# Effects of mean-field and the softening of equation of state on elliptic flow in Au+Au collision at $\sqrt{s_{NN}}$ = 5 GeV from JAM model

## Abstract

We perform a systematic study of elliptic flow () in Au+Au collision at GeV by using a microscopic transport model JAM. The centrality, pseudorapidity, transverse momentum and beam energy dependence of for charged as well as identified hadrons are studied. We investigate the effects of both the hadronic mean-field and the softening of equation of state (EoS) on elliptic flow. The softening of EoS is realized by imposing attractive orbits in two body scattering, which can reduce the pressure of the system. We found that the softening of EoS leads to the enhancement of , while the hadronic mean-field suppresses relative to the cascade mode. It indicates that elliptic flow at high baryon density regions is highly sensitive to the EoS and the enhancement of may probe the signature of a first-order phase transition in heavy-ion collisions at beam energies of a strong baryon stopping region.

###### pacs:

25.75.-q, 25.75.Ld, 25.75.Nq, 21.65.+f## I Introduction

Exploring the QCD phase transition is one of the main interests in current heavy-ion physics. Calculations from lattice QCD have shown that the transition from hadronic matter to quark-gluon plasma (QGP) is a crossover nature2006 (); PRD2005 () at vanishing baryon chemical potential (), while a first-order phase transition is expected for finite baryon chemical potentials nuclear1989 (); PRD77 (); PRD2008 (). The first-order phase transition of QCD matter is related to the existence of a “softest point” in the equation of state (EoS), where the “softest point” in the EoS represents a local minimum of the ratio of the pressure to the energy density as a function of energy density Hung75 (); Rischke1996 (). The collective flows have been frequently used to explore the properties of hot and dense matter science2002 (); nuclear2005 (), since, it can reflect the properties of the matter created in early stages of heavy-ion collisions and is expected to be sensitive to the EoS. Hydrodynamical calculations show the minimum in the excitation function of the directed flow around the softest point of the EoS, and this collapse of the directed flow is proposed as a possible signal of a first-order phase transition Rischke9505 (); Brachmann:1999xt ().

Elliptic flow is also one of the most important observables which measures the momentum anisotropy of produced particles. In relativistic heavy-ion collisions at finite impact parameters, the particle momentum distribution measured with respect to the reaction plane is not isotropic and it is usually expanded in a Fourier series Poskanzer58 (); Voloshin0809 ():

(1) |

where is the emission azimuthal angle of the particles and is the reaction plane angle. The flow coefficients are a quantitative characterization of the event anisotropy, where the symbol indicates an average over all particles and all events. Elliptic flow parameter is defined as the second Fourier coefficient of the particle momentum distributions and it can be expressed as

(2) |

where, and are the (the impact parameter direction on the reaction plane), and (the direction perpendicular to the reaction plane) components of the particle momenta, respectively. Elliptic flow is expected to arise out of the pressure gradient and subsequent interactions among the constituents in non-zero impact parameter collisions. Thus it provides a plenty of information about the early-time thermalization and it is a good tool to study the system formed in the early stages of high-energy nuclear collisions Sorge78 (); PRL82 (); Teaney86 (); Heinz0907 (). The elliptic flow is one of the most extensively studied observables in relativistic nucleus-nucleus collisions (for a review see ref. Voloshin0809 ()). The elliptic flow as a function of transverse momentum (), pseudorapidity (), and centrality have been widely measured at different experiments in these decades E895 (); Alt:2003ab (); Afanasiev:2009wq (); Adare:2010ux (); Adler0206 (); Abelev77 (); Abelev81 (); Adamczyk86 (); Adamczyk110 (); Adamczyk1301 (). Transport theoretical models are used to analyze the experimental data Isse0502 (); Nasim82 (); Nasim93 (); Xu1407 (); Xu:2016ihu (); Auvinen:2013sba (); Ivanov:2014zqa ().

