Dependence on beam energy and nuclear equation of state dependence of anisotropic flow and particle production in low-energy heavy-ion collisions
Sumit Kumar Kundu, Yoshini Bailung, Sudhir Pandurang Rode, Partha Pratim Bhaduri, Ankhi Roy
BBeam energy and nuclear equation of state dependence of anisotropicflow and particle production in low energy heavy-ion collisions
Sumit Kumar Kundu, Yoshini Bailung, Sudhir Pandurang Rode, Partha Pratim Bhaduri, and Ankhi Roy Discipline of Physics, School of Basic Sciences,Indian Institute of Technology Indore, Indore 453552 India Variable Energy Cyclotron Centre, HBNI, 1/AF Bidhan Nagar, Kolkata 700 064, India (Dated: February 12, 2021)We analyse various flow coefficients of anisotropic momentum distribution of final state particlesin mid-central ( b = 5–9 fm ) Au + Au collisions in the beam energy range E Lab = 1 A − A GeV.Different variants of the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) model, namelythe pure transport (cascade) mode and the hybrid mode, are employed for this investigation. In thehybrid UrQMD model, the ideal hydrodynamical evolution is integrated with the pure transportcalculation for description of the evolution of the fireball. We opt for the different available equationsof state (EoS) replicating the hadronic as well as partonic degrees of freedom together with possiblephase transitions, viz. hadron gas, chiral + deconfinement EoS and bag model EoS, to investigatetheir effect on the properties of the final state particles. We also attempt to gain insights aboutthe dynamics of the medium by studying different features of particle production such as particleratios and net-proton rapidity distribution. The results and conclusions drawn here would be usefulto understand the response of various observables to the underlying physics of the model as wellas to make comparisons with the upcoming measurements of the future experiments at Facility forAntiproton and Ion Research (FAIR) and Nuclotron-based Ion Collider fAcility (NICA).
I Introduction
One of the main objectives of the modern day rela-tivistic heavy-ion physics research is to understand thephase structure of the strongly interacting matter at ex-treme conditions of temperatures and net baryon densi-ties in the laboratory [1, 2]. Possible existence of criti-cal point of QCD matter along with phase transition todeconfined state motivates the high energy communityto continue the efforts in this direction. Exploration ofthe QCD matter at finite baryon densities is relativelyless extensive compared to the one created at negligi-ble baryon densities. Ample amount of investigationshave been performed in the latter direction in past twodecades with various experiments at Relativistic HeavyIon Collider (RHIC) [3, 4] and Large Hadron Collider(LHC) [5–7]. Upcoming experiments at future accelera-tor facilities such as Nuclotron-based Ion Collider fAcility(NICA) [8] and Facility for Antiproton and Ion Research(FAIR) [9, 10] aim to probe the baryon rich matter withgood precision. However an optimal use of these facili-ties demand an extensive analysis of the available dataand model based studies of different observables in thesimilar energy domain.The anisotropic flow of the particles emitted in non-central relativistic heavy-ion collisions is considered as apromising observable to investigate the collective effectsof the produced medium. Originated due to the pressuregradient as a result of the multiple scatterings among theconstituents of the medium, it is vulnerable to the un-derlying nuclear equation of state. Azimuthal anisotropyin momentum distribution of the final state particles isquantified in terms of various harmonic coefficients us-ing the Fourier series. These different anisotropic flow coefficients can be expressed as, v n = < cos[ n ( φ − Ψ)] > where azimuthal angle of the particle and reaction planeangle are indicated by φ and Ψ, respectively. Moreover, v n is defined as directed flow ( v ), elliptic flow ( v ), tri-angular flow ( v ), quadrangular flow ( v ) for n = 1, 2, 3,4 and so on, respectively. These coefficients are believedto provide an insight on dynamics of the fireball. For in-stance, significant magnitude of the elliptic flow has shedlight on the possibility that the bulk of the produced mat-ter achieve close to local thermal equilibrium conditions.The pressure gradient developed due to rescatterings inthe early stage of the collisions converts the initial statespatial anisotropy to final state momentum anisotropyand elliptic flow ( v ). Several experiments [11, 12] atdifferent energies have examined v for the possible sig-nature of thermalization of the produced medium. Sub-stantial amount of study has been performed to inspectelliptic flow in low energy collisions at various beam en-ergy ranges [13–15] availing variety of microscopic trans-port models [16–19]. At low beam energies, change ofsign, i.e. transition from out-of-plane to in-plane flowhas been observed [20, 21].On the other hand, the directed flow, v , quantifies thedeflection of the produced