# Testing the phase transition parameters inside neutron stars with the
production of protons and lambdas in relativistic heavy-ion collisions

Ang Li1, Gao-Chan Yong2,3, Ying-Xun Zhang4,5 1Department of Astronomy, Xiamen
University, Xiamen, Fujian 361005, China
2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China
3School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China
4China Institute of Atomic Energy, Beijing 102413, China
5Guangxi Key Laboratory Breeding Base of Nuclear Physics and Technology,
Guilin 541004, China

###### Abstract

We demonstrate the consistency of the quark deconfinement phase transition
parameters in the beta-stable neutron star matter and in the nearly symmetric
nuclear matter formed in heavy-ion collisions (HICs). We investigate the
proton and $\Lambda$ flow in Au+Au collisions at 3 and 4.5 GeV/nucleon
incident beam energies with the pure hadron cascade version of a multi-phase
transport model. The phase transition in HICs and neutron stars is described
based on a class of hybrid equations of state from the quark mean-field model
for the hadronic phase and a constant-speed-of-sound parametrization for the
high-density quark phase. The measurements of the anisotropic proton flow at 3
GeV/nucleon by the STAR collaboration favor a relatively low phase transition
density lower than $\sim 2.5$ times saturation density indicated by the
gravitational wave and electromagnetic observations of neutron stars. And the
proton flow data at the higher energy of 4.5 GeV/nucleon can be used to
effectively constrain the softness of high-density quark matter equations of
state. Finally, compared to the proton flow, the $\Lambda$ flow is found to be
less sensitive and not constraining to the equations of state.

##  I Introduction

It is generally believed that the degree of freedom of dense matter is hadron
around nuclear saturation (number) density $n_{0}\approx 0.16$ fm-3, and
quarks are liberated at asymptotically high densities. The phase structure and
the equation of state [EoS; i.e., pressure-density relation $p(\varepsilon)$]
of cold QCD matter for intermediate densities ($\sim 1$-$10~{}n_{0}$) are
unfortunately unknown. The EoS, however, serves as one of the important inputs
not only for the evolution of heavy-ion collisions (HICs) but also in the
study of neutron stars, core-collapse supernovae and binary neutron star
mergers. Presently we focus on one of the key problems, i.e. what is the
critical density for the phase transition to quark matter? It is a matter of
extensive debate not only within the theory of HICs 1996NuPhA.606..320B ;
2021PhLB..81536138G ; 2022EPJC…82..911S ; 2022arXiv220811996O ;
2019NuPhA.982..883S ; 2019ApJ…871..157S ; 2019EPJWC.20403001B , but also in
astrophysics 2018RPPh…81e6902B ; 2019PhRvD.100j3022H ; 2020ApJ…904..103M ;
2021PhRvC.103c5802X .

In high-energy HICs, the two colliding nuclei produce collective motion of
nucleons and fragments due to the large compression, which is named collective
expands in time and reaches the surrounding detectors as baryons and mesons.
The baryon flow, which is measured, depends sensitively on the gradient of
pressure developed in the fireball at the moment of maximum compression during
the collision 1996NuPhA.606..320B . For example, flow observables at
$\sqrt{s_{NN}}=2.5$-$5$ GeV are shown very sensitive to the dense nuclear
matter EoS at $2$-$4n_{0}$ 2022arXiv220811996O . In our previous work, the
properties of phase transition of dense nuclear matter nearly symmetric
nuclear matter formed in relativistic HICs is investigated 2021PhLB..81536138G
, by employing a density-dependent mean-field potential at hadronic phase and
a bag-model description of quark matter. It was shown that the results with
the first-order phase transition were overall well in agreement with
experimental data of proton sideward flow and proton directed flows, while
those with hadronic EoSs and crossover deviated from experimental data.

Many high-energy heavy-ion experiments have been directed toward the goal of
inferring properties of the nuclear EoS 2019NuPhA.982..163G . In parallel with
this effort, the observations of neutron stars and binary neutron stars have
long been used to infer the neutron star EoS 2007PhR…442..109L . Different
from the HICs, in neutron stars, nuclear matter is present in beta equilibrium
from very low density to several times the saturation density and is extremely
neutron-rich. The EoS of neutron star matter gives rise to a unique sequence
of stellar configurations under hydrostatic equilibrium through the Tolman-
Oppenheimer-Volkoff (TOV) equations and can be directly connected to the
current and proposed observations of neutron stars.

