.Rev.Lett. 106, 159902 (2011)].
  * Adare et al. [2007] Adare, A., et al. (PHENIX Collaboration), 2007.  Energy Loss and Flow of Heavy Quarks in Au+Au Collisions at s(NN)**(1/2) = 200-GeV.  Phys. Rev. Lett. 98, 172301\.  doi:10.1103/PhysRevLett.98.172301, arXiv:nucl-ex/0611018.
  * Acharya et al. [2021] Acharya, S., et al. (ALICE Collaboration), 2021.  Prompt D0, D+, and D∗+ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02 TeV arXiv:2110.09420.
  * Sirunyan et al. [2018] Sirunyan, A.M., et al. (CMS Collaboration), 2018.  Measurement of prompt and nonprompt charmonium suppression in PbPb collisions at 5.02 TeV.  Eur. Phys. J. C 78, 509\.  doi:10.1140/epjc/s10052-018-5950-6, arXiv:1712.08959.
  * Aad et al. [2020] Aad, G., et al. (ATLAS Collaboration), 2020.  Measurement of azimuthal anisotropy of muons from charm and bottom hadrons in Pb+Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV with the ATLAS detector.  Phys. Lett. B 807, 135595\.  doi:10.1016/j.physletb.2020.135595, arXiv:2003.03565.
  * Aad et al. [2021] Aad, G., et al. (ATLAS Collaboration), 2021.  Measurement of the nuclear modification factor for muons from charm and bottom hadrons in Pb+Pb collisions at 5.02 TeV with the ATLAS detector arXiv:2109.00411.
  * Acharya et al. [2021a] Acharya, S., et al. (ALICE Collaboration), 2021a.  Elliptic Flow of Electrons from Beauty-Hadron Decays in Pb-Pb Collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.  Phys. Rev. Lett. 126, 162001\.  doi:10.1103/PhysRevLett.126.162001, arXiv:2005.11130.
  * Acharya et al. [2021b] Acharya, S., et al. (ALICE Collaboration), 2021b.  Transverse-momentum and event-shape dependence of D-meson flow harmonics in Pb–Pb collisions at $\sqrt{s_{NN}}$ = 5.02 TeV.  Phys. Lett. B 813, 136054\.  doi:10.1016/j.physletb.2020.136054, arXiv:2005.11131.
  * Si et al. [2020] Si, F., Chen, X.L., Zhou, L., Zhang, Y.F., Zhang, S.H., Ju, X.Y., Li, X.J., Dong, X., Xu, N., 2020.  Charm and beauty isolation from heavy flavor decay electrons in Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV at RHIC.  Phys. Lett. B 805, 135465\.  doi:10.1016/j.physletb.2020.135465, arXiv:1906.08974.
  * Acharya et al. [2020] Acharya, S., et al. (ALICE Collaboration), 2020.  Measurement of electrons from semileptonic heavy-flavour hadron decays at midrapidity in pp and Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.  Phys. Lett. B 804, 135377\.  doi:10.1016/j.physletb.2020.135377, arXiv:1910.09110.
  * Acharya et al. [2018] Acharya, S., et al. (ALICE Collaboration), 2018.  Measurement of D0, D+, D∗+ and D${}_{s}^{+}$ production in Pb-Pb collisions at $\sqrt{{\mathrm{s}}_{\mathrm{NN}}}=5.02$ TeV.  JHEP 10, 174.  doi:10.1007/JHEP10(2018)174, arXiv:1804.09083.
  * Acharya et al. [2022] Acharya, S., et al. (ALICE Collaboration), 2022.  Measurement of prompt $D_{s}^{+}$-meson production and azimuthal anisotropy in Pb–Pb collisions at $\sqrt{s_{NN}}$=5.02TeV.  Phys. Lett. B 827, 136986\.  doi:10.1016/j.physletb.2022.136986, arXiv:2110.10006.
  * Acharya et al. [2021] Acharya, S., et al. (ALICE Collaboration), 2021.  Constraining hadronization mechanisms with $\rm\Lambda_{\rm c}^{+}$/D0 production ratios in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV arXiv:2112.08156.
