11institutetext: University of Kashmir, Srinagar 190006, India
22institutetext: Instituto de Alta Investigación, Universidad de Tarapacá,
Casilla 7D Arica 1000000, Chile

# Elliptic flow of identified hadrons in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ using the PHSD model

B. Towseef 11 M. Farooq 11 V. Bairathi _Present address:_
<EMAIL_ADDRESS>B. Waseem 11 S. Kabana 22 S. Ahmad 11

(Received: date / Revised version: date)

###### Abstract

We present elliptic flow ($v_{2}$) of identified hadrons at mid-rapidity
($|y|<1.0$) in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ using the
Parton Hadron String Dynamics (PHSD) model. Transverse momentum
($p_{\mathrm{T}}$) dependence of identified hadron $v_{2}$ in minimum bias
(0-80%) and three different centrality intervals (0-10%, 10-40%, and 40-80%)
are presented. A clear centrality dependence of $v_{2}(p_{\mathrm{T}})$ is
observed for particles and anti-particles in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$. We also present $p_{\mathrm{T}}$ dependence
of $v_{2}$ difference ($\mathrm{\Delta}v_{2}$) between particles and
corresponding antiparticles. A significant difference in $v_{2}$ values for
baryons and anti-baryons is observed. The number of constituent quark (NCQ)
scaling of $v_{2}$ is discussed in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$. We also present ratio of
$v_{2}(p_{\mathrm{T}})$ between the HSD and PHSD modes to explore the effect
of hadronic and partonic interactions in the medium. We observe higher $v_{2}$
in PHSD mode than the HSD mode, which suggests the formation of partonic
medium in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. These
predictions are useful for the interpretation of data measured in Beam Energy
Scan (BES) program at RHIC and will be useful for the future Compressed
Baryonic Matter (CBM) experiment at the Facility for Antiproton and Ion
Research (FAIR).

###### pacs:

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##  1 Introduction

The study of quark-gluon plasma (QGP) and the quantitative mapping of the QCD
phase diagram from low to high temperatures and baryon densities are major
goals of the relativistic heavy-ion experiments r1 ; r2 . The experiments at
Relativistic Heavy-Ion Collider (RHIC) r3 ; r4  and Large Hadron Collider
(LHC) r5 ; r6 ; r7  have explored the QCD phase diagram in the region of high
temperatures and vanishing baryon densities. RHIC also performed a beam energy
scan including fixed target heavy-ion collisions that accessed high baryon
density regions at small collision energies. The upcoming CBM experiment r8 ;
r9  at FAIR aims to investigate the region of the QCD phase diagram at high
net baryon densities and moderate temperatures. It will cover the beam energy
range of 2-35 A GeV in the laboratory frame for Au+Au collisions. The
anisotropic flow of produced particles has long been considered as a
deterministic probe to study properties of the QCD matter created in heavy-ion
collisions r10 ; r11 . The anisotropic flow appears as a momentum-space
anisotropy in the final states and develops due to pressure gradient resulting
from the initial spatial anisotropy of the collision. Therefore, it is
sensitive to the very early stages of the collision. Azimuthal anisotropy can
be studied by the Fourier expansion of azimuthal angle distribution of
produced particles with respect to the reaction plane angle r12 :

$E\frac{d^{3}N}{dp^{3}}=\frac{1}{2\pi}\frac{d^{2}N}{p_{\mathrm{T}}dp_{T}dy}\left(1+2\sum_{n=1}^{\infty}v_{n}\cos[n(\phi-\psi_{RP})]\right).$
(1)

Here, $n$ denotes order of the harmonics, $\phi$ is the particle’s azimuthal
angle, and $\psi_{RP}$ is the reaction plane angle made by the impact
parameter vector and the beam direction. The $n^{th}$ order flow coefficient
$v_{n}$ is given by the equation,