The mass measurements of massive pulsars establish a firm lower bound on the
neutron star’s maximum mass ($M_{\rm TOV}$). Only the EoSs that support a
$M_{\rm TOV}$ larger than this lower bound can pass this constraint. In recent
years, the masses of several neutron stars are known with good precision from
radio Shapiro-delay measurements 2010Natur.467.1081D ; 2021ApJ…915L..12F , for
example, one of the highest mass measured is $M\sim 2.07\,M_{\odot}$ of PSR
J0740+6620 2021ApJ…915L..12F . Furthermore, the improved accuracy for both
masses and radii of millisecond period X-ray pulsars is reached in the
measurements by NICER (Neutron Star Interior Composition Explorer)
2019ApJ…887L..21R ; 2019ApJ…887L..24M ; 2021ApJ…918L..27R ; 2021ApJ…918L..28M
and the recent observation of gravitational waves (GWs) emitted during a
coalescing neutron star binary has opened the door to new possibilities of
obtaining information on the neutron star EoS by means of the measurement of
the tidal deformability 2017PhRvL.119p1101A ; 2018PhRvL.121p1101A ;
2020ApJ…892L…3A . In addition, measurements are planned for the moment of
inertia of PSR J0737-3039 A, the $1.338\,M_{\odot}$ primary component of the
first double pulsar system PSR J0737-3039, based on the long-term pulsar
timing to determine the periastron advance due to relativistic spin-orbit
coupling, which will provide complementary constraints on the EoS
2005MNRAS.364..635B .

Generally, the neutron star radius is controlled mainly by the density
dependence of the nuclear symmetry energy around the nuclear saturation
density $n_{0}$ 2009AIPC.1128..131L , while the maximum mass is a reflection
of the EoS stiffness at several times the nuclear saturation density (for
example, $\gtrsim 5n_{0}$) possible in the neutron star inner core, where a
phase transition to quark matter might be present 2002PhLB..526…19B ;
2020ApJ…904..103M . Previously, incorporating the possible first-order phase
transition within the QMF+CSS framework (explained later in Sec. II)
2021ApJ…913…27L , we obtained the neutron star EoS constrained from available
multimessenger astrophysical observations of LIGO/Virgo 2017PhRvL.119p1101A ;
2020ApJ…892L…3A  and NICER 2019ApJ…887L..21R ; 2019ApJ…887L..24M ;
2021ApJ…918L..27R ; 2021ApJ…918L..28M , as well as the mocked data of a future
moment of inertia measurement of PSR J0737-3039 A 2022MNRAS.515.5071M , within
the Bayesian statistical approach. The analysis of the observational data is
found to prefer the hadron-quark phase transition taking place at not-too-high
densities ($n_{\rm trans}/n_{0}\approx 2$) (see also in Refs.
2021PhRvC.103c5802X ; 2022arXiv220809085I ), and the critical transition
pressure is about $16.8\;\text{MeV}\;\text{fm}^{-3}$ 2022MNRAS.515.5071M ,
corresponding to a critical baryon chemical potential around
$1050\;\text{MeV}$ for such zero-temperature beta-stable stellar matter. Also,
the sound speed squared $c_{\rm QM}^{2}$ above the transition should be larger
than $\approx 0.4$ (we work in units where $\hbar=c=1$). Nevertheless, the
strength of the phase transition (namely, how strongly the transition happens)
can not be well-determined by the neutron star global properties
2021ApJ…913…27L ; 2022EPJWC.26004001L .