Although, the characteristics of at high incident energies have been extensively investigated where one expects the creation of almost baryon free QGP, it is also of great interest to perform a corresponding research for high baryon density regions, and new experiments are planned such as BES II at RHIC BESII (), FAIR CBM (), J-PARC JPARC (), and NICA NICA (). In this work, we utilized a microscopic transport model JAM Nara61 (); Nara1512 (); Nara1601 () to systematically study the centrality, transverse momentum and pseudo-rapidity dependence of in Au+Au collision at GeV, which is the top center of mass energy of Compress Baryonc Matter (CBM) at SIS100 CBMSIS100 () heavy-ion collision experiment at FAIR. In the following, we shall investigate the effects of the mean field potential and the softening of EoS on the elliptic flow by employing the JAM transport model.

This paper is organized as follows. We provide a brief description of the JAM model based on which our studies were carried out in the section II. In Sec. III, we show the transverse mass spectra of negative pion, nucleons and charged particle in GeV Au+Au collisions. On the other hand, we present our results on the centrality, transverse momentum, pseudorapidity, and beam energy dependence of elliptic flow for charged hadrons as well as protons, pions, kaons and their corresponding anti-particles. Finally, a summary of our work will be given in Sec. IV.

## Ii JAM Model

Several microscopic transport models, such as RQMD PRC52 (), UrQMD UrQMD1 (); UrQMD2 (), AMPT PRC72 (), PHSD PHSD (), and JAM Nara61 (), have been frequently used to explore (ultra-) relativistic heavy-ion collisions. JAM (Jet AA Microscopic Transport Model) has been developed based on resonance and string degrees of freedom Nara61 () similar to the RQMD and UrQMD models, in order to simulate (ultra-) relativistic nuclear collisions from initial stages of reaction to final state interactions in hadronic gas stage. In JAM, particles are produced via the resonance or string formations followed by their decays. Hadrons and their excited states are explicitly propagated in space-time by the cascade method JAM2 ().

We study the effect of hadronic mean-field potentials on elliptic flow by employing the JAM mean-field mode in which hadronic mean-field potentials are implemented based on the framework of the simplified version of the Relativistic Quantum Molecular dynamics (RQMD/S) Isse0502 (). The Skyrme type density dependent and Lorentzian-type momentum dependent mean-field potentials PRC35 () for baryons are adopted in the RQMD/S approach and the single-particle potential has the form

(3) | |||||

where is the phase space distribution function and is the baryon density. The parameters , , , , , , , are taken from Ref. Nara1512 ().

We also study the effect of the softening of EoS on elliptic flow by the method of choosing attractive orbit in two-body scattering Nara1601 (). It is well known from the virial theorem Daniel53 () that attractive orbits in each two-body hadron-hadron scattering reduce the pressure of the system. We impose attractive orbit for all two-body scatterings, thus there is no free parameter in terms of the implementation of attractive orbit mode in JAM.

Fig. 1 displays the time evolution of the local pressure and energy density extracted from energy-momentum tensor for mid-central Au+Au collisions at GeV to see the difference of EoS in the three different modes in JAM. We observe that mean-field mode in JAM shows harder EoS, while attractive orbit mode significantly lower the pressure of the matter. We showed in Ref. Nara1601 () that attractive orbit simulation yields the compatible amount of softening of EoS as EOS-Q EOS-Q () first-order phase transition. It is also seen that highest maximum energy density is achieved in the attractive orbit mode in JAM due to a soft compression of the matter, while mean-field mode yields the lowest energy density due to repulsive potential effects.

## Iii Results

In Fig. 2, rapidity distributions of protons and negative pions in mid-central collisions ( fm) are shown. Those spectra are calculated by using three different modes in JAM, including the standard cascade, mean-field, and attractive orbit. There is no significant difference among three modes except a suppression of pion yield (5%) by the mean-field as is well-known. As expected from the time evolutions of EoS, attractive orbit mode in JAM enhances slightly the yields of both protons (8%) and pions (2%) at mid-rapidity, while at , the yields are less than the cascade mode, and integrated yield over rapidity remains the same.