  * Zhao and Rapp [2010] Zhao, X., Rapp, R., 2010.  Charmonium in Medium: From Correlators to Experiment.  Phys. Rev. C 82, 064905\.  doi:10.1103/PhysRevC.82.064905, arXiv:1008.5328.
  * Zhao and Rapp [2011] Zhao, X., Rapp, R., 2011.  Medium Modifications and Production of Charmonia at LHC.  Nucl. Phys. A 859, 114--125.  doi:10.1016/j.nuclphysa.2011.05.001, arXiv:1102.2194.
  * Du and Rapp [2015] Du, X., Rapp, R., 2015.  Sequential Regeneration of Charmonia in Heavy-Ion Collisions.  Nucl. Phys. A 943, 147--158.  doi:10.1016/j.nuclphysa.2015.09.006, arXiv:1504.00670.
  * Khachatryan et al. [2017] Khachatryan, V., et al. (CMS Collaboration), 2017.  Charged-particle nuclear modification factors in PbPb and pPb collisions at $\sqrt{s_{\mathrm{N}\;\mathrm{N}}}=5.02$ TeV.  JHEP 04, 039.  doi:10.1007/JHEP04(2017)039, arXiv:1611.01664.
  * Wilk and Wlodarczyk [2000] Wilk, G., Wlodarczyk, Z., 2000\.  On the interpretation of nonextensive parameter q in Tsallis statistics and Levy distributions.  Phys. Rev. Lett. 84, 2770\.  doi:10.1103/PhysRevLett.84.2770, arXiv:hep-ph/9908459.
  * Acharya et al. [2021a] Acharya, S., et al. (ALICE Collaboration), 2021a.  Measurement of beauty and charm production in pp collisions at $\sqrt{s}$ = 5.02 TeV via non-prompt and prompt D mesons.  JHEP 05, 220.  doi:10.1007/JHEP05(2021)220, arXiv:2102.13601.
  * Acharya et al. [2021b] Acharya, S., et al. (ALICE Collaboration), 2021b.  $\Lambda^{+}_{c}$ Production and Baryon-to-Meson Ratios in pp and p-Pb Collisions at $\sqrt{s_{NN}}$=5.02 TeV at the LHC.  Phys. Rev. Lett. 127, 202301\.  doi:10.1103/PhysRevLett.127.202301, arXiv:2011.06078.
  * Acharya et al. [2022] Acharya, S., et al. (ALICE Collaboration), 2022.  Prompt and non-prompt J/$\psi$ production cross sections at midrapidity in proton-proton collisions at $\sqrt{\mathrm{s}}$ = 5.02 and 13 TeV.  JHEP 03, 190.  doi:10.1007/JHEP03(2022)190, arXiv:2108.02523.
  * Aaboud et al. [2018] Aaboud, M., et al. (ATLAS Collaboration), 2018.  Measurement of quarkonium production in proton–lead and proton–proton collisions at $5.02~{}\mathrm{TeV}$ with the ATLAS detector.  Eur. Phys. J. C 78, 171\.  doi:10.1140/epjc/s10052-018-5624-4, arXiv:1709.03089.
  * Sirunyan et al. [2020] Sirunyan, A.M., et al. (CMS Collaboration), 2020.  Production of $\Lambda_{\mathrm{c}}^{+}$ baryons in proton-proton and lead-lead collisions at $\sqrt{s_{\mathrm{NN}}}=$ 5.02 TeV.  Phys. Lett. B 803, 135328\.  doi:10.1016/j.physletb.2020.135328, arXiv:1906.03322.
  * Acharya et al. [2020] Acharya, S., et al. (ALICE Collaboration), 2020.  Centrality and transverse momentum dependence of inclusive J/$\psi$ production at midrapidity in Pb–Pb collisions at sNN=5.02 TeV.  Phys. Lett. B 805, 135434\.  doi:10.1016/j.physletb.2020.135434, arXiv:1910.14404.
  * Acharya et al. [2018] Acharya, S., et al. (ALICE Collaboration), 2018.  Centrality determination in heavy ion collisions.  URL: http://cds.cern.ch/record/2636623.
  * Sjostrand et al. [2006] Sjostrand, T., Mrenna, S., Skands, P.Z., 2006.  PYTHIA 6.4 Physics and Manual.  JHEP 05, 026.  doi:10.1088/1126-6708/2006/05/026, arXiv:hep-ph/0603175.
  * Liu et al. [2006] Liu, H.d., Zhang, Y.f., Zhong, C., Xu, Z.b., 2006\.  Extract charm cross section from its semileptonic decay at RHIC.  Phys. Lett. B 639, 441--446.  doi:10.1016/j.physletb.2006.07.008, arXiv:nucl-ex/0601030.