Since one usually assumes that there is one theoretical model of EoS that can
correctly explain the nuclear matter data of different physical situations
obtained in both laboratory nuclear experiments and astronomical observations
2018Univ….4…67B ; 2020JHEAp..28…19L ; 2021arXiv211208157S ;
2022PhRvD.105b3018T ; 2022Natur.606..276H . It is extremely interesting and
helpful if the information on the nuclear EoS from HICs can be related to the
physics of neutron star interiors 2022Natur.606..276H ; 2022arXiv220113150M ;
2022arXiv220901413L ; 2022arXiv221102224L ; 2022FrASS…964294G . The objective
of the present study is to test the phase transition parameters inside neutron
stars, constrained by the available multimessenger astrophysical observations,
with the production of nucleons and lambdas in relativistic HICs. For both
studies of HICs and neutron stars, we employ a class of QMF + CSS EoSs, which
has an explicit first-order deconfinement phase transition anticipated at
finite baryon densities. Moreover, important progress has been made recently
in modelling HICs, and we utilize the so-called directed flow ($v_{1}$) and
elliptic flow ($v_{2}$) in the present analysis using an extended version of
relativistic transport model, namely the AMPT (A Multi-Phase Transport Model)
2005PhRvC..72f4901L .

The paper is organized as follows. Section II introduces the employed QMF +
CSS framework for the EoS modelling and the AMPT for the HIC modelling, as
well as the available EoS constraints for symmetric nuclear matter and
neutron-rich, beta-stable stellar matter. The results and discussions are
detailed in Sec. III and the conclusion and perspective are presented in Sec.
IV.

##  II Equations of state with deconfinement phase transition at symmetric and
beta-stable nuclear matter

In the present study, the EoS $p(\varepsilon)$ of nuclear matter is obtained
within the quark mean-field (QMF) model (see Ref. 2020JHEAp..28…19L  for a
review). We first adopt a harmonic oscillator potential to confine quarks in a
nucleon, with its parameters determined by the mass and radius of free
nucleon, and then connect the nucleon in the medium with a system of many
nucleons. The EoS is then constructed within the widely used relativistic
mean-field approach, based on an effective Lagrangian with meson fields
mediating strong interactions between quarks. Compatibility with ab-initio
calculations as well as available experimental and observational constraints
at sub-nuclear and higher densities is ensured 2018ApJ…862…98Z ;
2019AIPC.2127b0010L . For example, from the data of FOPI experiments on the
elliptic flow in Au+Au collisions between 400 MeV and 1.5 GeV per nucleon,
constraints can be made on the symmetric nuclear matter EoS. Additionally, the
subthreshold $K^{+}$ production in heavy-ion reactions measured by the Kaon
Spectrometer (KaoS) collaboration has been exploited for exploring the EoS and
its incompressibility. We include both constraints 2002Sci…298.1592D  on the
EoS for symmetric nuclear matter from HIC data of the KaoS experiment and flow
data in Fig. 1. Specifically, within QMF, the saturation density is
$n_{0}=0.16\;\text{fm}^{-3}$ and the corresponding values at saturation point
for the binding energy $E/A=-16\;\text{MeV}$, the incompressibility
$K=240\;\text{MeV}$, the symmetry energy $E_{\rm sym}=31\;\text{MeV}$, the
symmetry energy slope $L=40\;\text{MeV}$.

Figure 1: Pressure as a function of baryon density (in units of saturation
density) of symmetric nuclear matter for QMF hadronic EoS. The shaded area at
lower (higher) density corresponds to constraints 2002Sci…298.1592D  inferred
from KaoS (flow) experiment.  Figure 2: Unified QMF hadronic EoS employed for
the inner crust and outer core of neutron stars, along with various cases of
CSS parameter sets: $n_{\rm trans}/n_{0}=1.5,2.0,2.5,3.0,3.5,4.0$,
$\Delta\varepsilon/\varepsilon_{\rm trans}=0.1,0.5$ and $c_{\rm
QM}^{2}=1/3,1$, for the high-density quark matter and the hadron-quark
deconfinement phase transition. The orange/grey shaded background are the 90%
credible regions from joint analysis 2022MNRAS.515.5071M  of the present
astrophysical data, including two gravitational wave data and two NICER’s
mass-radius measurements, with or without a mocked moment of inertia
measurement for PSR J0737-3039 A with $\sim 10\%$ accuracy possible within the
next decade.  Figure 3: Radius (upper panel), $I/M^{3/2}$ (middle panel),
$\Lambda$ (lower panel) as functions of the stellar mass, corresponding to the
24 EoSs within QMF+CSS in Fig. 2. The color coding is the same as in Fig. 2.
And the black curves represent the results of pure hadronic stars
corresponding to the QMF EoS. In the radius-mass plot, the mass-radius
measurements of the NICER telescope for PSR J0030+0451 2019ApJ…887L..24M ;
2019ApJ…887L..21R  and PSR J0740+6620 2021ApJ…918L..27R ; 2021ApJ…918L..28M
are also shown. In the middle and lower panels, also compared are the binary
tidal deformability measurement from GW170817 by LIGO/Virgo
2018PhRvL.121p1101A  and a mocked moment of inertia measurement
2022MNRAS.515.5071M  for PSR J0737-3039 A to be $I/M_{\rm A}^{3/2}=40\,{\rm
km^{2}}/\,M_{\odot}^{1/2}$ (with $\sim 10\%$ accuracy), respectively.

