††thanks: Corresponding author

# Baryon density dependence of viscosities of the quark-gluon plasma at
hadronization

Zhidong Yang  School of Physics and Astronomy, Shanghai Key Laboratory for
Particle Physics and Cosmology, and Key Laboratory for Particle Astrophysics
and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai 200240, China
Yifeng Sun  School of Physics and Astronomy, Shanghai Key Laboratory for
Particle Physics and Cosmology, and Key Laboratory for Particle Astrophysics
and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai 200240, China
Lie-Wen Chen<EMAIL_ADDRESS>School of Physics and Astronomy, Shanghai Key
Laboratory for Particle Physics and Cosmology, and Key Laboratory for Particle
Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University, Shanghai
200240, China

###### Abstract

The $\phi$ meson and $\Omega$ baryon provide unique probes of the properties
of the quark-gluon plasma (QGP) at hadronization in relativistic heavy-ion
collisions. Using the quark recombination model with the quark phase-space
information parameterized in a viscous blastwave, we perform Bayesian
inference of the shear and bulk viscosities of the QGP at hadronization with a
temperature of $T\sim 160$ MeV by analyzing the $\phi$ and $\Omega$ data in
Au+Au collisions at $\sqrt{s_{\rm NN}}=$ 19.6-200 GeV and Pb+Pb collisions at
$\sqrt{s_{\rm NN}}=$ 2.76 TeV, corresponding to a baryon chemical potential
variation from $\mu_{B}\approx 0$ (at $\sqrt{s_{\rm NN}}=2.76$ TeV) to $200$
MeV (at $\sqrt{s_{\rm NN}}=19.6$ GeV). We find that the shear viscosity to
enthalpy ratio $\eta T/(\epsilon+P)$ of the QGP at hadronization decreases as
$\mu_{B}$ increases, with $\eta T/(\epsilon+P)\approx 0.18$ at $\mu_{B}=0$ and
$\eta T/(\epsilon+P)\approx 0.08$ at $\mu_{B}=200$ MeV, while the
corresponding specific bulk viscosity is essentially constant with $\zeta
T/(\epsilon+P)=0.02\sim 0.04$ for $\mu_{B}<200$ MeV. Our results suggest that
the QGP at hadronization ($T\sim 160$ MeV) with finite baryon density is more
close to perfect fluid than that with zero baryon density.

Heavy Ion Collisions, Quark-gluon Plasma, Shear viscosity, Bulk Viscosity

_Introduction.—_ Lattice Quantum Chromodynamics (QCD) calculations predict a
transition from ordinary hadronic matter to a new state of matter that
consists of deconfined quarks and gluons, called quark gluon plasma (QGP) [1].
The QGP is believed to have existed in the early universe $10^{-6}s$ after the
big bang and can be created in relativistic heavy-ion collisions (HICs) at the
BNL Relativistic Heavy Ion Collider (RHIC) and the CERN Large Hadron Collider
(LHC). Exploring the QCD phase diagram as well as the transport properties of
the QGP is one of the most fundamental problems in high-energy nuclear physics
(See, e.g., Ref. [2]). It is known that at zero baryon chemical potential
($\mu_{B}=0$), the transition between QGP and hadronic matter is a smooth
crossover [3, 4]. Further calculations from lattice QCD suggest that the
crossover line extends up to $\mu_{B}\sim$ 250-300 MeV [5]. However, it is not
yet known if there exists a critical point where the crossover transforms into
a first-order phase transition at higher baryon densities. In fact, the main
goal of the beam energy scan (BES) program at RHIC is to investigate the phase
diagram of QCD and locate the critical point [6].

One of the most significant discoveries made at RHIC and LHC is that the QGP
behaves like a near-perfect fluid, characterized by an exceptionally small
shear viscosity to entropy density ratio $\eta/s$, close to the universal
lower bound $1/4\pi$ based on the anti-de Sitter/conformal field theory
(AdS/CFT) correspondence [7]. This is surprising since the QGP was initially
expected to be a weakly interacting gas of quarks and gluons but turned out to
be a strongly coupled fluid. This discovery has attracted intense interest in
the transport properties of the QGP fluid, which are closely related to the
underlying strong interactions between quarks and gluons. In recent years,
theoretical calculations on the shear ($\eta$) and bulk ($\zeta$) viscosity
have been extensively explored [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19],
usually considering $\mu_{B}=0$ and focusing on the temperature dependence of
viscosities.

The viscosities of the QGP have significant effects on the final observables
of HICs [20, 21], allowing us to constrain their values with experimental
data. Early studies employing viscous hydrodynamics usually assumed a constant
$\eta/s$ over the entire evolution and found $\eta/s=0.08\sim 0.2$ [22, 23,
24, 25]. Recent researches using Bayesian statistical analysis with multi-
stage models that integrate initial conditions, viscous hydrodynamics and
hadronic transport obtained constraints on the temperature dependence of the
shear and bulk viscosities of the baryon-free QGP [26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36]. Overall, most recent analyses have yielded consistent results
and found $\eta/s\approx 0.16$ at at pseudo-critical temperature $T_{\rm
pc}\approx 160$ MeV for the baryon-free QGP [33, 36].

Studies have also found that the viscosities of QCD matter depend on baryon
chemical potential [37, 38, 18, 39, 40, 19]. Hybrid models that incorporate
hadronic transport and viscous hydrodynamics show that different values of
effective shear viscosity are required to describe the data at different
collision energies by assuming a constant value of $\eta/s$ for each collision
energy [41, 42]. The baryon chemical potential dependence of $\eta/s$ is
explored in Refs. [43, 44, 45]. In general, due to large uncertainties in the
initial conditions and equation of state, hydrodynamics simulations at finite
$\mu_{B}$ are under development [46, 47, 43, 48, 45] and a quantitative
estimate for QGP’s viscosities at finite $\mu_{B}$ is still challenging.

In this work, we present a novel approach to constrain the shear and bulk
viscosities of QGP at finite $\mu_{B}$ with a temperature of $T\sim 160$ MeV.
In a recent work [49], we proposed using $\phi$ meson and $\Omega$ baryon to
explore the viscosities of the baryon-free QGP at hadronization. Here we
generalize the approach to collisions at lower energies and study the
viscosities of QGP at finite $\mu_{B}$. In our approach, hadrons are produced
through quark recombination [50, 51, 52, 53, 54, 55] with the phase-space
distribution of quarks at hadronization parameterized in a viscous blastwave
[20, 56, 57, 58, 59] which includes non-equilibrium deformations of thermal
distributions due to shear and bulk stresses. The viscous effects for the QGP
at hadronization are then imported into $\phi$ and $\Omega$ through the
recombination process. Since the $\phi$ and $\Omega$ have relatively small
hadronic interaction cross sections [60], they thus carry direct information
of QGP at hadronization with negligible hadronic effects [60, 61, 62, 63, 64,
65, 66, 67, 68]. We find that the QGP at $T\sim 160$ MeV with finite baryon
density is more close to perfect fluid than that with zero baryon density. It
should be pointed out that blastwave models obtain their parameters through
fits to data and are independent of initial conditions and equation of state,
thus providing a complementary way to hydrodynamic simulations [57, 58].

