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

TL;DR

This paper investigates the elliptic flow ($v_2$) of identified hadrons in Au+Au collisions at 35A GeV using the PHSD model, revealing centrality dependence, particle-antiparticle differences, and the influence of partonic interactions.

Contribution

It provides detailed predictions of $v_2$ behavior, including NCQ scaling and mode comparisons, highlighting the role of partonic medium formation at this energy.

Findings

Higher $v_2$ in PHSD mode indicates partonic medium formation.

Significant $v_2$ difference between particles and antiparticles.

Centrality dependence of $v_2(p_T)$ observed.

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).

Equations

$$E\frac{d^{3}N}{d{p}^3}=\frac{1}{2\pi }\frac{d^{2}N}{p_{\mathrm{T}}d{p}_{T}dy}\left(1+2\sum _{n=1}^{\infty }{v}_n\cos [n(\varphi -{\psi }_{RP})]\right).$$

$$v_n=⟨\cos [n(\varphi -{\psi }_{RP})]⟩,$$

$${\psi }_n=\frac{1}{n}{\tan }^{-1}\left[\frac{{\sum }_i{\omega }_i\sin (n{\varphi }_i)}{{\sum }_i{\omega }_i\cos (n{\varphi }_i)}\right].$$

$$R_2=\sqrt{⟨\cos [2({\psi }_2^a-{\psi }_2^b)]⟩},$$

Full text

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 :

[TABLE]

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,

[TABLE]

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%.

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,

[TABLE]

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,

[TABLE]

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 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.

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 .

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 $\mathrm{\Delta}v_{2}$.

4.4 Number of constituent quark scaling

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 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.