For high-density quark matter and the hadron-quark deconfinement phase
transition, making use of the feature that for a considerable class of
microscopic quark matter models the speed of sound turns out weakly density-
dependent, we here assume the speed of sound in quark matter is density
independent (see e.g., in 2021ChPhC..45e5104X  for detailed discussions on the
density dependence of the speed of sound in quark matter), i.e., the
“constant-speed-of-sound” (CSS) parameterization 2013PhRvD..88h3013A  is
applied. Consequently, the EoS from the onset of the phase transition up to
the maximum central density of a star is determined by three dimensionless
parameters: the transition density $n_{\rm trans}/n_{0}$, the transition
strength $\Delta\varepsilon/\varepsilon_{\rm trans}$, and the sound speed
squared in quark matter $c^{2}_{\rm QM}$. The full EoS is

$\varepsilon(p)=\left\\{\\!\begin{array}[]{ll}\varepsilon_{\rm
HM}(p),&p<p_{\rm trans}\\\ \varepsilon_{\rm HM}(p_{\rm
trans})+\Delta\varepsilon+c_{\rm QM}^{-2}(p-p_{\rm trans}),&p>p_{\rm
trans}\end{array}\right.$

where $n_{\rm trans}\equiv n_{\rm HM}(p_{\rm trans})$ and $\varepsilon_{\rm
trans}\equiv\varepsilon_{\rm HM}(p_{\rm trans})$, and
$\Delta\varepsilon/\varepsilon_{\rm trans}$ is essentially the finite
discontinuity in the energy density at the phase boundary. The high-pressure
CSS EoS after the onset of the phase transition can be written as
2013PhRvD..88h3013A ; 2020ApJ…904..103M ,

$\displaystyle p(\mu_{B})$ $\displaystyle=$ $\displaystyle
A\mu_{B}^{1+1/c_{\rm QM}^{2}}-B$ (1) $\displaystyle\mu_{B}(p)$
$\displaystyle=$ $\displaystyle[(p+B)/A]^{c_{\rm QM}^{2}/(1+c_{\rm QM}^{2})}$
(2) $\displaystyle n(\mu_{B})$ $\displaystyle=$ $\displaystyle(1+1/c_{\rm
QM}^{2})A\mu_{B}^{1/c_{\rm QM}^{2}}$ (3)

where $A$ is a parameter with energy dimension $3-c_{\rm QM}^{-2}$ and
$B=(\varepsilon_{\rm trans}+\Delta\varepsilon-c_{\rm QM}^{-2}\,p_{\rm
trans})/(1+c_{\rm QM}^{-2})$. To construct a first-order transition from some
low-pressure EoS to a high-pressure EoS of Eq. (1), $A$ is chosen such that
the pressure is monotonically increasing with $\mu_{B}$ and the baryon number
density does not decrease at the transition. In practice, our study is limited
to first-order phase transitions with a sharp interface (Maxwell
construction). Within CSS, the matching of hadron matter to quark matter is
done either in the case of beta-stable stellar matter (as shown in Fig. 2 for
deducing the neutrons star observational properties; See Sec. II.1) or
symmetric nuclear matter (serving as input of the HIC transport simulation;
See Sec. II.2). To avoid uncertainties due to the crust, for the computation
of neutrons star properties, we also utilize unified EoSs with the consistent
treatment of neutron star crusts and the crust-core transition properties
along with their cores (see details in e.g., 2021ApJ…913…27L ).