_Theoretical model.—_ Quark recombination or coalescence models were initially
proposed to explain the baryon-over-meson enhancement and valence quark number
scaling observed in RHIC Au+Au collisions [50, 51, 52, 53]. In a recent work
[49], we introduce viscous corrections into quark recombination, and here give
a brief overview of the formalism. In the following $\mathbf{r}$ is the
3-space position, $\mathbf{p}$ is the 3-momentum and $m$ is particle mass. The
4-momentum of hadrons are denoted as $p^{\mu}=(E,~{}\mathbf{p})$ with
$E=\sqrt{m^{2}+p^{2}}$. Following Refs. [53, 52, 54], the momentum
distribution of mesons is given by

$\displaystyle E\frac{dN_{M}}{d^{3}\bf p}$ $\displaystyle=$ $\displaystyle
C_{M}\int_{\Sigma}\frac{p^{\mu}\cdot
d\mathbf{\sigma_{\mu}}}{(2\pi)^{3}}\int_{0}^{1}dx_{1}dx_{2}\Phi_{M}(x_{1},x_{2})$
(1) $\displaystyle\times f_{a}({\bf r},x_{1}\mathbf{p})f_{b}({\bf
r},x_{2}\mathbf{p}).\ $

where $C_{M}$ is the spin degeneracy factor of a given meson species, $\Sigma$
is the hypersurface of hadronization, $\varPhi_{M}$ is the effective wave
function squared of mesons, $x_{1,2}$ are light cone coordinates defined as
${\bf p}_{1,2}=x_{1,2}\bf p$, and $f_{a,b}$ are the parton phase-space
distributions. The $\varPhi_{M}$ is parameterized as Gaussian type

$\Phi_{M}=\frac{2}{\sqrt{2\pi}\sigma_{M}}\exp(-\frac{(x_{1}-x_{a})^{2}+(x_{2}-x_{b})^{2}}{\sigma_{M}^{2}})\delta(x_{1}+x_{2}-1)$
(2)

where $\sigma_{M}$ is the variance, $x_{a,b}=m_{1,2}/(m_{1}+m_{2})$ are the
peak values, and $m_{1,2}$ are the masses of the constituent partons. Similar
expression can be derived for baryons.

The quark phase-space distribution is parameterized in a viscous blastwave
[57, 58, 59], based on the Retiere and Lisa blastwave [69]. The quark
distribution is given by

$f(r,p)=f_{0}(r,p)+\delta f_{\rm{shear}}(r,p)+\delta f_{\rm{bulk}}(r,p)$ (3)

where $f_{0}$ is the equilibrium Bose/Fermi distribution depending on the flow
field $u^{\mu}$, particle momentum $p^{\mu}$, the local temperature $T$ and
chemical potentials $\mu_{i}=b_{i}\mu_{B}+s_{i}\mu_{S}+q_{i}\mu_{Q}$ with
baryon number $b_{i}$, strangeness $s_{i}$ and electric charge $q_{i}$ (we
assume $\mu_{Q}=0$ in this work).

$f_{0}(r,p)=\frac{1}{e^{(u^{\mu}p_{\mu}-\mu_{i})/T}\mp 1}$ (4)

$\delta f_{\rm{shear}}$ and $\delta f_{\rm{bulk}}$ denote corrections from the
shear and bulk viscosities, respectively. For the shear viscous corrections,
we use the Grad’s method [70, 71]

$\delta
f_{\rm{shear}}=\frac{1}{2T^{2}}\frac{p_{\mu}p_{\nu}}{\epsilon+P}\pi^{\mu\nu}f_{0}(1\pm
f_{0})$ (5)

where $\epsilon$ is the energy density, $P$ is the pressure, $\pi^{\mu\nu}$ is
the shear stress tensor and $+(-)$ for bosons (fermions). In the Navier-Stokes
approximation $\pi^{\mu\nu}=2\eta\sigma^{\mu\nu}$ where $\eta$ is the shear
viscosity and $\sigma^{\mu\nu}$ is the shear gradient tensor defined as
$\sigma^{\mu\nu}=\frac{1}{2}\left(\nabla^{\mu}u^{\nu}+\nabla^{\nu}u^{\mu}\right)-\frac{1}{3}\Delta^{\mu\nu}\nabla_{\lambda}u^{\lambda}$
with flow field $u^{\mu}$, $\nabla^{\mu}=\Delta^{\mu\nu}\partial_{\nu}$ and
$\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$. For the bulk viscous corrections,
we use the 14-moment approximation [72, 73]

$\delta f_{\rm bulk}=-f_{0}(1\pm
f_{0})\Pi\frac{\tau_{\Pi}}{\zeta}\left[\frac{1}{3}\frac{m^{2}}{T}\frac{1}{p^{\mu}u_{\mu}}+\frac{p^{\mu}u_{\mu}}{T}\left(c_{s}^{2}-\frac{1}{3}\right)\right]$
(6)

where $\zeta$ is the bulk viscosity, $\Pi$ is the bulk viscous pressure and
$\tau_{\Pi}$ is the bulk relaxation time. At the first order approximation,
one has $\Pi=-\zeta\partial_{\mu}u^{\mu}$. The expression for
$\frac{\tau_{\Pi}}{\zeta}$ is given in [72]. We note that the fluidity of a
system at finite chemical potential should be evaluated by the ratio of shear
viscosity over the enthalpy multiplied by the temperature, $\eta
T/(\epsilon+P)$ [74], where $\epsilon+P=Ts+\mu_{B}n_{B}$. When $\mu_{B}=0$,
this quantity reduces to $\eta/s$.

Let us now describe the blastwave parameterization for the flow field
$u^{\mu}$. Here $R_{x,y}$ are the semi axes of the fireball at freeze-out,
$\rho=\sqrt{x^{2}/R_{x}^{2}+y^{2}/R_{y}^{2}}$ is the reduced radius,
$\eta_{s}=\frac{1}{2}\ln\frac{t+z}{t-z}$ is the space-time rapidity. The
hypersurface is assumed to be constant $\tau=\sqrt{t^{2}-z^{2}}$. The flow
field is parameterized as

$\displaystyle u^{\mu}$ $\displaystyle=$
$\displaystyle(\cosh\eta_{s}\cosh\eta_{T},\sinh\eta_{T}\cos\phi_{b},$ (7)
$\displaystyle\qquad\sinh\eta_{T}\sin\phi_{b},\sinh\eta_{s}\cosh\eta_{T})$

where $\eta_{T}$ is the transverse flow rapidity and $\phi_{b}$ is the
azimuthal angle of $u^{\mu}$ in the transverse plane. $\eta_{T}$ is given by
the transverse velocity $v_{T}=\tanh\eta_{T}$ with

$v_{T}=\rho^{n}(\alpha_{0}+\alpha_{2}\cos 2\phi_{b})$ (8)

where $\alpha_{0}$ is the average surface velocity, $\alpha_{2}$ is an
elliptic deformation of the flow field and $n$ is a power term. In this work,
we use a linear expression and set $n=1$. The transverse flow vector is chosen
to be perpendicular to the elliptic surface at $\rho=1$. The ratio
$R_{y}/R_{x}$ significantly influences elliptical flow, so we choose
$R_{y}/R_{x}$ as a fit parameter and constrain $R_{x}$, $R_{y}$ and $\tau$ by
adding a simple geometric estimate
$R_{x}\approx(R_{0}-b/2)+0.65\tau(\alpha_{0}+\alpha_{2})$ where $R_{0}$ is the
radius of the colliding nucleus and $b$ is the impact parameter. The values of
$b$ used for each centrality bin are based on Glauber Monte Carlo simulations
for related experiments [75, 76].