$v_{n}=\langle\cos[n(\phi-\psi_{RP})]\rangle,$ (2)

where $\langle\rangle$ represents an average over particles and events. The
second Fourier coefficient, also known as the elliptic flow ($v_{2}$), is
essential for understanding the QCD matter created in heavy-ion collisions.
Due to the self-quenching effect of $v_{2}$ r13 ; r14 ; r15 , it offers
details on the dynamics at the beginning of the collision. However, hadronic
re-scattering at the later stages could alter the early dynamics of the
collision. The identified hadrons $v_{2}$ can be used to investigate the bulk
properties of the medium produced in heavy-ion collisions. The elliptic flow
has been studied previously in the following Ref. r4 ; r5 ; r16 ; r17 ; r18 ;
r19 ; r20 ; r21 . In comparison to non-strange hadrons, multi-strange hadrons
($\Xi$ and $\Omega$) are expected to have smaller hadronic interaction cross-
section r22 . Also, the freeze-out temperatures of multi-strange hadrons are
close to the quark-hadron transition temperature suggested by the lattice QCD
r21 ; r23 . Therefore, it is believed that information from the partonic stage
of the collision can be predominantly provided by these multi-strange hadrons
r24 ; r25 ; r26 .

In this paper, we report the first observations on the elliptic flow of
identified hadrons in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$
using the PHSD model r27 ; r28 . This beam energy is one of the energies from
the future CBM experiment at the FAIR facility and it corresponds to the
center of mass energy $\sqrt{s_{\mathrm{NN}}}\approx 8.0\mathrm{~{}GeV}$.
Predictions of the elliptic flow for the FAIR energy are presented and the
results are compared with the Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$ published by the STAR experiment
r29 ; r30 . The elliptic flow is obtained as a function of transverse momentum
in different centrality classes. Centrality dependence of the identified
hadron elliptic flow is discussed. The constituent quark number scaling
behavior of the elliptic flow is also studied. The effect of parton and hadron
dynamics on the elliptic flow of identified hadrons will also be discussed.

The paper is organized as follows. The PHSD model and analysis method for
calculation of $v_{2}$ are briefly discussed in section 2 and 3, respectively.
The transverse momentum and centrality dependence of identified hadron $v_{2}$
are presented in section 4. The difference in $v_{2}$ between particles and
anti-particles is presented in sub-section 4.3. The number of constituent
quark scaling of identified hadron $v_{2}$ is discussed in sub-section 4.4. In
sub-section 4.5, the effect of hadronic and partonic interactions on $v_{2}$
is discussed. Finally, we summarize the results reported in this paper in
section 5.

##  2 PHSD model

The PHSD model is a microscopic off-shell transport approach for strongly
interacting systems in and out of equilibrium. PHSD incorporates both
effective partonic and hadronic degrees of freedom and includes a dynamical
description of the hadronization process from partonic to hadronic matter. It
is based on a dynamical quasi-particle model (DQPM) r31 ; r32  for partons
developed to reproduce lattice quantum chromodynamics (lQCD) results,
including the partonic equation of state, by fitting the three DQPM
parameters, such as energy density, pressure, and entropy density, in
thermodynamic equilibrium r33 ; r34 . The hadronic phase is controlled via the
Hadron String Dynamics (HSD) part of the transport method r35 ; r36 . The PHSD
approach completely realises the entire evolution of a relativistic heavy-ion
collision from the initial hard NN collisions out of equilibrium up to the
hadronization and final interactions of the resulting hadronic particles. At
energies ranging from SIS to LHC, PHSD has been effectively used for p+p, p+A,
and A+A reactions r37 .

For this work, version 4.1 of the PHSD model is employed. Our measurements are
carried out with different modes of the model, namely the partonic (PHSD) and
the hadronic (HSD) mode. We have simulated 50 million minimum bias Au+Au
collision events at $E_{lab}=35\mathrm{~{}A~{}GeV}$
($\sqrt{s_{\mathrm{NN}}}\approx 8.0\mathrm{~{}GeV}$). The data is generated
for the impact parameter range chosen randomly between $b_{min}$ = 0 $fm$ and
$b_{max}$ = 15 $fm$. Measurements are made in the central rapidity region
($|y|<1.0$) and different centrality intervals, which cover central to
peripheral collisions. The centrality is calculated using the multiplicity
within $|\eta|<0.5$ from the PHSD model. The reference multiplicity
distribution is shown in Fig. 1. The multiplicity is divided into three
centrality classes 0-10%, 10-40%, and 40-80%.

Figure 1: (Color online) Reference multiplicity distribution ($|\eta|<0.5$) in
Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model.