###  II.1 Astrophysical observations of neutron star and neutron star merger

Neutron stars are ideal cosmic laboratories for the study of dense matter
above the nuclear saturation density. Their global properties, such as the
mass $M$, radius $R$, tidal deformability $\Lambda$ and moment of inertia $I$,
have a one-to-one correspondence to the underlying EoS of neutron-rich matter
in beta equilibrium. The calculated neutron star properties with the chosen
QMF+CSS EoSs are reported in Fig. 3, together with the available or
forthcoming astrophysical constraints, respectively. See above in Fig. 2 for
the resulting regions of the EoS jointly constrained by these observational
data, shown as shaded background. For more detailed discussions on the
dependence of the hybrid star mass-radius relations with the parameters of the
deconfinement phase transition, we refer to Ref. 2020ApJ…904..103M .

Among the adopted observational data, the simultaneous mass-radius
measurements are for PSR J0740+6620 2021ApJ…918L..27R ; 2021ApJ…918L..28M  and
PSR J0030+0451 2019ApJ…887L..21R ; 2019ApJ…887L..24M  from the NICER
collaboration. The dimensionless tidal deformability $\Lambda$ is related to
the compactness $M/R$ and the Love number $k_{2}$ through
$\Lambda=\frac{2}{3}k_{2}(M/R)^{-5}$, with the tidal Love number $k_{2}$
denoting the ratio of the induced quadruple moment $Q_{ij}$ to the applied
tidal field $E_{ij}$, $Q_{ij}=-k_{2}\frac{2R^{5}}{3G}E_{ij}$. For example, the
tidal deformability observations of LIGO/Virgo for the merging binary of
GW170817 2017PhRvL.119p1101A  disfavor very stiff EoSs and obtained as
$\Lambda(1.4)=190^{+390}_{-120}$ for a typical $1.4M_{\odot}$ star at the
$90\%$ credible level 2018PhRvL.121p1101A . Finally, a moment of inertia
measurement of PSR J0737-3039 A, with a 90% upper limit $I_{A}<3\times
10^{45}\,{\rm g\ cm^{2}}$ is reported 2021PhRvX..11d1050K , by fitting the
post-Keplerian parameters in the binary system. Such a measurement gives a
loose upper limit for PSR J0737-3039 A’ radius of 22 km (with 90% confidence)
and is regarded as not constraining, in comparison with the accurate
observations from, e.g., LIGO/Virgo, where $R_{\rm 1.4}\leq 13.5$ km is
obtained 2018PhRvL.121p1101A . In the near future, the moment of inertia of
PSR J0737-3039 A is expected to be measured with 11 percent precision at a 68%
confidence level in 2030 2020MNRAS.497.3118H  (indicated in Fig. 3 in the
middle panel). Both analyses with or without the forthcoming moment of inertia
data are performed 2022MNRAS.515.5071M , as reported in Fig. 2.

###  II.2 The production of nucleons and lambdas in relativistic heavy-ion
collisions within AMPT

Since the possible quark-gluon plasma produced in HICs is out of thermal and
dynamical equilibrium 2007PrPNP..59..365W , the study of nuclear EoS is
subject to the adopted framework of transport simulations 2022arXiv220113150M
. We use the AMPT to model relativistic HICs, and a pure hadron cascade plus
hadron mean-field potential mode AMPT-HC is used 2005PhRvC..72f4901L ;
2021PhLB..82036521Y . The AMPT model 2005PhRvC..72f4901L  is known consists of
four components, i.e., a fluctuating initial condition, partonic interactions,
conversion from the partonic to the hadronic matter, and hadronic
interactions. Within AMPT-HC, the dynamics of the resulting hadronic matter
are described by a relativistic transport model 2001IJMPE..10..267L . The
nucleon density distribution is provided by a Woods-Saxon parameterization and
the local Thomas-Fermi approximation is employed to initialize the position
and momentum of each nucleon in colliding projectile and target. In addition
to the usual elastic and inelastic collisions, hadron potentials with the
test-particle method are applied to nucleons, baryon resonances, strangenesses
as well as their antiparticles 2021PhLB..82036521Y ; 2022PhRvC.106b4902Y . For
strange baryons $\Lambda$, $\Sigma$, $\Xi$, the quark counting rule is used
1974PhRvD…9.1613M ; 2001PhRvL..86.2533C .