It is worth to mention that the viscous blastwave offers a simplified
representation of the flow field and freeze-out hypersurface, serving as an
approximate snapshot of a viscous hydrodynamic system at a fixed time [20, 56,
57, 58, 59]. The dissipative effects before freeze-out are incorporated into
the parameterized flow field $u^{\mu}$. Consequently, the viscous blastwave
carries information on the viscosities of the fluid at a specific time, e.g.,
the QGP at hadronization in our present work.

Table 1:  Experimental data of $\phi$ and $\Omega$ used in our analysis.
Values in parentheses are centralities of $\Omega$ when they are different
from $\phi$.

$\sqrt{s_{NN}}$(GeV) | Centrality
---|---
19.6,39 | $v_{2}(p_{T})$ | 0-10%, 10-40%, 40-80% [77, 78, 79]
$\frac{dN}{d^{2}p_{T}dy}$ | 0-10%, 20-30%(20-40%), 40-60% [80, 81]
54.4,62.4 | $v_{2}(p_{T})$ | 0-10%, 10-40%, 40-80% [77, 82]
$\frac{dN}{d^{2}p_{T}dy}$ | 0-20%, 20-40%, 40-60% [83, 84]
200 | $v_{2}(p_{T})$ | 0-30%, 30-80% [85]
$\frac{dN}{d^{2}p_{T}dy}$ | 10-20%(0-5%), 40-50%(40-60%) [86, 87]
2760 | $v_{2}(p_{T})$ | 10-20%, 20-30%, 30-40%, 40-50% [88, 89]
$\frac{dN}{d^{2}p_{T}dy}(\phi)$
$\frac{dN}{d^{2}p_{T}dy}(\Omega)$ | 10-20%, 20-40%, 40-60% [90]

_Experimental data and fit parameters.—_ We utilize the transverse-momentum
($p_{T}$) spectra and elliptic flows $v_{2}$ of $\phi$ and $\Omega$ as our
observables. The data we used are from the STAR collaboration, covering Au+Au
collisions at 19.6, 39, 54.4, 62.4 and 200 GeV [86, 87, 83, 84, 85, 77, 80,
81, 82, 78, 79], and the ALICE collaboration for Pb+Pb collisions at
$\sqrt{s_{NN}}$=2.76 TeV [90, 88, 89] around mid-rapidity, as listed in Tab.
1. Due to data availability, the centrality bins for spectra are slightly
different from that of $v_{2}$. For Au+Au at 19.6, 39, 54.4 and 62.4 GeV, we
use data of $\Omega^{-}$. For Au+Au at 200 GeV and Pb+Pb at 2.76 TeV, we use
data of $\Omega^{-}+\bar{\Omega}^{+}$. For Au+Au collisions at
$\sqrt{s_{NN}}=$ 7.7,11.5 and 27 GeV, the current data of $\phi$ and $\Omega$
have very few points and large uncertainties, so we will not use them in our
analysis. As for the spectra of $\phi$ and $\Omega$ at $\sqrt{s_{NN}}=$ 54.4
GeV, there is currently no available data and we use the spectra from
$\sqrt{s_{NN}}=$ 62.4 GeV instead.

For each collision energy, we perform a combined analysis of the available
centrality bins. The fitted $p_{T}$ ranges for $\phi$ and $\Omega$ are given
in Tab. 2. We have checked that the correction $\delta f(r,p)$ is small for
both $\phi$ and $\Omega$ with the above fit ranges, i.e., less than $20\%$ of
$f_{0}$ for the majority of points (very few points going up to $40\%$ of
$f_{0}$), which is much smaller than the commonly adopted upper bound $\sim 1$
[30, 20], ensuring the applicability of the viscous corrections.

The temperature $T$ at hadronization is taken from Ref. [91], assuming its
value is close to the chemical freeze-out temperature $T_{ch}$. The baryon
chemical potentials $\mu_{B}$ are energy-dependent and their values are also
taken from Ref. [91]. Besides, the strangeness chemical potentials $\mu_{S}$
are obtained by extrapolating the results from Ref. [81]. The values of
$(T,\mu_{B},\mu_{S})$ used in our calculation are listed in Tab. 2. To fit the
yields of $\phi$ and $\Omega$ and to determine the value of freeze-out time
$\tau$, we introduce a fugacity factor $\gamma_{s,\bar{s}}$ and assume
$\gamma_{s}=\gamma_{\bar{s}}$. We set $\gamma_{s}=0.65$ for 19.6-62.4 GeV and
$\gamma_{s}=0.8$ for 200, 2760 GeV. Regarding other constants, we specify the
hadron wave function variances as $\sigma_{M}=0.3$ and $\sigma_{B}=0.1$. We
use sound speed squared $c^{2}_{s}=0.15$ (see Eq. (6)) for the QGP at
hadronization [92, 93], quark mass $m_{s}=500$ MeV, spin degeneracy factor
$C_{M}=3$ for $\mathrm{\phi}$ and $C_{B}=4$ for $\Omega^{-}$.

Table 2:  Values of temperature ($T$), baryon chemical potential ($\mu_{B}$), strangeness chemical potential ($\mu_{S}$) and fit ranges for $\phi$ and $\Omega$ observables used in quark recombination for different collision energies. $\sqrt{s_{NN}}$ (GeV) | 19.6 | 39 | 54.4 | 62.4 | 200 | 2760
---|---|---|---|---|---|---
$T$ (MeV) | 155 | 157.5 | 158.5 | 158.5 | 160 | 160
$\mu_{B}$ (MeV) | 197 | 107 | 79 | 69 | 22 | 0
$\mu_{S}$ (MeV) | 46 | 29 | 21 | 18 | 8 | 0
$p_{T}$-range (GeV/$c$) | $\phi$ | $<2.3$ | $<2.4$ | $<2.6$ | $<2.7$ | $<2.6$ | $<2.8$
$\Omega$ | $<3.2$ | $<3.0$ | $<3.9$ | $<3.8$ | $<4.0$ | $<3.8$

The parameters left in our model are ($\tau$, $\alpha_{0}$, $\alpha_{2}$,
$R_{y}/R_{x}$, $\eta T/(\epsilon+P)$, $\zeta T/(\epsilon+P)$), which can be
determined by fitting experimental data. For each centrality bin at each
collision energy, the fluid has unique values for ($\tau$, $\alpha_{0}$,
$\alpha_{2}$, $R_{y}/R_{x}$) and shared values for ($\eta T/(\epsilon+P)$,
$\zeta T/(\epsilon+P)$) and thus we have 3$\times$4+2=14 parameters for
$\sqrt{s_{NN}}$= 19.6-62.4 GeV, 2$\times$4+2=10 parameters for
$\sqrt{s_{NN}}$= 200 GeV and 4$\times$4+2=18 parameters for $\sqrt{s_{NN}}$=
2.76 TeV.