  * Acharya et al. [2021] Acharya, S., et al. (ALICE Collaboration), 2021.  First measurement of quarkonium polarization in nuclear collisions at the LHC.  Phys. Lett. B 815, 136146\.  doi:10.1016/j.physletb.2021.136146, arXiv:2005.11128.
  * Tanabashi et al. [2018] Tanabashi, M., et al. (Particle Data Group), 2018.  Review of Particle Physics.  Phys. Rev. D 98, 030001\.  doi:10.1103/PhysRevD.98.030001.
  * Cacciari et al. [1998] Cacciari, M., Greco, M., Nason, P., 1998.  The P(T) spectrum in heavy flavor hadroproduction.  JHEP 05, 007.  doi:10.1088/1126-6708/1998/05/007, arXiv:hep-ph/9803400.
  * Park [2021] Park, J. (ALICE Collaboration), 2021\.  Nuclear modification factor of electrons from open beauty-hadron decays in PbPb collisions at $\sqrt{s_{NN}}$=5.02 TeV with ALICE.  PoS HardProbes2020, 034\.  doi:10.22323/1.387.0034.
  * Cao et al. [2013] Cao, S., Qin, G.Y., Bass, S.A., 2013.  Heavy-quark dynamics and hadronization in ultrarelativistic heavy-ion collisions: Collisional versus radiative energy loss.  Phys. Rev. C 88, 044907\.  doi:10.1103/PhysRevC.88.044907, arXiv:1308.0617.
  * Cao et al. [2015] Cao, S., Qin, G.Y., Bass, S.A., 2015.  Energy loss, hadronization and hadronic interactions of heavy flavors in relativistic heavy-ion collisions.  Phys. Rev. C 92, 024907\.  doi:10.1103/PhysRevC.92.024907, arXiv:1505.01413.
  * Li et al. [2020] Li, S.Q., Xing, W.J., Liu, F.L., Cao, S., Qin, G.Y., 2020\.  Heavy flavor quenching and flow: the roles of initial condition, pre-equilibrium evolution, and in-medium interaction.  Chin. Phys. C 44, 114101\.  doi:10.1088/1674-1137/abadee, arXiv:2005.03330.
  * Song et al. [2018] Song, T., Cassing, W., Moreau, P., Bratkovskaya, E., 2018\.  Open charm and dileptons from relativistic heavy-ion collisions.  Phys. Rev. C 97, 064907\.  doi:10.1103/PhysRevC.97.064907, arXiv:1803.02698.
  * Acharya et al. [2018] Acharya, S., et al. (ALICE Collaboration), 2018.  $D$-meson azimuthal anisotropy in midcentral Pb-Pb collisions at $\mathbf{\sqrt{s_{\rm NN}}=5.02}$ TeV.  Phys. Rev. Lett. 120, 102301\.  doi:10.1103/PhysRevLett.120.102301, arXiv:1707.01005.
  * Acharya et al. [2017] Acharya, S., et al. (ALICE Collaboration), 2017.  J/$\psi$ elliptic flow in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV.  Phys. Rev. Lett. 119, 242301\.  doi:10.1103/PhysRevLett.119.242301, arXiv:1709.05260.
  * Zhou et al. [2014] Zhou, K., Xu, N., Xu, Z., Zhuang, P., 2014.  Medium effects on charmonium production at ultrarelativistic energies available at the CERN Large Hadron Collider.  Phys. Rev. C 89, 054911\.  doi:10.1103/PhysRevC.89.054911, arXiv:1401.5845.
  * Moreira De Godoy [2017] Moreira De Godoy, D. (ALICE Collaboration), 2017.  Measurements of the nuclear modification factor and elliptic flow of leptons from heavy-flavour hadron decays in Pb-Pb collisions at $\sqrt{s_{\rm NN}}$ = 2.76 and 5.02 TeV with ALICE.  Nucl. Phys. A 967, 636--639.  doi:10.1016/j.nuclphysa.2017.05.050, arXiv:1705.02800.
  * Jain [1996] Jain, V. (CLEO Collaboration), 1996\.  Recent results from CLEO on charm and bottom hadrons.  Nucl. Phys. B Proc. Suppl. 50, 96--102.  doi:10.1016/0920-5632(96)00372-6, arXiv:hep-ex/9601002.