Cross sections among various kinds of hadronic scatterings and mean-field
potentials of hadrons are the main inputs of a hadronic transport model. In
the AMPT-HC model, cross sections of hadronic scatterings can be found in
Refs. 2001IJMPE..10..267L ; 2005PhRvC..72f4901L ; 2021PhLB..82036521Y  and
references therein. The single baryonic potential used in the transport model
is derived from the QMF EoS mentioned above. It is undeniable that the single
nucleon potential at high energies and high densities (above $\sim 2n_{0}$) is
still unknown to date 2020PhRvC.102b4913N . The experimental Hama potential
1990PhRvC..41.2737H  is effective only at the saturation density. To make
minimum assumption, in this study, we use a momentum-independent single
particle potential. Specifically, $U(n)=\partial\varepsilon_{\rm pot}/\partial
n$, where the potential energy density $\varepsilon_{\rm pot}$ = $\varepsilon$
\- $\varepsilon_{\rm kin}\cdot n$. The $\varepsilon$ is total energy density
as same as in Eq. (II) and the nucleon average kinetic energy is usually
derived from the Fermi-gas model: $\varepsilon_{\rm kin}$=${(8\pi p_{\rm
F}^{5})}/({5m_{N}h^{3}n})$, with $m_{N}$ = 0.938 GeV being nucleon mass and
$p_{\rm F}$ its Fermi momentum. After getting the single baryonic potential
$U(n)$, the baryonic dynamics equation is expressed as
$\dot{\overrightarrow{p}}$ = -$\nabla_{\overrightarrow{r}}U$. For strange
baryon $\Lambda$, its potential $U_{\Lambda}(n)$ = $\frac{2}{3}U(n)$ (see
e.g., 2022PhRvC.106b4902Y  for more details). We mention here that, for the
matter produced in HICs, the effect of temperature dependence has been taken
into account in the transport model simulations through the phase space
distributions during the reactions.

##  III Results and Discussions

Flow in general is sensitive to the EoS which governs the evolution of the
system created in the nuclear collision. At relatively higher energies, the
collective flow in nuclear collisions is driven by the pressure gradients in
the early thermalized stages of the reaction and hence encodes the information
about the underlying EoS, characterizing it as relatively “hard” or “soft”
2001nucl.th..10037T . The anisotropic flow is an important observable in
characterizing how the anisotropy in the initial coordinate space develops
into that in the initial momentum space, as a result of the strong interaction
in the QGP matter created in relativistic HICs.

The particle directed and elliptic flow in HICs can be derived from the
Fourier expansion of the azimuthal distribution at given rapidity $y$
1994hep.ph….7282V ; 2008PhLB..663..312L , i.e.,

$\frac{dN}{d\phi}\propto 1+2\displaystyle{\sum_{i=1}^{n}}v_{n}\cos(n\phi)\ .$
(4)

The directed flow $v_{1}$ and the elliptic flow $v_{2}$ can be, respectively,
expressed as:

$v_{1}=\langle\cos(\phi)\rangle=\langle\frac{p_{x}}{p_{t}}\rangle\ ;$ (5)
$v_{2}=\langle\cos(2\phi)\rangle=\langle\frac{p_{x}^{2}-p_{y}^{2}}{p_{t}^{2}}\rangle\
,$ (6)

where $p_{t}$ = $\sqrt{p_{x}^{2}+p_{y}^{2}}$, and the rapidity $y$ =
$\frac{1}{2}\ln\frac{E+p_{z}}{E-p_{z}}$.

Figure 4: Rapidity dependence of proton directed flow in 10-40% centrality for
Au+Au collisions at $\sqrt{s_{NN}}$ = 3 GeV by the AMPT-HC model with
($\Delta\varepsilon/\varepsilon_{\rm trans},c_{\rm QM}^{2})=(0.1,1/3)$ (solid
curves), $(0.1,1)$ (dashed curves), $(0.5,1/3)$ (dotted curves), $(0.5,1)$
(dash-dotted curves), at transition density $n_{\rm
trans}/n_{0}=1.5,2.0,2.5,3.0$ is compared with STAR experimental data
2022PhLB..82737003A .  Figure 5: Same with Fig. 4, but for proton elliptic
flow.  Figure 6: Rapidity dependence of proton directed flow in 10-25%
centrality for Au+Au collisions at $\sqrt{s_{NN}}$ = 4.5 GeV by the AMPT-HC
model with ($\Delta\varepsilon/\varepsilon_{\rm trans},c_{\rm
QM}^{2})=(0.1,1/3)$ (solid curves), $(0.1,1)$ (dashed curves), $(0.5,1/3)$
(dotted curves), $(0.5,1)$ (dash-dotted curves), at transition density $n_{\rm
trans}/n_{0}=1.5,2.0,2.5,3.0,3.5,4.0$ is compared with STAR experimental data
2021PhRvC.103c4908A .