_Bayesian method.—_ To determine the above parameters, we employ the Bayesian
analysis package from the Models and Data Analysis Initiative (MADAI) project
[94, 27]. The MADAI package includes a Gaussian process emulator and a
Bayesian analysis tool. Further information is available in Refs. [27, 30]. We
adopt a uniform prior distribution for all model parameters. For instance, for
10-40% centrality bin at $\sqrt{s_{NN}}$= 19.6 GeV, we set prior ranges
5.8-7.8 fm/$c$ for $\tau$, 0.44-0.6$c$ for $\alpha_{0}$, 0.012-0.042$c$ for
$\alpha_{2}$, 1.12-1.32 for $R_{y}/R_{x}$ and obtain posterior values
($\tau,\alpha_{0},\alpha_{2},R_{y}/R_{x}$)=(6.8 fm/$c$, 0.53$c$,
0.03$c$,1.24). Besides, we set prior ranges 0-0.2 for $\eta T/(\epsilon+P)$,
0-0.12 for $\zeta T/(\epsilon+P)$ at 19.6 GeV and obtain $\eta
T/(\epsilon+P)=0.076$ and $\zeta T/(\epsilon+P)=0.035$. The same procedure is
applied to other centralities and energies.

After setting prior ranges for each parameter, we generate a set of training
points within the parameter space and calculate all fitted observables at each
training point. The MADAI package then builds a Gaussian process emulator,
which can estimate the observables for random parameter values. Finally a
Markov Chain Monte Carlo (MCMC) provides a likelihood analysis and gives the
maximum likelihood or best-fit parameters. Here for each collision energy we
use N=500 training points. To validate proper functioning of the Bayesian
analysis, we perform a closure test and confirm the Bayesian framework
correctly reproduces model parameters within reasonable uncertainties. The
likelihood analysis has used ${\rm N}_{\star}=2\times 10^{6}$ predicted points
to search for the best-fit parameters, which is sufficient for MCMC to
converge.

Figure 1:  Transverse-momentum spectra and elliptic flows $v_{2}$ of $\phi$
mesons and $\Omega$ baryons (selected centrality bins). For Au+Au at 19.6-62.4
GeV, we use data of $\Omega^{-}$. For Au+Au at 200 GeV and Pb+Pb at 2.76 TeV,
we use data of $\Omega^{-}+\bar{\Omega}^{+}$. Solid lines are recombination
calculation using the best-fit parameters. The data from STAR [86, 87, 83, 84,
85, 77, 80, 81, 82, 78, 79] and ALICE [90, 88, 89] are included for
comparison.

_Results and discussions.—_ Using the data and parameters discussed earlier,
we perform a model-to-data comparison with MADAI package and obtain the best-
fit parameters, which are defined as the mean value given by the maximum
likelihood analysis. With the best-fit parameters provided by Bayesian
analysis, we can calculate transverse momentum spectra and elliptic flows of
$\phi$ and $\Omega$ and compare with experimental data. Fig. 1 shows our
results for $p_{T}$ spectra and $v_{2}$ of $\phi$ and $\Omega$ in selected
centralities at different collision energies. As seen from Fig. 1, our
calculations describe data rather well.

Fig. 2 shows our Bayesian inference of the $\eta T/(\epsilon+P)$ (a) and
$\zeta T/(\epsilon+P)$ (b) at 68.3% confidence level (C.L.) for the QGP at
hadronization or $T\sim 160$ MeV with different $\mu_{B}$. The range of baryon
chemical potentials is between $\mu_{B}=0$ and $\mu_{B}=200$ MeV, which
corresponds to collison energy varying from 2.76 TeV (most left) to 19.6 GeV
(most right) [91]. For the purpose of comparison, in Fig. 2 we also include
results from other approaches, i.e., Chapman-Enskog theory (Chap-Ensk)[37] for
$\eta$, hadron resonance gas model (HRG) [38] and holographic model (Holo)
[19] for both $\eta$ and $\zeta$. In these approaches, $\eta$ and $\zeta$ are
caculated as a function of temperature with different $\mu_{B}$. Here we take
their values at T=160 MeV for $\mu_{B}=0$ and 300 MeV (T=150 MeV with
$\mu_{B}=0$ and 500 MeV for Chap-Ensk). Note that the results from Chap-Ensk
and HRG are for hadronic matter.

Figure 2: Baryon chemical potential dependence of shear (a) and bulk (b)
viscosities for QGP/hadronic matter at hadronization (see text for details).

One sees from Fig. 2(a) that the shear viscosity has a significant dependency
on baryon chemical potential and decreases with $\mu_{B}$, which suggests that
the QGP with finite baryon chemical potential is more close to perfect fluid
than that with zero baryon chemical potential. We note that this trend of
shear viscosity, i.e., $\eta T/(\epsilon+P)$ decreases as $\mu_{B}$ increases,
aligns with observations in Chap-Ensk [37], HRG [38, 40] and holographic model
[19], but seems to be different from the findings in hybrid models [41, 42,
45], where a constant $\eta/s$ [41, 42] or temperature-independent $\eta
T/(\epsilon+P)$ [45] was assumed for each collision energy and thus the
effects of varying $T$ during the dynamical evolution are neglected. As found
in Refs. [37, 38], the behavior of $\eta T/(\epsilon+P)$ is primarily
attributed to the rapid increase in entropy density $s$ and very slow increase
in $\eta$ with $\mu_{B}$. We would like to emphasize that at $\mu_{B}=0$, the
most recent Bayesian statistical analysis obtained $\eta/s\approx 0.16$ at 160
MeV [33, 36], which is consistent with our result but at variance with [41].

On the other hand, Fig. 2(b) indicates rather small values for $\zeta$ at
hadronization for $\mu_{B}<200$ MeV. In particular, we find $\zeta
T/(\epsilon+P)$ is essentially constant within uncertainty with $\zeta
T/(\epsilon+P)=0.02\sim 0.04$ for $\mu_{B}<200$ MeV. Interestingly, our
results are quantitatively consistent with HRG and holographic model. However,
due to the accuracy of our results, we cannot draw any conclusions about the
trend for $\mu_{B}$ dependence of $\zeta$. We note that HRG gives an increase
of $\zeta T/(\epsilon+P)$ with $\mu_{B}$, while the holographic model predicts
an opposite behavior.