ENS Paris-Saclay, FR<EMAIL_ADDRESS>CNRS, IRIF, Université
de Paris<EMAIL_ADDRESS>University of Liverpool, UK
totzke@liverpool.ac.ukhttp://orcid.org/0000-0001-5274-8190 P. Totzke
[500]Theory of computation Markov decision processes [300]Theory of
computation Network games John Q. Open and Joan R. Access 2 42nd Conference on
Very Important Topics (CVIT 2016) CVIT 2016 CVIT 2016 December 24–27, 2016
Little Whinging, United Kingdom 42 23

# Controlling a Random Population is -hard

Corto Mascle  Mahsa Shirmohammadi  Patrick Totzke

###### Abstract

Bertrand et al. [1] describe two-player zero-sum games in which one player
tries to achieve a reachability objective in $n$ games (on the same finite
arena) simultaneously by broadcasting actions, and where the opponent has full
control of resolving non-deterministic choices. They show completeness for the
question if such games can be won for every number $n$ of games.

We consider the _almost-sure_ variant in which the opponent randomizes their
actions, and where the player tries to achieve the reachability objective
eventually with probability one. The lower bound construction in [1] does not
directly carry over to this randomized setting.111The construction in Theorem
6.1 would allow spurious wins for the controller by alternating actions init
and reset until an incorrect initialization of the ATM is reached. This must
happen eventually with probability one and can subsequently be exploited by
controller to win directly. In this note we show hardness for the almost-sure
problem by reduction from Countdown Games.

###### keywords:

Markov Decision Processes, Synchronization

###### category:

##  1 Definitions

### Population Control.

Write $\mathcal{M}=(Q,\Sigma,\delta)$ for a Markov Decision Process, where $Q$
and $\Sigma$, are finite sets of _states_ and _actions_ , and
$\delta:Q\times\Sigma\to\mathit{Dist}(Q)$ assigns to each state and action a
probability distribution over states. A _successor_ of $q$ on action $a$ is a
state $p$ with $\delta(q,a)(p)>0$. The $n$-fold synchronized product of
$\mathcal{M}$ is the MDP $\mathcal{M}^{(n)}=(Q^{(n)},\Sigma,\delta)$ whose
states, called _configurations_ here, are $n$-dimensional vectors with
components in $Q$, and $\delta$ is lifted to $Q^{(n)}$ in the natural way:
$\delta({\mathbf{q}},a)({\mathbf{p}})=\prod_{i=0}^{n-1}\delta({\mathbf{q}}(i),a)({\mathbf{p}}(i))$
for all ${\mathbf{q}},{\mathbf{p}}\in Q^{(n)}$ and $a\in\Sigma$. A _strategy_
$\sigma:Q^{(n)}\to\Sigma$ assigns to every configuration ${\mathbf{q}}$ an
action222 We consider here only memoryless deterministic strategies as those
suffice for almost-sure reachability problems [3]. . For every initial state
${\mathbf{q}}\in Q^{(n)}$, such a strategy induces a probability space
$(\Omega,\mathcal{P}_{{\mathbf{q}}}^{\sigma})$ over all infinite sequences in
$\Omega={\mathbf{q}}(Q^{(n)})^{\omega}$ (see [3] for details).