In Fig. 3, we show the transverse mass spectra, at mid-rapidity , for negative pion, nucleons and charged particles. By comparing with the standard JAM cascade, we found that both the mean-field and the attractive orbits mode enhance the transverse radial flow. Such enhancement of slope comes from different dynamical origin. The enhancement in the mean-field mode is due to the repulsive potential interactions, while in the case of attractive orbit mode, it is due to the creation of more compressed matter and soft expansion which result in the longer lifetime of the system. Namely, matter compressed and expand softer, and there are more interactions which create stronger radial flow. Note that the radial flow can be generated all the way from early to late stages of collisions unlike anisotropic flows which are more sensitive to the early pressure. In addition, radial flow is proportional to the work in the hydrodynamic approximation, thus essentially proportional to the system size. On the other hand, early and late pressures contribute with opposite signs to the elliptic flow Sorge78 () as we will address below. We note that the enhancement of proton collective radial transverse flow by a first-order phase transition is also seen in the hydrodynamical simulations Petersen:2009mz (); Ivanov:2013yla () as consistent with our attractive orbit simulation in JAM.

Various methods are proposed to extract the Fourier coefficients of the particle momentum distributions since the reaction plane is not known in heavy ion experiments. Before studying systematics of elliptic flow, we have compared two methods: the event plane Poskanzer58 () and two-particle cumulant method Adler0206 (); Zhou:2015eya (). These methods were already applied to the JAM simulations Nara2017 () and found that they agree with each other. Here we also check these methods for attractive orbit mode in JAM. As seen in Fig. 4, both event plane elliptic flow and the cumulant elliptic flow are in good agreement with the reaction plane elliptic flow . This is consistent with the observation by the STAR collaboration at the BES energy region Adamczyk86 () in which elliptic flow from four-particle cumulants method agrees with the values extracted from both two-particle cumulants and event plane methods for mid-central collisions at GeV. Since we do not see any significant differences among different methods in our beam energy range, we consider the reaction plane elliptic flow below.

We now present the JAM results for the centrality, transverse momentum and pseudorapidity dependence of in Au+Au collision at GeV. All results are computed directly from the formula Eq. (2) taking a true reaction plane from the JAM model. The collisions centrality is defined by the charged particle multiplicity within .

Figure 5 shows the centrality dependence of charged hadron at mid-rapidity () in Au+Au collisions at = 5 GeV. As we can see, the magnitude of the elliptic flow in semi-central collisions (20-30%) is the largest for all three modes, which are the cascade, mean-field and attractive orbit, respectively. The general trend of versus centrality for the mean-field and attractive orbit mode is similar to the cascade mode predictions. We observe that the mean-field reduces the values of charged hadron compared to the cascade mode as consistent with the previous studies by transport models science2002 (); Isse0502 (), while the attractive orbits enhance the elliptic flow of charged hadrons. In the case of the mean-field mode, higher pressures are generated in the system due to the repulsive interactions which accelerate the expansion of the participant matter. As a result, spectator matters squeeze participant matter out-of-plane more than the cascade mode which leads to the suppression of science2002 (); Sorge78 (). We note that recently different mechanism of the generation of negative has been proposed at lower beam energies around AGeV within the QMD approach LeFevre:2016vpp ().

On the other hand, the pressure is significantly reduced in the case of attractive orbit mode. Consequently, participant matter may expand much slower which reduces the interactions with the spectator matters that results in the strong in-plane emission. This might be the reason why we see the enhancement of in the attractive orbit mode.

To gain more information about the effects of mean field and the softening of EoS on the elliptic flow, we study the elliptic flow of identified hadrons (, , ) and their anti-particles in Au+Au collisions at GeV. Since the yield of anti-protons produced at this beam energy in JAM is very small, the measurement of for antiproton has large statistical error and we would not show the anti-proton in our results.