##  3 Flow Analysis Method

The standard event plane method described in r12  is used for the elliptic
flow analysis. The reaction plane angle $\psi_{RP}$ in Eq. 2 cannot be
measured because the direction of the impact parameter is impossible to
determine in the experiments. Therefore, an estimator of $\psi_{RP}$, namely
the event plane angle $\psi_{n}$ is used to measure $v_{2}$. The estimated
event plane angle is obtained from the azimuthal angle distribution of the
produced particles. The $n^{th}$ harmonic event plane angle is defined as,

$\psi_{n}=\frac{1}{n}\tan^{-1}\left[\frac{\sum_{i}\omega_{i}\sin(n\phi_{i})}{\sum_{i}\omega_{i}\cos(n\phi_{i})}\right].$
(3)

Here, $\phi_{i}$ is the azimuthal angle of $i^{th}$ particle and $\omega_{i}$
is the weight taken as the $p_{\mathrm{T}}$ of the particle to optimize the
event plane resolution r12 . The sum runs over all the particles used to
calculate event plane angle in an event. In this work, $\psi_{n}$ is
reconstructed using particles in the pseudo-rapidity range $|y|<1.0$ and
transverse momentum range $0.2<p_{\mathrm{T}}<2.0$ GeV/c. The elliptic flow is
obtained using the second-order event plane ($\psi_{2}$) and corrected for the
event plane resolution ($R_{2}$). To suppress non-flow effects, the $\eta$-sub
event plane method is used with an $\eta$ gap of 0.1 between the two sub-
events. The sub-events are defined in the negative ($-1<\eta<-0.05$) and
positive ($0.05<\eta<1$) pseudo-rapidity regions. $R_{2}$ is then estimated
using,

$R_{2}=\sqrt{\langle\cos[2(\psi_{2}^{a}-\psi_{2}^{b})]\rangle},$ (4)

where $\psi_{2}^{a}$ and $\psi_{2}^{b}$ are the sub-event plane angles in the
negative and positive pseudo-rapidity regions, respectively. To minimize auto-
correlation effect, the elliptic flow for particles in the positive pseudo-
rapidity window is measured with respect to the $\psi_{2}$ calculated in the
negative window and vice versa. Additionally, we will present $v_{2}$ measured
with respect to the participant plane angle $\psi_{2}^{PP}$ provided by the
PHSD model, and also using the reaction plane angle $\psi_{RP}$ set to zero.

Figure 2: (Color online) Event plane resolution as a function of centrality
using the $\eta$-sub event plane method in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model.

Figure 2 shows the event plane resolution as a function of centrality for
Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. The event plane
resolution peaks near the mid-central collisions and decreases towards the
central and peripheral collisions. It reaches a maximum of 34% for centrality
(20-30%) in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. The decrease
in resolution is attributed to the comparatively low multiplicity of
peripheral collisions, whereas for more central collisions, the effect is due
to the small flow magnitudes. The values of resolution from the PHSD model are
very close to the published STAR results in Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$ r29 .

##  4 Results

In this section, we report $p_{\mathrm{T}}$ and centrality dependence of the
identified hadrons $v_{2}$ at mid-rapidity ($|y|<1.0$) in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ ($\sqrt{s_{\mathrm{NN}}}\approx
8.0\mathrm{~{}GeV}$) using the PHSD model. We also compare $v_{2}$ results
with the measurements from the STAR experiment in Au+Au collisions at
$\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$. The results presented in this
work are first predictions of identified hadrons $v_{2}$ from the PHSD model
for upcoming CBM experiment at FAIR.

###  4.1 Differential elliptic flow $v_{2}(p_{\mathrm{T}})$

The elliptic flow $v_{2}$ as a function of $p_{\mathrm{T}}$ for identified
hadrons at mid-rapidity ($|y|<1.0$) in minimum bias (0-80%) Au+Au collisions
at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model is shown in Figs. 3 and
4. The identified hadron $v_{2}$ is increasing with $p_{T}$ for all the
particles studied. The obtained $v_{2}$ values are compared with the published
results in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$
from the STAR experiment r29 . The $v_{2}$ of $\pi^{\pm}$, $K^{\pm}$,
$K_{s}^{0}$, $p$, and $\Lambda$ are comparable to the STAR experiment data
whereas $\overline{p}$, $\overline{\Lambda}$, and $\overline{\Xi}^{+}$ show
larger deviations. The experimental value of $v_{2}$ for
$\overline{\Omega}^{+}$ is negative, therefore it is not shown in Fig. 4.