In a configuration ${\mathbf{q}}\in Q^{(n)}$, we say $m$ components _mark_
state $p\in Q$ if the number of different indices $0\leq i<n$ with
${\mathbf{q}}(i)=p$ is equal to $m$. In case $m>0$ we simply call the state
$p$ _marked_ in ${\mathbf{q}}$. Let $\mathbf{{Start}},\mathbf{End}\in Q^{(n)}$
denote the configurations in which all $n$ components mark a designated
initial (or final, resp.) state of $\mathcal{M}$. A configuration
${\mathbf{q}}\in Q^{(n)}$ _can be synchronized_ if there exists a strategy
$\sigma$ such that the probability
$\mathcal{P}_{{\mathbf{q}}}^{\sigma}((Q^{(n)})^{*}\textbf{End})$, of
eventually visiting $\mathbf{End}$, is one. $\mathcal{M}^{(n)}$ _can be
synchronized_ if Start can be synchronized.

Given an MDP $\mathcal{M}$ equipped with initial and final states, the
PopulationControl problem asks whether $\mathcal{M}^{(n)}$ can be synchronized
for every $n\in\mathbb{N}$.

### Countdown Games.

A _Countdown Game_ is given by a directed graph $\mathcal{G}=(V,E)$, where
edges carry positive integer weights, $E\subseteq(V\times\mathbb{N}_{>0}\times
V)$. For an initial pair $(v,c_{0})\in V\times\mathbb{N}$ of a vertex and a
number, two opposing players (Player 1 and 2) alternatingly determine a
sequence of such pairs as follows. In each round, from $(v,c)$, Player 1 picks
a number $d\leq c$ such that $E$ contains at least one edge
$(v,d,v^{\prime})$; then Player 2 picks one such edge and the game continues
from $(v^{\prime},c-d)$. Player 1 wins the game iff the play reaches a pair in
$V\times\\{0\\}$.

CountdownGame is the decision problem which asks if Player 1 has a strategy to
win a given game for a given initial pair $(v_{0},c_{0})$. All constants in
the input are written in binary.

######  Proposition 1.1 (Thm. 4.5 in [2]).

CountdownGame is -complete.

##  2 The Reduction

In order to reduce CountdownGame to PopulationControl we first observe that
the number of turns in a Countdown Game cannot exceed the initial value of the
counter, as the initial counter value decreases at each turn. Thus, if Player
2 has a winning strategy, choosing actions at random yields a positive
probability of applying that strategy, hence a positive probability of
winning. Therefore Player 1 wins the initial game if, and only if, she wins
with probability one against a randomized adversary.

The main idea for our further construction is to require Player 1 to move
components one-by-one away from a waiting state, first into the control graph
of the Countdown Game, and ultimately into the goal. To avoid a loss in the
intermediate phase she needs to win an instance of that game against a
randomizing opponent. This is enforced using a combination of gadgets,
including two binary counters that can effectively test for zero, be set to
specific numbers, and that are set up so that they can decrement at the same
rate. As a result, Player 1 has a winning strategy for the two-player
Countdown Game if, and only if, the controller can synchronize the $n$-fold
product of the constructed MDP for all $n$.

For a given Countdown Game $\mathcal{G}$ with an initial pair $(v_{0},c_{0})$
we construct an MDP $\mathcal{M}$ as follows. We write that action $a$ _takes
state_ $s$ to _successor_ $t$ to mean that $\delta(s,a)(t)>0$. The exact
probability distributions $\delta$ do not matter in our construction so we let
$\delta(s,a)$ be the uniform distribution over such successors.

Whenever action $a$ takes state $s$ only back to itself we say that $s$
_ignores_ $a$. There are states Heaven (the target) and Hell which ignore all
actions. For a given state $s$, an action $a$ is _angelic_ if it takes $s$
only to Heaven, and _daemonic_ if it takes $s$ to Hell. An action $a$ is
_safe_ in a configuration if it is not daemonic for any marked state (in any
gadget).

waitreadywaitwaitwait

$W$$G$$A$$B$waitgo$\Sigma_{\mathcal{G}}$next$\Sigma_{AC}$$\Sigma_{MC}$win

Figure 1: The waiting (on left) and the control gadgets (on right). Edges
labelled by $\Sigma_{X}$ are shorthand for several edges, one for each action
in $\Sigma_{X}$. All but the depicted actions are daemonic.

Besides the special states Heaven and Hell, $\mathcal{M}$ contains several
gadgets described below.

### Waiting.

The waiting gadget has two states Wait and Ready which react to the action
wait as depicted in Figure 1 (left). Whenever a configuration marks one of
these states, a strategy that continuously plays wait will almost-surely reach
a configuration in which exactly one component marks Ready.