particles in the reaction plane.Sensitivity to the longitudinal dynamics and possibilityof being developed prior to v [22–24], make v worthstudying in relativistic nuclear collisions. The magni-tude of directed flow is expected to vanish in the vicinityof the phase transition due to softening of the underly-ing EoS and this makes it an exciting observable for theanalysis at RHIC-BES, FAIR and NICA energies. Bulkamount of the activities has been carried out in this di-rection in a past few decades at various experiments. For a r X i v : . [ nu c l - t h ] F e b instance, the slope of the directed flow being the measureof the signal strength, shows linearity at the midrapidityat AGS [25–27] energies and below. However, this lin-earity at midrapidity is not expected to be maintainedat higher beam energies because of the slope at midra-pidity is found to be different than that at beam rapidityat energies above SPS [28–30]. Hydrodynamical modelcalculations indicate that the so called structure ”wig-gle” is sensitive to the underlying EoS [31–33]. Studyof higher order harmonics has gained some attention ina past few years and expected to provide insights aboutthe produced fireball. Fourth order harmonic coefficient, v has been known to be sensitive to intrinsic ellipticflow [34–36] and therefore, it is quite interesting to inves-tigate it over wider range of the beam energies which hasalso been attempted using microscopic transport model,JAM [23, 37]. It bears some crucial details about thecollision dynamics predicted by hydrodynamical calcula-tions [36].In this article, we make some efforts to address thenuclear equations of state dependence of the anisotropicflow coefficients and particle production in non-central( b = 5–9 f m ) Au–Au collisions in very wide ranges ofthe beam energies, E Lab = 1 A − A GeV which spanover existing GSI-SIS energy of HADES experiment upto top SPS energy. For this purpose, we employ thepublicly available version 3.4 of the UrQMD model withdifferent configurations of hybrid UrQMD for the inter-mediate hydrodynamical stage viz., Hadron Gas (HG),Chiral + deconfinement EoS and Bag Model EoS alongwith pure transport approach. The latter two mimic thepartonic degrees of freedom and phase transition in themedium, however, the first one includes hadronic degreesof freedom only. The reaction plane angle (Ψ) is takento be zero within this model. It is important to notethat this is not first time UrQMD model with hydro-dynamical plug-in is used to study anisotropic flow atlow beam energies. In [38], the authors have calculatedthe transverse momentum and rapidity dependence of v and v at 40A and 160A GeV in Pb + Pb collisions us-ing standard UrQMD model at various centralities whichshowed disagreement with NA49 experimental measure-ment. In addition, v and v were also studied as afunction of beam energy in the range of E Lab = 90AMeV to E cm = 200A GeV and also, showed disagree-ment with the available data. In Ref [39], the excitationfunction of v was examined in the range of GSI-SIS toCERN-SPS energies using UrQMD with HG EoS withinhybrid approach and other harmonics such as, v and v are studied with Chiral EoS in Au–Au systems between √ s NN = 5–200 GeV [15]. The collision energy depen-dence of v is tested using the hybrid model for nuclearreactions between √ s NN = 3 −
20 GeV [40]. In our pre-vious work [41], study on nuclear equations of state de-pendence of anisotropic flow was performed using hybridUrQMD model within 6A–25A GeV with HG and chiralEoS. All these results seem to suggest the applicabilityof this model to real scenario is rather limited. How- ever, in this paper, we qualitatively aim to understandthe effect of various nuclear equations of state on theflow harmonics and hence gain some insights about thedynamics leading to their development at various beamenergies ranging from 1A–158A GeV.Besides anisotropic flow, we also attempt to study theEoS dependence of particle production in non-central col-lisions. The particle ratios of various species are exam-ined for this purpose. We also look at the net-protonrapidity distributions. The structure of the net-protonrapidity spectra at the mid rapidity is expected to besensitive to the underlying EoS of the nuclear fireball. Incentral collisions, adequate studies have been performedin this direction. In Refs. [42–46], the authors have quan-tified the structure of net proton rapidity distribution atmid rapidity in central collisions, in terms of reduced cur-vature. It was studied as a function of beam energy andcompared with predictions incorporating various possiblescenarios of fireball expansion. In the present article weextend these studies to the mid central collisions.This article is arranged in the following order. In sec-tion II, basic principle of UrQMD model and its differ-ent variants are briefly introduced. The obtained resultson anisotropic flow coefficients and particle productionproperties over a very wide range of colliding energiesare presented in section III. Finally we summarize theresults in section IV.