In Fig. 6, we show the centrality dependence of for particles (, , , , ) in Au+Au collisions at GeV from the JAM model in the three different modes. We observe that calculated from the attractive orbit mode show larger values for pion and kaon compared to the cascade mode, but proton is similar to the cascade mode. On the other hand, we find that the magnitude of the from the mean-field mode is smaller than the results from the cascade mode for all particles at mid-central. Thus, the enhancement of charged hadron in the attractive orbit mode observed in Fig. 5 comes mainly from the changes of pion and kaon flows. We note that the JAM mean-field result for seems to be in good agreement with the experimental data from the top AGS energy GeV E895 ().

Experimentally, the measured antiparticle is lower than the corresponding particle and the difference in between particles and their antiparticles should increase with decreasing beam energy Adamczyk110 (). However, JAM predicts that the values of for particles are similar to the results of their antiparticles. The similarity of the values of between particles and their anti-particles in JAM is due to the scalar type baryonic mean-field potentials implemented for all baryons, and no mean-field for pions and kaons. The Skyrme type density dependent potentials have been tested for a long time by QMD and BUU microscopic transport models, and it is a reasonable approximation at the beam energies under consideration from the view point of tiny number of anti-baryons produced in the colliion. In Ref. Xu1407 (); Xu:2016ihu (), it is found that the different mean-field potentials among particles and their anti-particles in the hadronic as well as partonic phases improve the description of the data on the difference of between particles and their anti-particles observed in the STAR Beam Energy Scan (BES) program.

We have also studied the pseudo-rapidity and transverse momentum dependence of the in mid-central (20-40%) Au+Au collisions. In Fig 7, the dependence of for the particles (, , ) and corresponding antiparticles (, ) are presented. The results of the JAM model for particles and antiparticles show a similar decreasing trend of with increase in . We observe that the values of for pions and kaons from the attractive orbit mode are larger than those from the cascade mode, while for protons is similar to the cascade mode prediction at mid-pseudorapidity. At the same time, from our results it is clear to see that from mean-field mode is always smaller than the results from the cascade and attractive orbit modes for all the particles.

In Figure 8, we show for identified particles as a function of the transverse momentum for in 20-40% mid-central Au+Au collisions at = 5 GeV. The results from three different modes show a similar transverse momentum dependence in . It is also observed that the proton from JAM standard cascade and JAM with attractive orbit modes are similar for low range. The results of from the cascade and attractive orbit mode are larger than the result from JAM with the mean-field mode for pions. Although the statistical error on the kaon and antikaon is relatively large, the general increasing trend of with increasing is still obvious. The difference in between the particles and corresponding antiparticles from JAM is small as expected from the integrated results.

Finally, in Figure 9, we compute a beam energy dependence of the elliptic flow for charged hadrons at mid-rapidity. It is seen that from JAM attractive mode is always greater for all beam energies up to GeV, and the effect of mean-field is to suppress . We note that for charged hadrons above GeV from the JAM attractive mode does not show any enhancement relative to the JAM standard cascade results Nara1601 (), and the effects of hadronic mean-field on is very small at SPS energies Isse0502 (). Thus an enhancement of is predicted only at the beam energy lower than 7 GeV in JAM, which is due to the suppression of squeeze-out effect by the softening of the EoS. It is known that microscopic hadronic transport model predictions including hadronic mean-field are consistent with the data up to the top AGS energy 4.7 GeV science2002 (); Pinkenburg:1999ya (); Isse0502 (), thus the scenario of the phase transition seems to be ruled out at the beam energies less than 5.0 GeV. However, there is no data between 5.0 and 7.7 GeV, and it is still interesting to measure the elliptic flow by experiment in this beam energy region in order to investigate a possible phase transition signal of a strongly interacting matter created in heavy ion collisions.

## Iv Summary

We have studied the effects of the hadronic mean-field and the softening of the EoS on the elliptic flow in Au+Au collision at GeV within the JAM model. The calculations of are performed within three different modes, which are cascade, mean-field, and attractive orbit, respectively. We observed that both mean-field and attractive orbit modes enhance the spectrum slope of nucleons and charged particles. On the other hand, we found that the value of from the attractive orbit mode is larger than the one from the cascade mode, while the mean-field mode yields less than the results from cascade mode. We have also presented the centrality, and dependence of for identified particles (, , ) and corresponding antiparticles (, ), respectively. The magnitude of from the JAM model for identified particles are similar to those for their antiparticles.