We also compare identified hadrons $v_{2}$ measured with respect to the event
plane angle ($\psi_{2}$) and the participant plane angle ($\psi_{2}^{PP}$) in
minimum bias Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ as shown in
Figs. 3 and 4. The participant plane angle is provided by the PHSD model,
which is calculated using the positions of participant nucleons. The $v_{2}$
values calculated with respect to $\psi_{2}^{PP}$ for all the particles and
anti-particles show larger deviations above $p_{\mathrm{T}}>$ 1.0 GeV/c. This
could be due to the event-by-event fluctuations in the positions of
participating nucleons used for the calculation of $\psi_{2}^{PP}$. We also
discuss $v_{2}$ measured with respect to $\psi_{2}=0$. The $v_{2}$ values for
mesons ($\pi$, $K$) above $p_{\mathrm{T}}>$ 1.0 GeV/c are lower when
calculated using the $\psi_{2}=0$, while the magnitude of $v_{2}$ for baryons
is similar within statistical uncertainties.

Figure 3: (Color online) $v_{2}(p_{\mathrm{T}})$ for identified hadrons
($\pi^{+}$, $K^{+}$, $p$, $K_{s}^{0}$, $\Lambda$, $\Xi^{-}$, and $\Omega^{-}$)
in 0-80% central Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the
PHSD model (open and solid circles). The $v_{2}$ results from the STAR
experiment in Au+Au collisions at $\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$
(open and solid stars) is also shown r29 . The error bars shown are the
statistical uncertainties.

Figure 4: (Color online) $v_{2}(p_{\mathrm{T}})$ for identified hadrons
($\pi^{-}$, $K^{-}$, $K_{s}^{0}$, $\overline{p}$, $\overline{\Lambda}$,
$\overline{\Xi}^{+}$, and $\overline{\Omega}^{+}$) in 0-80% central Au+Au
collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model (open and
solid circles). The $v_{2}$ results from the STAR experiment in Au+Au
collisions at $\sqrt{s_{\mathrm{NN}}}=7.7\mathrm{~{}GeV}$ (open and solid
stars) is also shown r29 . The error bars shown are the statistical
uncertainties.

###  4.2 Centrality dependence of the elliptic flow

We report $v_{2}(p_{\mathrm{T}})$ for particles ($\pi^{+}$, $K^{+}$, $p$,
$K_{s}^{0}$, $\Lambda$, and $\Xi^{-}$) and anti-particles ($\pi^{-}$, $K^{-}$,
$\overline{p}$, $K_{s}^{0}$, $\overline{\Lambda}$, and $\overline{\Xi}^{+}$)
at mid-rapidity ($|y|<1.0$) for different centralities (0-10%, 10-40%, and
40-80%) in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD
model as shown in Figs. 5 and 6, respectively. The centrality dependence of
$v_{2}$ is observed for all the measured hadron species. The magnitude of
$v_{2}$ rising from central (0-10%) to peripheral (40-80%) collisions. A
similar trend has been observed for $v_{2}$ of identified hadrons measured by
the STAR experiment r30 ; r41 ; r42 . This centrality dependence is expected
due to increase in eccentricity of the initial overlap region of the colliding
nuclei from central to peripheral collisions. The above observation is in line
with the hydrodynamic model, which predicts that the momentum anisotropy in
the final state is driven by the initial spatial anisotropy r43 .

Figure 5: (Color online) $v_{2}$ as a function of $p_{\mathrm{T}}$ for
identified particles ($\pi^{+}$, $K^{+}$, $p$, $K_{s}^{0}$, $\Lambda$, and
$\Xi^{-}$) in 0-10%, 10-40%, and 40-80% centralities in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model. The error bars are
statistical uncertainties.

Figure 6: (Color online) $v_{2}$ as a function of $p_{\mathrm{T}}$ for
identified anti-particles ($\pi^{-}$, $K^{-}$, $\overline{p}$, $K_{s}^{0}$,
$\overline{\Lambda}$, and $\overline{\Xi}^{+}$) in 0-10%, 10-40%, and 40-80%
centralities in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the
PHSD model. The error bars are statistical uncertainties.