_Conclusions._ — We have performed Bayesian inference of the shear and bulk
viscosities of the QGP at hadronization (with $T\sim 160$ MeV) by analyzing
the $\phi$ and $\Omega$ data in Au+Au collisions at $\sqrt{s_{\rm NN}}=$
19.6-200 GeV and Pb+Pb collisions at $\sqrt{s_{\rm NN}}=$ 2.76 TeV, based on
the quark recombination model with the quark phase-space information
parameterized in a viscous blastwave. We find that the shear viscosity to
enthalpy ratio $\eta T/(\epsilon+P)$ of the QGP at hadronization decreases as
$\mu_{B}$ increases, while the corresponding specific bulk viscosity is
essentially constant for $\mu_{B}<200$ MeV, suggesting that the QGP at $T\sim
160$ MeV with finite baryon density is more close to perfect fluid than that
with zero baryon density.

Our work provides valuable reference for future theoretical calculations as
well as parameterization of viscosities in hydro simulations at finite
$\mu_{B}$. Our work is also useful for exploring the dynamics of binary
neutron star mergers, given that quark matter is expected to exist in the
cores of massive neutron stars and a transition from nuclear to quark matter
may happen [95]. Furthermore, dramatic changes in transport properties may be
considered as a signal of phase transition, and a thorough understanding of
the baryon density dependence of viscosities of QCD matter is of great
significance for the exploration of phase transition [40]. Our study
represents a step forward in achieving this objective, especially if high
quality $\phi$ and $\Omega$ data in Au+Au collisions at $\sqrt{s_{NN}}=$ 7.7
and 11.5 GeV or even lower energies would be provided at RHIC in future.

_Acknowledgements._ — The authors would like to thank Rainer J. Fries, Defu
Hou, Jie Pu and Kaijia Sun for useful discussions. This work was supported in
part by the National Natural Science Foundation of China under Grant Nos.
12205182, 12235010, 11625521 and 12375124, and the National SKA Program of
China No. 2020SKA0120300. Y.S. thanks the sponsorship from Yangyang
Development Fund.