Our results indicate a high sensitivity of the elliptic flow on the pressure of the system. Hadronic mean-field generates more pressure which leads to stronger squeeze-out effect. On the other hand, the enhancement of the elliptic flow is predicted for the attractive orbit mode which leads to the softening of the EoS within the non-equilibrium microscopic simulations for the first time. The enhancement of is caused by a suppression of squeeze-out effects due to a less pressure of the system. Our results suggest that the enhancement of the elliptic flow in Au+Au collision at highest baryon density region may be used as a signal of a first-order phase transition. For the further investigations in this direction, a study of the EoS dependence of the elliptic flow by the transport approach with the EoS modified collision term Nara:2016hbg () may provide a useful information.

## Acknowledgments

This work is supported by the MoST of China 973-Project No.2015CB856901, NSFC under grant No. 11575069, 11221504. Y. N. is supported by the Grants-in-Aid for Scientific Research from JSPS (Nos.15K05079 and 15K05098).

### References

- Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo, Nature 443, 675 (2006).
- C. Bernard, T.Burch, C. DeTar et al. Phys. Rev. D 71, 034504 (2005).
- A. Masayuki, Y. Koichi, Nucl. Phys. A, 504, 668 (1989).
- P. Costa, M. C. Ruivo, and C. A. de Sousa, Phys. Rev. D77, 096001 (2008).
- S. Ejiri, Phys. Rev. D 78 074507 (2008).
- C. M. Hung and E.V. Shuryak, Phys. Rev. Lett. 75, 4003 (1995).
- D. H. Rischke and M. Gyulassy, Nucl. Phys. A 608, 479 (1996).
- H. Petersen, J. Steinheimer, M. Bleicher and H. Stoecker, J. Phys. G 36, 055104 (2009).
- Y. B. Ivanov, Phys. Rev. C 89, no. 2, 024903 (2014).
- P. Danielewicz, R. Lacey, W. G. Lynch1, Science 298, 1592 (2002).
- H. Stoecker, Nucl. Phys. A 750, 121 (2005).
- D. H. Rischke, Y. Puersuen, J. A. Maruhn, H. Stoecker and W. Greiner, Heavy Ion Phys. 1, 309 (1995).
- J. Brachmann, S. Soff, A. Dumitru, H. Stoecker, J. A. Maruhn, W. Greiner, L. V. Bravina and D. H. Rischke, Phys. Rev. C 61, 024909 (2000).
- A. M. Poskanzer and S. A. Voloshin, Phys. Rev. C 58, 1671 (1998).
- S. A. Voloshin, A. M. Poskanzer, and R. Snellings. In Relativistic Heavy Ion Physics, pages . Springer, 2010. [arXiv:0809.2949].
- H. Sorge, Phys. Rev. Lett. 78, 2309 (1997).
- H.Sorge, Phys. Rev. Lett. 82, 2048 (1999).
- D. Teaney and J. Lauret and E. V. Shuryak, Phys. Rev. Lett. 86, 4783 (2001).
- U. Heinz, Nucl. Phys. A 830, 287 (2009).
- C. Pinkenburg et al. [E895 Collaboration], Phys. Rev. Lett. 83, 1295 (1999).
- C. Alt et al. [NA49 Collaboration], Phys. Rev. C 68, 034903 (2003).
- S. Afanasiev et al. [PHENIX Collaboration], Phys. Rev. C 80, 024909 (2009).
- A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 105, 062301 (2010).
- C. Adler et al. (STAR Collaboration), Phys. Rev. C 66, 034904 (2002).