A special action go (to indicate successful isolation of one component) takes
Ready to the initial state $v_{0}$ of the game $\mathcal{G}$. All other
actions (in gadgets described below) are ignored.

### Game.

The game $\mathcal{G}=(G,E)$ is directly interpreted as MDP: For every edge
$(s,d,s^{\prime})\in E$ there is an action $(s,d)$ which takes $s$ to
$s^{\prime}$ and which is daemonic for all states $s^{\prime}\neq s$.

The action win is is angelic for every state of $G$. All other actions are
ignored.

### Binary Counters.

A ($k$-bit) Counter consists of states (i:j) for all $0\leq i<k$ and
$j\in\\{0,1\\}$. For every bit $i$ there is a decrement action (deci) which

  * •

takes (j:0) only to (j:1) for all $0\leq j<i$,

  * •

takes (i:1) only to (i:0),

  * •

is daemonic for (i:0), and

  * •

is ignored by all (j:l), for all $i<j$ and $l\in\\{0,1\\}$.

We say that a configuration _holds_ the number $c<2^{k}$ in this counter if it
marks those states that represent the binary expansion of $c$: for all $0\leq
i\leq k-1$, state (i:j) is marked iff the $i$th bit in the binary expansion of
$c$ is $j$. An action $a$ _sets_ the counter to number $d$ if for all $0\leq
i<k$, it takes (i:0) to only (i:j) where $j\in\\{0,1\\}$ is the $i$th bit in
the binary expansion of $d$, and is daemonic for all (i:1) (to ensure that the
counter can only be set if it holds $0$). Observe that if a counter holds $c$
then there is a unique maximal sequence of safe decrement actions, that has
length $c$ and after which the counter holds $0$.

Additionally, for every bit $i$ the gadget has an error action errori, which
is daemonic for (i:0) and (i:1), and angelic for every other state (of
$\mathcal{M}$). These actions can be used to quickly synchronize any
configuration in which the counter is not correctly initialized, i.e., does
not hold a number. See Figure 2 for a depiction of a $4$-bit counter.

The MDP $\mathcal{M}$ will contain two distinct counter gadgets. A main
counter $MC$ has $\log_{2}(n_{0})$ bits to hold possible counter values of the
Countdown Game. An auxiliary counter $AC$ has $\log_{2}(d_{\texttt{max}})$
many bits to hold the largest edge weight $d_{\texttt{max}}$ in $\mathcal{G}$.
These have distinct sets of states and actions, so for clarity, we write
${C}.{x}$ to refer to state (or action) $x$ in gadget $C$. We connect some new
actions to these two counters as follows.

  * •

The action go sets $MC$ to $n_{0}$; this ensures that $MC$ holds $n_{0}$ when
starting to simulate $\mathcal{G}$.

  * •

The action win is daemonic for every state ${MC}.{\texttt{({{i}:{1}})}}$. This
enforces that the $MC$ must hold $0$ when a strategy claims Player 1 wins
$\mathcal{G}$.

  * •

Any action $(v,d)\in\Sigma_{\mathcal{G}}$ sets $AC$ to $d$;

  * •

The action next is daemonic for every state ${AC}.{\texttt{({{i}:{1}})}}$.
This enforces that a strategy must first count down from $d$ to $0$ before it
can simulate the next move in $\mathcal{G}$.

0:01:02:03:00:11:12:13:1dec1,dec2,dec3
dec0dec2,dec3dec1dec0dec0dec2dec0,dec1dec0,dec1dec3
dec3dec0,dec1,dec2dec0,dec1,dec2 Figure 2: A (4-bit) Binary Counter. Not
displayed are edges labelled by (deci) that make the respective actions
daemonic for state (i:0), and error actions errori, which are daemonic for
(i:0) and (i:1), for all bits $i\in\\{0,1,2,3\\}$.

### Control.

The control gadget will enforce that a synchronizing strategy proposes actions
in a proper order; see Figure 1. It consists of states $W,G,A,B$, and contains
actions of all gadgets above (including go, win, $\texttt{next})$ and a new
error action, which is angelic for all states except $W$, for which it is
daemonic. All omitted edges in Figure 1 are daemonic.