## References

  * Fodor and Katz [2004] Z. Fodor and S. D. Katz, Critical point of QCD at finite T and mu, lattice results for physical quark masses, JHEP 04, 050, arXiv:hep-lat/0402006 .
  * Shen [2021] C. Shen, Studying QGP with flow: A theory overview, Nucl. Phys. A 1005, 121788 (2021), arXiv:2001.11858 [nucl-th] .
  * Bhattacharya  _et al._ [2014] T. Bhattacharya _et al._ , QCD Phase Transition with Chiral Quarks and Physical Quark Masses, Phys. Rev. Lett. 113, 082001 (2014), arXiv:1402.5175 [hep-lat] .
  * Borsanyi  _et al._ [2020] S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, QCD Crossover at Finite Chemical Potential from Lattice Simulations, Phys. Rev. Lett. 125, 052001 (2020), arXiv:2002.02821 [hep-lat] .
  * Borsányi  _et al._ [2021] S. Borsányi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pásztor, C. Ratti, and K. K. Szabó, Lattice QCD equation of state at finite chemical potential from an alternative expansion scheme, Phys. Rev. Lett. 126, 232001 (2021), arXiv:2102.06660 [hep-lat] .
  * An  _et al._ [2022] X. An _et al._ , The BEST framework for the search for the QCD critical point and the chiral magnetic effect, Nucl. Phys. A 1017, 122343 (2022), arXiv:2108.13867 [nucl-th] .
  * Kovtun  _et al._ [2005] P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94, 111601 (2005).
  * Meyer [2007] H. B. Meyer, A Calculation of the shear viscosity in SU(3) gluodynamics, Phys. Rev. D 76, 101701 (2007), arXiv:0704.1801 [hep-lat] .
  * Meyer [2008] H. B. Meyer, A Calculation of the bulk viscosity in SU(3) gluodynamics, Phys. Rev. Lett. 100, 162001 (2008), arXiv:0710.3717 [hep-lat] .
  * Meyer [2009] H. B. Meyer, Transport properties of the quark-gluon plasma from lattice QCD, Nucl. Phys. A 830, 641C (2009), arXiv:0907.4095 [hep-lat] .
  * Mages  _et al._ [2015] S. W. Mages, S. Borsányi, Z. Fodor, A. Schäfer, and K. Szabó, Shear Viscosity from Lattice QCD, PoS LATTICE2014, 232 (2015).
  * Astrakhantsev  _et al._ [2017] N. Astrakhantsev, V. Braguta, and A. Kotov, Temperature dependence of shear viscosity of $SU(3)$–gluodynamics within lattice simulation, JHEP 04, 101, arXiv:1701.02266 [hep-lat] .
  * Astrakhantsev  _et al._ [2018] N. Y. Astrakhantsev, V. V. Braguta, and A. Y. Kotov, Temperature dependence of the bulk viscosity within lattice simulation of $SU(3)$ gluodynamics, Phys. Rev. D 98, 054515 (2018), arXiv:1804.02382 [hep-lat] .
  * Borsányi  _et al._ [2018] S. Borsányi, Z. Fodor, M. Giordano, S. D. Katz, A. Pasztor, C. Ratti, A. Schafer, K. K. Szabo, and B. C. Tóth, High statistics lattice study of stress tensor correlators in pure $SU(3)$ gauge theory, Phys. Rev. D 98, 014512 (2018), arXiv:1802.07718 [hep-lat] .
  * Christiansen  _et al._ [2015] N. Christiansen, M. Haas, J. M. Pawlowski, and N. Strodthoff, Transport Coefficients in Yang–Mills Theory and QCD, Phys. Rev. Lett. 115, 112002 (2015), arXiv:1411.7986 [hep-ph] .
  * Liu and Rapp [2020] S. Y. F. Liu and R. Rapp, Spectral and transport properties of quark–gluon plasma in a nonperturbative approach, Eur. Phys. J. A 56, 44 (2020), arXiv:1612.09138 [nucl-th] .
  * Ghiglieri  _et al._ [2018] J. Ghiglieri, G. D. Moore, and D. Teaney, QCD Shear Viscosity at (almost) NLO, JHEP 03, 179, arXiv:1802.09535 [hep-ph] .
  * Rougemont  _et al._ [2017] R. Rougemont, R. Critelli, J. Noronha-Hostler, J. Noronha, and C. Ratti, Dynamical versus equilibrium properties of the QCD phase transition: A holographic perspective, Phys. Rev. D 96, 014032 (2017), arXiv:1704.05558 [hep-ph] .
  * Grefa  _et al._ [2022] J. Grefa, M. Hippert, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, Transport coefficients of the quark-gluon plasma at the critical point and across the first-order line, Phys. Rev. D 106, 034024 (2022), arXiv:2203.00139 [nucl-th] .
  * Teaney [2003] D. Teaney, The Effects of viscosity on spectra, elliptic flow, and HBT radii, Phys. Rev. C 68, 034913 (2003), arXiv:nucl-th/0301099 .
  * Ryu  _et al._ [2015] S. Ryu, J. F. Paquet, C. Shen, G. S. Denicol, B. Schenke, S. Jeon, and C. Gale, Importance of the Bulk Viscosity of QCD in Ultrarelativistic Heavy-Ion Collisions, Phys. Rev. Lett. 115, 132301 (2015), arXiv:1502.01675 [nucl-th] .
  * Romatschke and Romatschke [2007] P. Romatschke and U. Romatschke, Viscosity Information from Relativistic Nuclear Collisions: How Perfect is the Fluid Observed at RHIC?, Phys. Rev. Lett. 99, 172301 (2007), arXiv:0706.1522 [nucl-th] .
  * Schenke  _et al._ [2011] B. Schenke, S. Jeon, and C. Gale, Elliptic and triangular flow in event-by-event (3+1)D viscous hydrodynamics, Phys. Rev. Lett. 106, 042301 (2011), arXiv:1009.3244 [hep-ph] .
  * Gale  _et al._ [2013] C. Gale, S. Jeon, B. Schenke, P. Tribedy, and R. Venugopalan, Event-by-event anisotropic flow in heavy-ion collisions from combined Yang-Mills and viscous fluid dynamics, Phys. Rev. Lett. 110, 012302 (2013), arXiv:1209.6330 [nucl-th] .
  * Niemi  _et al._ [2016] H. Niemi, K. J. Eskola, and R. Paatelainen, Event-by-event fluctuations in a perturbative QCD + saturation + hydrodynamics model: Determining QCD matter shear viscosity in ultrarelativistic heavy-ion collisions, Phys. Rev. C 93, 024907 (2016), arXiv:1505.02677 [hep-ph] .
  * Bernhard  _et al._ [2015] J. E. Bernhard, P. W. Marcy, C. E. Coleman-Smith, S. Huzurbazar, R. L. Wolpert, and S. A. Bass, Quantifying properties of hot and dense QCD matter through systematic model-to-data comparison, Phys. Rev. C 91, 054910 (2015), arXiv:1502.00339 [nucl-th] .
  * Bernhard  _et al._ [2016] J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu, and U. Heinz, Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and quark-gluon plasma medium, Phys. Rev. C 94, 024907 (2016), arXiv:1605.03954 [nucl-th] .
  * Bernhard  _et al._ [2019] J. E. Bernhard, J. S. Moreland, and S. A. Bass, Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma, Nature Phys. 15, 1113 (2019).
  * Everett  _et al._ [2021a] D. Everett _et al._ (JETSCAPE), Phenomenological constraints on the transport properties of QCD matter with data-driven model averaging, Phys. Rev. Lett. 126, 242301 (2021a), arXiv:2010.03928 [hep-ph] .
  * Everett  _et al._ [2021b] D. Everett _et al._ (JETSCAPE), Multisystem Bayesian constraints on the transport coefficients of QCD matter, Phys. Rev. C 103, 054904 (2021b), arXiv:2011.01430 [hep-ph] .
  * Nijs  _et al._ [2021a] G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Transverse Momentum Differential Global Analysis of Heavy-Ion Collisions, Phys. Rev. Lett. 126, 202301 (2021a), arXiv:2010.15130 [nucl-th] .
  * Nijs  _et al._ [2021b] G. Nijs, W. van der Schee, U. Gürsoy, and R. Snellings, Bayesian analysis of heavy ion collisions with the heavy ion computational framework Trajectum, Phys. Rev. C 103, 054909 (2021b), arXiv:2010.15134 [nucl-th] .