- B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 77, 054901 (2008).
- B. I. Abelev et al. (STAR Collaboration), Phys. Rev. C 81, 024911 (2010).
- L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C 86, 054908 (2012).
- L. Adamczyk et al. (STAR Collaboration), Phys. Rev. Lett. 110, 142301 (2013).
- L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C 88, 014902 (2013).
- M. Isse, A. Ohnishi, N. Otuka, P. K. Sahu and Y. Nara, Phys. Rev. C 72, 064908 (2005).
- Md. Nasim, L. Kumar, P. K. Netrakanti, and B. Mohanty, Phys. Rev. C 82, 054908 (2010).
- Md. Nasim, R. Esha, and H. Z. Huang, Phys. Rev. C 93, 044920 (2016).
- J. Xu, C. M. Ko, F. Li and T. Song, H. Liu Nucl. Phys. Rev 32, 146, (2015)
- J. Xu and C. M. Ko, Phys. Rev. C 94, no. 5, 054909 (2016).
- J. Auvinen and H. Petersen, Phys. Rev. C 88, no. 6, 064908 (2013).
- Y. B. Ivanov and A. A. Soldatov, Phys. Rev. C 91, no. 2, 024914 (2015).
- G. Odyniec, EPJ Web Conf. 95, 03027 (2015).
- C. HÃ¶hne [CBM Collaboration], J. Phys. Conf. Ser. 420, 012016 (2013).
- H. Sako et al., Nucl. Phys. A 931, 1158 (2014); H. Sako et al. [J-PARC Heavy-Ion Collaboration], Nucl. Phys. A 956, 850 (2016).
- V. Kekelidze, A. Kovalenko, R. Lednicky, V. Matveev, I. Meshkov, A. Sorin and G. Trubnikov, Nucl. Phys. A 956, 846 (2016).
- Y. Nara, N. Otuka, A. Ohnishi, K. Niita and S. Chiba, Phys. Rev. C 61, 024901 (1999).
- Y. Nara and A. Ohnishi, Nucl. Phys. A 956, 284 (2016).
- Y. Nara, H. Niemi, A. Ohnishi and H. Stoecker, Phys. Rev. C 94, 034906 (2016).
- T. Ablyazimov, et al. [CBM Collaboration], arXiv:1607.01487 [nucl-ex].
- H.Sorge, Phys. Rev. C 52, 3291 (1995); Phys. Lett. B 402, 251 (1997).
- S. A. Bass et al., Prog. Part. Nucl. Phys. 41, 255 (1998).
- M. Bleicher et al., J. Phys. G 25, 1859 (1999).
- Z. W. Lin, C. M. Ko, B. An. Li, B. Zhang and S. Pal, Phys. Rev. C 72 064901 (2005).
- W. Cassing and E. L. Bratkovskaya, Nucl. Phys. A 831, 215 (2009).
- T. Hirano and Y. Nara, PTEP 2012, 01A203 (2012).
- C. Gale, G. Bertsch, S. Das Gupta, Phys. Rev. C 35 1666 (1987).
- P. Danielewicz and S. Pratt, Phys. Rev. C 53, 249 (1996).
- P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Lett. B 459, 667 (1999); P. F. Kolb, J. Sollfrank and U. W. Heinz, Phys. Rev. C 62, 054909 (2000); H. Song and U. W. Heinz, Phys. Rev. C 77, 064901 (2008).
- Y. Nara, H. Niemi, A. Ohnishi, J. Steinheimer, X. Luo and H. Stöcker, arXiv:1708.05617 [nucl-th].
- Y. Zhou, K. Xiao, Z. Feng, F. Liu and R. Snellings, Phys. Rev. C 93, no. 3, 034909 (2016).
- A. Le Fevre, Y. Leifels, C. Hartnack and J. Aichelin, arXiv:1611.07500 [nucl-th].
- C. Pinkenburg et al. [E895 Collaboration], Phys. Rev. Lett. 83, 1295 (1999).
- K. Filimonov et al. [CERES/NA45 Collaboration], AIP Conf. Proc. 610, 556 (2002).
- Y. Nara, H. Niemi, J. Steinheimer and H. Stoecker, Phys. Lett. B 769, 543 (2017).