We first show in Fig. 4 (Fig. 5) the $v_{1}$ ($v_{2}$) of protons as a
function of rapidity in 10-40% centrality for Au+Au collisions at
$\sqrt{s_{NN}}$ = 3 GeV simulated by the AMPT-HC model, for various cases of
transition properties. The full dots represent the experimental data from the
STAR experiment 2022PhLB..82737003A . The collision energy is defined as
$\sqrt{s_{NN}}$ = $\sqrt{2m_{N}\times(E_{\rm lab}+2m_{N})}$ with $E_{\rm lab}$
representing the kinetic energy per nucleon of incident beam with fixed target
mode. At low transition density $n_{\rm trans}/n_{0}=1.5$ (shown in the upper-
left panels of both figures), one observes an interplay between the transition
strength and sound speed parameters ($\Delta\varepsilon/\varepsilon_{\rm
trans},c_{\rm QM}^{2}$) when explaining both the $v_{1}$ and $v_{2}$ data.
Neither the combination of weak first order phase transition and high speed of
sound $\Delta\varepsilon/\varepsilon_{\rm trans}=0.1,c_{\rm QM}^{2}=1$ or
strong first order phase transition and low speed of sound
$\Delta\varepsilon/\varepsilon_{\rm trans}=0.5,c_{\rm QM}^{2}=1/3$ can explain
well the experimental data. The case of $\Delta\varepsilon/\varepsilon_{\rm
trans}=0.1,c_{\rm QM}^{2}=1$ for $n_{\rm trans}/n_{0}=1.5$ is especially not
favored by the proton $v_{2}$ since the simulated $v_{2}$ shows no dip at
central rapidities and deviate evidently from the STAR data. This is
consistent with the astrophysical constraints on the phase transition in the
beta-stable matter shown previously in Fig. 2 (upper-left panel), where the
$(0.1,1)$ results are outside $90\%$ credible boundaries indicated by the
available neutron star properties.

When comparing the results at different transition densities, it can be also
seen that the experimental data points gradually shift outside of the
theoretical predictions since different CSS parameters actually lead to
increasingly similar results with the increase of the transition density.
Consequently, the anisotropic flow of protons at beam energy $\sqrt{s_{NN}}$ =
3 GeV can be regarded as a good probe to the transition density of possible
hadron-quark phase transition and the current STAR data favor a phase
transition density of $n_{\rm trans}/n_{0}\lesssim 2.5$. In addition, the data
seem to support a relatively modest phase transition with
$\Delta\varepsilon/\varepsilon_{\rm trans}=0.1$, but tend to exclude a stiff
quark matter EoS with a speed of sound as high as the speed of light ($c_{\rm
QM}^{2}=1$).

Figure 7: Same with Fig. 6, but for $\Lambda$ directed flow.

Fig. 6 and Fig. 7 present the $v_{1}$ as a function of rapidity at increased
beam energy $\sqrt{s_{NN}}$ = 4.5 GeV, for protons and lambdas, respectively.
The full dots represent the experimental data from the STAR experiment
2021PhRvC.103c4908A . The density in this case can reach $\gtrsim 4n_{0}$
2021PhLB..81536138G . It is seen that, for both types of hadrons, there is a
change of sign of the directed flow at mid-rapidity and the $v_{1}$ values
show in most cases the well-known bounce-off behaviour for larger rapidities,
which are consistent with previous studies 1991PhR…202..233A . The $c_{\rm
QM}^{2}=1$ results again fail to explain the slope of $v_{1}$ (no bounce-off)
for low transition densities $n_{\rm trans}/n_{0}\leq 2.0$. Nevertheless, the
flow data at $\sqrt{s_{NN}}$ = 4.5 GeV is not a good probe of the phase
transition onset properties and show no sensitivity to the transition strength
$\Delta\varepsilon/\varepsilon_{\rm trans}$. Since both the proton and lambda
$v_{1}$ STAR data support a soft quark matter close to $1/3$, the measurement
at such high beam energy can be used to indicate the EoS stiffness (or
softness) of high-density quark matter. It should be noted that the lambda
$v_{1}$ simulations are more alike under different EoSs in comparison to those
of protons, therefore the lambda production is less constraining than the
nucleon ones, regarding to the EoS 2022PhRvC.106d4902N .