  * Nijs and van der Schee [2022] G. Nijs and W. van der Schee, Hadronic Nucleus-Nucleus Cross Section and the Nucleon Size, Phys. Rev. Lett. 129, 232301 (2022), arXiv:2206.13522 [nucl-th] .
  * Parkkila  _et al._ [2021] J. E. Parkkila, A. Onnerstad, and D. J. Kim, Bayesian estimation of the specific shear and bulk viscosity of the quark-gluon plasma with additional flow harmonic observables, Phys. Rev. C 104, 054904 (2021), arXiv:2106.05019 [hep-ph] .
  * Parkkila  _et al._ [2022] J. E. Parkkila, A. Onnerstad, S. F. Taghavi, C. Mordasini, A. Bilandzic, M. Virta, and D. J. Kim, New constraints for QCD matter from improved Bayesian parameter estimation in heavy-ion collisions at LHC, Phys. Lett. B 835, 137485 (2022), arXiv:2111.08145 [hep-ph] .
  * Heffernan  _et al._ [2023] M. R. Heffernan, C. Gale, S. Jeon, and J.-F. Paquet, Bayesian quantification of strongly-interacting matter with color glass condensate initial conditions, (2023), arXiv:2302.09478 [nucl-th] .
  * Denicol  _et al._ [2013] G. S. Denicol, C. Gale, S. Jeon, and J. Noronha, Fluid behavior of a baryon-rich hadron resonance gas, Phys. Rev. C 88, 064901 (2013), arXiv:1308.1923 [nucl-th] .
  * Kadam and Mishra [2015] G. P. Kadam and H. Mishra, Dissipative properties of hot and dense hadronic matter in an excluded-volume hadron resonance gas model, Phys. Rev. C 92, 035203 (2015), arXiv:1506.04613 [hep-ph] .
  * Soloveva  _et al._ [2021] O. Soloveva, D. Fuseau, J. Aichelin, and E. Bratkovskaya, Shear viscosity and electric conductivity of a hot and dense QGP with a chiral phase transition, Phys. Rev. C 103, 054901 (2021), arXiv:2011.03505 [nucl-th] .
  * McLaughlin  _et al._ [2022] E. McLaughlin, J. Rose, T. Dore, P. Parotto, C. Ratti, and J. Noronha-Hostler, Building a testable shear viscosity across the QCD phase diagram, Phys. Rev. C 105, 024903 (2022), arXiv:2103.02090 [nucl-th] .
  * Karpenko  _et al._ [2015] I. A. Karpenko, P. Huovinen, H. Petersen, and M. Bleicher, Estimation of the shear viscosity at finite net-baryon density from $A+A$ collision data at $\sqrt{s_{\mathrm{NN}}}=7.7-200$ GeV, Phys. Rev. C 91, 064901 (2015), arXiv:1502.01978 [nucl-th] .
  * Auvinen  _et al._ [2018] J. Auvinen, J. E. Bernhard, S. A. Bass, and I. Karpenko, Investigating the collision energy dependence of $\eta$/s in the beam energy scan at the BNL Relativistic Heavy Ion Collider using Bayesian statistics, Phys. Rev. C 97, 044905 (2018), arXiv:1706.03666 [hep-ph] .
  * Shen and Alzhrani [2020] C. Shen and S. Alzhrani, Collision-geometry-based 3D initial condition for relativistic heavy-ion collisions, Phys. Rev. C 102, 014909 (2020), arXiv:2003.05852 [nucl-th] .
  * Götz and Elfner [2022] N. Götz and H. Elfner, Temperature and net baryochemical potential dependence of $\eta$/s in a hybrid approach, Phys. Rev. C 106, 054904 (2022), arXiv:2207.05778 [hep-ph] .
  * Shen  _et al._ [2023] C. Shen, B. Schenke, and W. Zhao, Viscosities of the Baryon-Rich Quark-Gluon Plasma from Beam Energy Scan Data, (2023), arXiv:2310.10787 [nucl-th] .
  * Akamatsu  _et al._ [2018] Y. Akamatsu, M. Asakawa, T. Hirano, M. Kitazawa, K. Morita, K. Murase, Y. Nara, C. Nonaka, and A. Ohnishi, Dynamically integrated transport approach for heavy-ion collisions at high baryon density, Phys. Rev. C 98, 024909 (2018), arXiv:1805.09024 [nucl-th] .
  * Du and Heinz [2020] L. Du and U. Heinz, (3+1)-dimensional dissipative relativistic fluid dynamics at non-zero net baryon density, Comput. Phys. Commun. 251, 107090 (2020), arXiv:1906.11181 [nucl-th] .
  * Wu  _et al._ [2022] X.-Y. Wu, G.-Y. Qin, L.-G. Pang, and X.-N. Wang, (3+1)-D viscous hydrodynamics at finite net baryon density: Identified particle spectra, anisotropic flows, and flow fluctuations across energies relevant to the beam-energy scan at RHIC, Phys. Rev. C 105, 034909 (2022), arXiv:2107.04949 [hep-ph] .
  * Yang and Chen [2023] Z. Yang and L.-W. Chen, Bayesian inference of the specific shear and bulk viscosities of the quark-gluon plasma at crossover from $\phi$ and $\Omega$ observables, Phys. Rev. C 107, 064910 (2023), arXiv:2207.13534 [nucl-th] .
  * Greco  _et al._ [2003a] V. Greco, C. M. Ko, and P. Levai, Parton coalescence and anti-proton / pion anomaly at RHIC, Phys. Rev. Lett. 90, 202302 (2003a), arXiv:0301093 .
  * Greco  _et al._ [2003b] V. Greco, C. M. Ko, and P. Levai, Parton coalescence at RHIC, Phys. Rev. C 68, 034904 (2003b), arXiv:0305024 [nucl-th] .
  * Fries  _et al._ [2003a] R. J. Fries, B. Muller, C. Nonaka, and S. A. Bass, Hadronization in heavy ion collisions: Recombination and fragmentation of partons, Phys. Rev. Lett. 90, 202303 (2003a), arXiv:0301087 [nucl-th] .
  * Fries  _et al._ [2003b] R. J. Fries, B. Muller, C. Nonaka, and S. A. Bass, Hadron production in heavy ion collisions: Fragmentation and recombination from a dense parton phase, Phys. Rev. C 68, 044902 (2003b), arXiv:0306027 [nucl-th] .
  * Fries  _et al._ [2008] R. J. Fries, V. Greco, and P. Sorensen, Coalescence Models For Hadron Formation From Quark Gluon Plasma, Ann. Rev. Nucl. Part. Sci. 58, 177 (2008), arXiv:0807.4939 [nucl-th] .
  * He  _et al._ [2010] M. He, R. J. Fries, and R. Rapp, Scaling of Elliptic Flow, Recombination and Sequential Freeze-Out of Hadrons in Heavy-Ion Collisions, Phys. Rev. C 82, 034907 (2010), arXiv:1006.1111 [nucl-th] .
  * Jaiswal and Koch [2015] A. Jaiswal and V. Koch, A viscous blast-wave model for relativistic heavy-ion collisions, (2015), arXiv:1508.05878 [nucl-th] .
  * Yang and Fries [2017] Z. Yang and R. J. Fries, A Blast Wave Model With Viscous Corrections, J. Phys. Conf. Ser. 832, 012056 (2017), arXiv:1612.05629 [nucl-th] .
  * Yang and Fries [2020] Z. Yang and R. J. Fries, Parameterizing Smooth Viscous Fluid Dynamics With a Viscous Blast Wave, (2020), arXiv:2007.11777 [nucl-th] .
  * Yang and Fries [2022] Z. Yang and R. J. Fries, Shear stress tensor and specific shear viscosity of hot hadron gas in nuclear collisions, Phys. Rev. C 105, 014910 (2022), arXiv:1807.03410 [nucl-th] .
  * Shor [1985] A. Shor, Phi meson production as a probe of the quark-gluon plasma, Phys. Rev. Lett. 54, 1122 (1985).
  * van Hecke  _et al._ [1998] H. van Hecke, H. Sorge, and N. Xu, Evidence of early multistrange hadron freezeout in high-energy nuclear collisions, Phys. Rev. Lett. 81, 5764 (1998), arXiv:9804035 .
  * Chen and Ko [2006] L.-W. Chen and C. M. Ko, phi and omega production from relativistic heavy ion collisions in a dynamical quark coalescence model, Phys. Rev. C 73, 044903 (2006), arXiv:nucl-th/0602025 .
  * Chen  _et al._ [2008] J. H. Chen, F. Jin, D. Gangadharan, X. Z. Cai, H. Z. Huang, and Y. G. Ma, Parton Distributions at Hadronization from Bulk Dense Matter Produced at RHIC, Phys. Rev. C 78, 034907 (2008), arXiv:0801.2265 [nucl-th] .
  * Auvinen  _et al._ [2017] J. Auvinen, K. Redlich, and S. A. Bass, Multi-strange hadrons and the precision extraction of QGP properties in the RHIC-BES domain, J. Phys. Conf. Ser. 779, 012045 (2017), arXiv:1610.00589 [nucl-th] .