###  4.3 $v_{2}(p_{T})$ difference between particles and anti-particles

In Fig. 7, we present $v_{2}$ difference ($\mathrm{\Delta}v_{2}$) between the
particles and their corresponding anti-particles. $\mathrm{\Delta}v_{2}$ for
pions and kaons do not exhibit any significant difference within statistical
uncertainties for the measured $p_{\mathrm{T}}$ range in minimum bias Au+Au
collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. This observation is expected
for particles with same mass and constituent quarks. There is a noticeable
difference between $v_{2}(p_{\mathrm{T}})$ of baryons and anti-baryons, where
$v_{2}(p_{\mathrm{T}})$ of anti-baryons is more compared to the baryons in
PHSD model for minimum bias Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$. The $\mathrm{\Delta}v_{2}$ for protons shows
$p_{\mathrm{T}}$ dependence and the difference increases with
$p_{\mathrm{T}}$. This difference in $v_{2}$ of protons could be due to the
baryon stopping at this lower energy. The same behavior is observed for
$\Lambda$ and $\Xi$, which indicates that the exchange of one or more u-quark
by s-quark does not have any significance on $\mathrm{\Delta}v_{2}$.

Figure 7: (Color online) The elliptic flow $v_{2}$ as a function of
$p_{\mathrm{T}}$ for identified hadrons in 0-80% central Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model. The difference in
$v_{2}(p_{\mathrm{T}})$ between particles and anti-particles is displayed in
the lower rows of each panel. The error bars are statistical uncertainties.

###  4.4 Number of constituent quark scaling

Figure 8: (Color online) Number of quark scaled elliptic flow
($v_{2}$/$n_{q}$) as a function of $p_{\mathrm{T}}/n_{q}$ and
($m_{T}-m_{0}$)/$n_{q}$ for identified particles (panel a, c) and anti-
particles (panel b, d) in minimum bias Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ from the PHSD model. The error bars are
statistical uncertainties.

The elliptic flow $v_{2}(p_{\mathrm{T}})$ of identified hadrons follows the
mass-ordering at lower $p_{\mathrm{T}}$ predicted by the hydrodynamic model
r44 . This mass-ordering could be caused by the interplay of elliptic and
radial flow, which reduces the $v_{2}$ magnitude for heavier particles. The
baryon-meson splitting at intermediate $p_{\mathrm{T}}$ is also shown in the
experimental measurements r29 ; r30 . The separation of $v_{2}$ for mesons and
baryons at intermediate $p_{\mathrm{T}}$ implies that $v_{2}$ is proportional
to the number of constituent quarks ($n_{q}$). In order to investigate the
number of constituent quark (NCQ) scaling of $v_{2}$ in the PHSD model, we
have plotted $v_{2}$/$n_{q}$ versus $p_{\mathrm{T}}/n_{q}$ and transverse
kinetic energy ($m_{T}-m_{0}$)/$n_{q}$ as shown in Fig. 8 for minimum bias
(0-80%) Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. The transverse
kinetic energy removes the dependence of rest mass of particles on $v_{2}$ at
lower $p_{\mathrm{T}}$. The $v_{2}$ scaled with number of constituent quarks
for all the particles follows a single curve within the statistical
uncertainties. The NCQ scaling holds better for the particles than anti-
particles in the PHSD model for Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$. The observed NCQ scaling of $v_{2}$ at this
energy indicates that $v_{2}$ has been built up in early stages of the
collision where the dominating degrees of freedom are partonic in nature as
suggested by the quark coalescence model for the QGP r45 ; r46 ; r47 .

###  4.5 Hadronic and partonic interactions

Figure 9: (Color online) $v_{2}$ as a function of $p_{\mathrm{T}}$ for
identified (a) particles ($\pi^{+}$, $K^{+}$, $p$, $\Lambda$, $\Xi^{-}$, and
$\Omega^{-}$) and (b) anti-particles ($\pi^{-}$, $K^{-}$, $\overline{p}$,
$\overline{\Lambda}$, $\overline{\Xi}^{+}$, and $\overline{\Omega}^{+}$) in
0-80% central Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ from the
partonic (PHSD) and hadronic (HSD) modes of the PHSD model. The ratio of
$v_{2}(p_{\mathrm{T}})$ between HSD and PHSD is also shown in the lower rows
of each panel. The error bars are statistical uncertainties.