##  IV Conclusions

The dense matter EoS plays a major role not only in the astrophysical study
related to neutron stars but also in the hydrodynamical evolution of HICs. In
this work, recent astrophysical observations of (binary)neutron stars and
heavy-ion data are confronted with our present understanding of the dense QCD
matter and the possible quark deconfinement phase transition. The advantages
of using HICs to test the phase transition of neutron star matter is that HICs
have been operated in terrestrial laboratories for decades of years and
nuclear properties have been extensively studied from different angles. We
here make use of Au+Au collision data (directed flow and elliptic flow for
both protons and lambdas) with the STAR (Solenoidal Tracker At RHIC)
experiment at the beam energy $\sqrt{s_{NN}}$ = 3 and 4.5 GeV.

The dense matter in both HICs and neutron stars are described by one model of
QMF+CSS, keeping consistency with available experimental and observational
constraints at sub-nuclear and higher densities. In our previous work, the EoS
parameter space encapsulating a first-order hadron-quark phase transition has
been jointly constrained by the neutron-star observational data. We here use
the obtained EoS space and the phase transition parameters (here mainly the
transition density $n_{\rm trans}/n_{0}$, the transition strength
$\Delta\varepsilon/\varepsilon_{\rm trans}$ within CSS) as a baseline for
comparisons.

We find that the flow data at $\sqrt{s_{NN}}$ = 3 GeV can be used to
effectively constrain the transition density, while the data at
$\sqrt{s_{NN}}$ = 4.5 GeV can be used to effectively constrain the high-
density quark matter EoS. The current data support a relatively low phase
transition density $\lesssim 2.5n_{0}$ and exclude a very stiff high-density
quark matter EoS, which further supports the similar phase transition
properties indicated by the astrophysical neutrons star observations. In
particular, although the transition strength can not be well signified by the
neutron star properties, it might be revealed by the flow data with beam
energies around $\sqrt{s_{NN}}$ = 3 GeV. Finally, although high-energy HICs
indicate a soft high-density (quark matter) EoS with the sound speed squared
close to the non-interacting limit of $1/3$ (i.e., the conformal limit), there
is almost no possibility to probe the possible transition from heavy-ion data
with energy as high as $\sqrt{s_{NN}}=4.5$ GeV, at least based on the analysis
of flow observables, due to its insensitivity of the transition threshold
properties; Also, the lambda flow is found to be less sensitive than the
proton flow.

There are many aspects we can explore further in the future. First, since we
focus on the possible hadron-quark phase transition at the high-density
nuclear matter, we here freeze the energy contribution of the isospin
asymmetry part of nuclear matter following the one choice of the symmetry
energy parameters at $n_{0}$ built in the hadronic QMF EoS. Although the
choice can comfortably reconcile with various empirical constraints from low-
energy nuclear physics 2014EPJA…50…37X ; 2021PhRvC.103a4616L , there is still
ambiguity regarding their ranges and possible intrinsic correlations between
them 2020PhRvL.125t2702D . Therefore, it is desired to incorporate the
uncertainty of nuclear symmetry energy in a future study, to understand better
the relevance of the phase transition taking place in symmetric nuclear matter
and asymmetric matter. Also, more reliable constraints on EoS and phase
transition parameters should be obtained in a Bayesian framework for the HIC
and neutron star observational data, especially that more and higher accuracy
data for both kinds are expected in the near future.

We are thankful to Xinle Shang, Chang Xu, Zhenyu Zhu and the XMU neutron star
group for helpful discussions. The work is supported by National SKA Program
of China (No. 2020SKA0120300), the National Natural Science Foundation of
China (Grant Nos. 11873040, 12273028, 12275322, 11875323, 12275359) and the
Youth Innovation Fund of Xiamen (No. 3502Z20206061).

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