  * Hwa and Zhu [2017] R. C. Hwa and L. Zhu, Phenomenological Implications of the $p_{T}$ Spectra of $\phi$ and $\Omega$ produced at LHC and RHIC, J. Phys. Conf. Ser. 779, 012049 (2017), arXiv:1609.06365 [nucl-ex] .
  * Ye  _et al._ [2017] Y. J. Ye, J. H. Chen, Y. G. Ma, S. Zhang, and C. Zhong, $\Omega$ and $\phi$ in Au $+$ Au collisions at $\sqrt{s_{NN}}=$ 200 and 11.5 GeV from a multiphase transport model, Chin. Phys. C 41, 084101 (2017), arXiv:1706.00894 [nucl-th] .
  * Pu  _et al._ [2018] J. Pu, K.-J. Sun, and L.-W. Chen, Extracting strange quark freeze-out information in Pb+Pb collisions at $\sqrt{s_{NN}}$=2.76 TeV from $\phi$ and $\Omega$ production, Phys. Rev. C 98, 064905 (2018), arXiv:1808.04053 [nucl-th] .
  * Song  _et al._ [2020] J. Song, F.-l. Shao, and Z.-t. Liang, Quark number scaling of $p_{T}$ spectra for $\Omega$ and $\phi$ in relativistic heavy-ion collisions, Phys. Rev. C 102, 014911 (2020), arXiv:1911.01152 [nucl-th] .
  * Retiere and Lisa [2004] F. Retiere and M. A. Lisa, Observable implications of geometrical and dynamical aspects of freeze out in heavy ion collisions, Phys. Rev. C 70, 044907 (2004), arXiv:0312024 [nucl-th] .
  * Song and Heinz [2008] H. Song and U. W. Heinz, Causal viscous hydrodynamics in 2+1 dimensions for relativistic heavy-ion collisions, Phys. Rev. C 77, 064901 (2008), arXiv:0712.3715 [nucl-th] .
  * Damodaran  _et al._ [2020] M. Damodaran, D. Molnar, G. G. Barnaföldi, D. Berényi, and M. F. Nagy-Egri, Improved single-particle phase-space distributions for viscous fluid dynamic models of relativistic heavy ion collisions, Phys. Rev. C 102, 014907 (2020), arXiv:1707.00793 [nucl-th] .
  * Denicol  _et al._ [2014] G. S. Denicol, S. Jeon, and C. Gale, Transport Coefficients of Bulk Viscous Pressure in the 14-moment approximation, Phys. Rev. C 90, 024912 (2014), arXiv:1403.0962 [nucl-th] .
  * Paquet  _et al._ [2016] J.-F. Paquet, C. Shen, G. S. Denicol, M. Luzum, B. Schenke, S. Jeon, and C. Gale, Production of photons in relativistic heavy-ion collisions, Phys. Rev. C 93, 044906 (2016), arXiv:1509.06738 [hep-ph] .
  * Liao and Koch [2010] J. Liao and V. Koch, On the Fluidity and Super-Criticality of the QCD matter at RHIC, Phys. Rev. C 81, 014902 (2010), arXiv:0909.3105 [hep-ph] .
  * Abelev  _et al._ [2013] B. Abelev _et al._ (ALICE), Centrality determination of Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV with ALICE, Phys. Rev. C 88, 044909 (2013), arXiv:1301.4361 [nucl-ex] .
  * Adamczewski-Musch  _et al._ [2018] J. Adamczewski-Musch _et al._ (HADES), Centrality determination of Au + Au collisions at 1.23A GeV with HADES, Eur. Phys. J. A 54, 85 (2018), arXiv:1712.07993 [nucl-ex] .
  * Adamczyk  _et al._ [2016a] L. Adamczyk _et al._ (STAR), Centrality dependence of identified particle elliptic flow in relativistic heavy ion collisions at $\sqrt{s_{NN}}$=7.7–62.4 GeV, Phys. Rev. C 93, 014907 (2016a), arXiv:1509.08397 [nucl-ex] .
  * Liu [2023] L.-K. Liu (STAR), Elliptic and triangular flow of (multi-)strange hadrons and phi mesons in BES-II energies at STAR, EPJ Web Conf. 276, 01018 (2023), arXiv:2208.14604 [nucl-ex] .
  * Dixit [2022] P. Dixit (STAR), Anisotropic flow of (multi-)strange hadrons in Au+Au collisions at BES-II energies., PoS ICHEP2022, 496 (2022), arXiv:2211.12973 [nucl-ex] .
  * Adamczyk  _et al._ [2016b] L. Adamczyk _et al._ (STAR), Probing parton dynamics of QCD matter with $\Omega$ and $\phi$ production, Phys. Rev. C 93, 021903 (2016b), arXiv:1506.07605 [nucl-ex] .
  * Adam  _et al._ [2020] J. Adam _et al._ (STAR), Strange hadron production in Au+Au collisions at $\sqrt{s_{{}_{\mathrm{NN}}}}$ = 7.7, 11.5, 19.6, 27, and 39 GeV, Phys. Rev. C 102, 034909 (2020), arXiv:1906.03732 [nucl-ex] .
  * Abdallah  _et al._ [2023] M. Abdallah _et al._ (STAR), Azimuthal anisotropy measurement of (multi)strange hadrons in Au+Au collisions at $\sqrt{S_{NN}}=54.4GeV$, Phys. Rev. C 107, 024912 (2023), arXiv:2205.11073 [nucl-ex] .
  * Abelev  _et al._ [2009] B. I. Abelev _et al._ (STAR), Measurements of phi meson production in relativistic heavy-ion collisions at RHIC, Phys. Rev. C 79, 064903 (2009), arXiv:0809.4737 [nucl-ex] .
  * Aggarwal  _et al._ [2011] M. M. Aggarwal _et al._ (STAR), Strange and Multi-strange Particle Production in Au+Au Collisions at $\sqrt{s_{NN}}$ = 62.4 GeV, Phys. Rev. C 83, 024901 (2011), [Erratum: Phys.Rev.C 107, 049903 (2023)], arXiv:1010.0142 [nucl-ex] .
  * Adamczyk  _et al._ [2016c] L. Adamczyk _et al._ (STAR), Centrality and transverse momentum dependence of elliptic flow of multistrange hadrons and $\phi$ meson in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV, Phys. Rev. Lett. 116, 062301 (2016c), arXiv:1507.05247 [nucl-ex] .
  * Adams  _et al._ [2007] J. Adams _et al._ (STAR), Scaling Properties of Hyperon Production in Au+Au Collisions at s**(1/2) = 200-GeV, Phys. Rev. Lett. 98, 062301 (2007), arXiv:nucl-ex/0606014 .
  * Abelev  _et al._ [2007] B. I. Abelev _et al._ (STAR), Partonic flow and phi-meson production in Au + Au collisions at s(NN)**(1/2) = 200-GeV, Phys. Rev. Lett. 99, 112301 (2007), arXiv:nucl-ex/0703033 .
  * Abelev  _et al._ [2015] B. B. Abelev _et al._ (ALICE), Elliptic flow of identified hadrons in Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, JHEP 06, 190, arXiv:1405.4632 [nucl-ex] .
  * Adam  _et al._ [2017] J. Adam _et al._ (ALICE), K${}^{*}(892)^{0}$ and $\phi(1020)$ meson production at high transverse momentum in pp and Pb-Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ = 2.76 TeV, Phys. Rev. C 95, 064606 (2017), arXiv:1702.00555 [nucl-ex] .
  * Abelev  _et al._ [2014] B. B. Abelev _et al._ (ALICE), Multi-strange baryon production at mid-rapidity in Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV, Phys. Lett. B 728, 216 (2014), [Erratum: Phys.Lett.B 734, 409–410 (2014)], arXiv:1307.5543 [nucl-ex] .
  * Andronic  _et al._ [2018] A. Andronic, P. Braun-Munzinger, K. Redlich, and J. Stachel, Decoding the phase structure of QCD via particle production at high energy, Nature 561, 321 (2018), arXiv:1710.09425 [nucl-th] .
  * Huovinen and Petreczky [2010] P. Huovinen and P. Petreczky, QCD Equation of State and Hadron Resonance Gas, Nucl. Phys. A 837, 26 (2010), arXiv:0912.2541 [hep-ph] .
  * Monnai  _et al._ [2019] A. Monnai, B. Schenke, and C. Shen, Equation of state at finite densities for QCD matter in nuclear collisions, Phys. Rev. C 100, 024907 (2019), arXiv:1902.05095 [nucl-th] .
  * Bass  _et al._ [2016] S. A. Bass _et al._ , MADAI collaboration: http://madai.phy.duke.edu/ (accessed June, 2016).
  * Most  _et al._ [2019] E. R. Most, L. J. Papenfort, V. Dexheimer, M. Hanauske, S. Schramm, H. Stöcker, and L. Rezzolla, Signatures of quark-hadron phase transitions in general-relativistic neutron-star mergers, Phys. Rev. Lett. 122, 061101 (2019), arXiv:1807.03684 [astro-ph.HE] .