Figure 9 shows the $p_{\mathrm{T}}$ dependence of $v_{2}$ for identified (a)
particles and (b) anti-particles in minimum bias Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$ from the partonic and hadronic modes of the
PHSD model. The $v_{2}$ ratio between HSD and PHSD modes are shown in the
lower rows of each panel. The partonic mode (PHSD) incorporates both the
hadronic and partonic interactions, whereas the hadronic mode (HSD) only
includes the hadronic interactions. $v_{2}$ increases with $p_{\mathrm{T}}$
for all hadron species in both the modes. The $v_{2}$ values in HSD mode are
lower than the PHSD mode for all the particles except $p$, $\Lambda$,
$\Xi^{-}$, and $\Omega^{-}$ which might contain protons from the colliding
nuclei. The value of ratio less than one shows the effect of partonic degrees
of freedom and might indicate the formation of QGP in Au+Au collisions at
$E_{lab}=35\mathrm{~{}A~{}GeV}$. The ratio of $v_{2}$ between HSD and PHSD
modes for $p$, $\Lambda$, $\Xi^{-}$, and $\Omega^{-}$ is greater than one at
lower $p_{\mathrm{T}}$. This could be due to the baryon stopping process at
lower beam energies such as $E_{lab}=35\mathrm{~{}A~{}GeV}$.

##  5 Summary and Conclusions

In this paper, we reported first measurements on elliptic flow of identified
hadrons ($\pi^{+}$, $K^{+}$, $p$, $K_{s}^{0}$, $\Lambda$, $\Xi^{-}$, and
$\Omega^{-}$) and their anti-particles ($\pi^{-}$, $K^{-}$, $\overline{p}$,
$K_{s}^{0}$, $\overline{\Lambda}$, $\overline{\Xi}^{+}$, and
$\overline{\Omega}^{+}$) at mid-rapidity for minimum bias (0-80%) and
different centrality intervals (0-10%, 10-40%, and 40-80%) in Au+Au collisions
at FAIR energy ($E_{lab}=35\mathrm{~{}A~{}GeV}$) from the PHSD model. The
$v_{2}$ measurements are carried out using the $\eta$-sub event plane method.
Non-flow correlations in the $v_{2}$ measurements were removed by introducing
an $\eta$ gap between the positive and negative sub-events. The elliptic flow
of the identified hadrons in the measured $p_{\mathrm{T}}$ range is found to
be an increasing function of $p_{\mathrm{T}}$. For all the particle species in
the measured $p_{\mathrm{T}}$ range, the $v_{2}$ increases from central to
peripheral collisions. The observed centrality dependence of $v_{2}$ is
consistent with the published results from the STAR experiment at RHIC. A
significant difference in $v_{2}$ of baryons and anti-baryons is observed,
this difference increases with $p_{\mathrm{T}}$. The magnitude of $v_{2}$ for
anti-baryons is larger compared to baryons at mid-rapidity for minimum bias
Au+Au collisions in the PHSD model. The same behavior observed in the
$\mathrm{\Delta}v_{2}$ for $p(uud)$, $\Lambda(uds)$, and $\Xi^{-}(dss)$ shows
that the exchange of one or more u-quarks by s-quarks is not responsible for
this difference. The NCQ scaling is observed for all the measured particles
and anti-particles in Au+Au collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$ in
the PHSD model. However, the NCQ scaling seems better for particles than the
anti-particles. The observed NCQ scaling of $v_{2}$ suggests that parton
recombination might be responsible for the particle production and
collectivity during the partonic stage of the medium created in Au+Au
collisions at $E_{lab}=35\mathrm{~{}A~{}GeV}$. The PHSD mode which includes
partonic scatterings show larger $v_{2}$ values compared to the HSD mode. This
observation also suggests that the initially produced matter in the Au+Au
collisions at FAIR energy ($E_{lab}=35\mathrm{~{}A~{}GeV}$) is partonic in
nature. These measurements are useful for understanding the data measured in
the STAR BES program and also serve as the predictions for the CBM experiment
at FAIR, which can be verified once data become available.

## Acknowledgements

S. Kabana acknowledge the financial support received by ANID PIA/APOYO
AFB220004. This research was supported in part by the cluster computing
resource provided by the IT Division at the GSI Helmholtzzentrum für
Schwerionenforschung, Darmstadt, Germany. The authors acknowledge helpful
advices from the PHSD group members E. L. Bratkovskaya, V. Voronyuk, W.
Cassing, P. Moreau, O. E. Soloveva, and L. Oliva.

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