### Start/End.

To complete the construction of $\mathcal{M}$, we introduce an initial state
Init and actions start and end. The action start takes Init to Wait (Waiting
gadget), $W$ (Control gadget), and all (i:0) states of counters $AC$ and $MC$.
It is daemonic for every other state.

The action end is daemonic for Wait and Ready, and angelic for every other
state in $\mathcal{M}$.

######  Lemma 2.1.

$\mathcal{M}^{(n)}$ is synchronizable for all $n\in\mathbb{N}$ iff Player 1
wins $\mathcal{G}$.

######  Proof 2.2.

Suppose Player 1 wins the game $\mathcal{G}$. Fix $n$. Recall that in
$\mathcal{M}^{(n)}$ all components of the initial configuration mark Init. A
synchronizing strategy proceeds as follows:

  * •

Play start to initialize the Waiting and Control gadgets, and to set $AC$ and
$MC$ to $0$. If any of the gadgets is not correctly initialized afterwards,
play the respective error action to win directly. For instance, if $W$ is
unmarked, play error to synchronize.

  * •

Reduce the number of components marking Wait one by one until a configuration
is reached in which Wait is not marked. Once this is true, play end to
synchronize.

  * •

To reduce the number of components marking Wait, isolate one of them, and move
it to Heaven by simulating the Countdown Game:

    1. 1.

Play wait until only a single component marks Ready, then play go. This will
mark $v_{0}$ in the game gadget and sets $MC$ to $n_{0}$. Recall that
$(v_{0},n_{0})$ is the initial pair of $\mathcal{G}$.

    2. 2.

Simulate rounds of the game $\mathcal{G}$: assume state $v$ in the game gadget
is marked and the counter $MC$ holds $c$, then let $d$ be the the number
Player $1$ plays to win from the pair $(v,c)$ in $\mathcal{G}$. Play $(v,d)$.
This action will set $AC$ to $d$. Alternate between (safe) decrement actions
in $AC$ and $AB$ until they hold $0$ and $c-d$, respectively. Play next.

    3. 3.

The above simulation of rounds in $\mathcal{G}$ is repeated until both $AC$
and $AB$ hold $0$, by assumption that Player 1 wins $\mathcal{G}$ this is
possible. At this point it is safe to play win.

Conversely, assume that Player 1 cannot win $\mathcal{G}$. Suppose that after
the (only possible) initial move start, all gadgets are correctly initialized.
Clearly, for every $n$, this event has strictly positive probability. We argue
that no strategy can synchronize such a configuration. Indeed, a successful
strategy had to play a sequence in $\texttt{wait}^{*}\cdot\texttt{go}$ first,
followed by actions in
$(\Sigma_{\mathcal{G}}\cdot(\Sigma_{AC}\cdot\Sigma_{MC}\cdot\texttt{next})^{*})^{*}$,
by construction of the control gadget. If after playing go, more than one
component mark $v_{0}$, there is a non-zero chance that these will diverge,
making subsequent actions in $\Sigma_{\mathcal{G}}$ unsafe. If exactly one
component marks $v_{0}$ then the second sequence of actions (assuming all
actions are safe) corresponds to a play of $\mathcal{G}$. This inevitably
leads to a configuration in which counter $MC$ holds $0$ and the control
enforces that the next action is in $\Sigma_{MC}$. But any such action will be
daemonic for some state in $MC$ and thus not be safe. We conclude that every
strategy will lead to a configuration that at least one component marks Hell
and thus cannot be synchronized.

Our claim follows immediately from Propositions 1.1 and 2.1.

######  Theorem 2.3.

PopulationControl is -hard.

## References

  * [1] Nathalie Bertrand, Miheer Dewaskar, Blaise Genest, Hugo Gimbert, and Adwait Amit Godbole.  Controlling a population.  Logical Methods in Computer Science, 2019.
  * [2] Marcin Jurdziński, François Laroussinie, and Jeremy Sproston.  Model checking probabilistic timed automata with one or two clocks.  In Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 2007.
  * [3] Martin L. Puterman.  Markov Decision Processes: Discrete Stochastic Dynamic Programming.  1st edition, 1994.