II Model description
For detailed description of the Ultra-relativistic Quan-tum Molecular Dynamics (UrQMD) model, the reader isreferred to Refs. [16, 17, 47]. The purpose of the UrQMDmodel is to simulate high energy nucleus-nucleus colli-sions. The initializations of the target and projectile nu-clei in co-ordinate and momentum space are done withthe help of Woods-Saxon profile and Fermi gas model,respectively. Together with the various experimental in-puts such as cross-sections, decay widths, the collisionsin the model are narrated in terms of interactions amongresonances, hadrons and their excited states at low en-ergies and in terms of excitations of color strings withtheir subsequent fragmentation into hadrons at higherenergies [17]. The propagation of hadrons is taken placeon straight line trajectories amid subsequent collisions.In the hybrid version of the UrQMD, the 3D relativisticfluid dynamical evolution is combined with pure trans-port approach for a better modelling of the intermediatehot and dense stages of the collision. The calculation ofinitial state of the hydrodynamical evolution is crucialto account for non-equilibrium nature of the early stage,moreover, this also incorporate event-by-event fluctua-tions of the initial states. The hydrodynamical evolu-tion is commenced upon crossing of the two Lorentz-contracted nuclei [47]. Thereafter, the mapping of parti-cles which are treated as ”point-like” in the initial stage,to hydrodynamic grid is performed while the spectatorsare propagated in the cascade. Right after this, thehydrodynamical evolution is performed for which equa-tion of state (EoS) serves as one of the important in-puts. After dropping of energy density (cid:15) below five timesthe ground state energy density (cid:15) in all cells [47], thehydrodynamical evolution ceases and the hadronizationon an iso-energy density hypersurface is performed bymapping the hydrodynamical fields to the hadrons usingCooper-Frye prescription [48]. Thereafter, the hadronsare evolved through rescatterings and decays until thedecoupling of the system.In hydro mode, there are several available EoS that canbe employed. One of them is Hadron Gas (HG) EoS [49]which has similar underlying degrees of freedom as puretransport approach. It consists of non-interacting gas ofhadrons described by grand canonical ensemble and doesnot incorporate any type of phase transition. This givesan excellent opportunity to compare the hydrodynamicaland pure transport approach on equal footings.The other possible choice includes the BAG modelEoS [50]. It has an in built first order deconfinementphase transition anticipated at finite baryon densities.In this EoS, an improved version of σ − ω − model withrealistic effective nucleon mass and ground state incom-pressibility values is employed in case of hadronic matterwhereas, standard MIT bag model is recruited for theQGP phase. During the transition, both these phasesare matched with the help of Gibbs’ conditions for phaseequilibrium [50].Moving on, there is another available EoS named Chi-ral + deconfinement EoS [51] employed in this investiga-tion. Both chiral as well as deconfinement phase transi-tions are included in this EoS while the latter is a contin-uous cross over all finite net baryon densities ( µ B ). Thechiral phase transition is administrated by hadronic in-teractions whereas, deconfinement transition via quarksand Polyakov potential. The partonic degrees of freedomonly show up at higher temperatures where hadrons dis-appear. At vanishing µ B this EoS matches well with thelattice QCD simulations. Independent use of three dif-ferent EoS within the hybrid UrQMD model enables usto compare three distinct fireball evolution scenarios overthe entire enrgy domain investigated in this work. III Results and Discussion
In this section, we present the results of our investiga-tions on various anisotropic flow coefficients at differentbeam energies for charged and identified hadrons. Allthe three EoS mentioned above are employed for thispurpose.Then we move on to investigate the sensitivityof underlying EoS to the different particle productionmechanisms such as strange to non-strange ratio, baryonto meson ratio and so on. Finally, we also look at thenet proton rapidity spectra for different EoS to look forpossible insights into the longitudinal dynamics of themedium.
A. Anisotropic flow coefficients
Among various harmonic coefficients, directed flow isbelieved to hold sensitivity against the longitudinal dy-namics of the QCD medium. Therefore, we start by esti-mating the directed flow of charged hadrons as a functionof rapidity at different beam energies and for pure trans-port and hybrid versions of UrQMD model. The resultsare shown in Fig. 1. In presence of hydrodynamic ex-pansion, the slope at mid-rapidity remains positive at allinvestigated energies. For a pure transport approach, theslope initially remains positive and eventually becomesnegative.Directed flow of pions and protons at 40A and 158AGeV are compared with the existing measurement byNA49 experiment at SPS in 10–40% central Au + Aucollisions as shown in Fig. 2. Hybrid mode fails to ex-plain the slope of directed flow except for pions at 40AGeV. The pure transport approach is seen to do a betterjob explaining the proton v reasonably well at both en-ergies at midrapidity an observation in line with previousstudies [38, 52].Slopes of directed flow of charged hadrons, pions, pro-tons and net-protons as a function of beam energy arequantified in Fig. 3. The slope is obtained by fitting dif-ferential directed flow ( v ( y )) using first order polynomialat mid-rapidity. Similar values of slopes are spotted in allthree cases of hydro mode up to 10A GeV for all species.The slope using cascade mode is smaller compared to hy-dro mode. For pions, the slope obtained in cascade modelalways remain negative for all investigated energies andshow transition from negative to positive value between30A to 80A GeV in case of hydro mode. The slope doesnot show any sensitivity to underlying dofs brought byHG and chiral EoS in charged hadrons case which wasalso observed in our previous study between beam ener-gies 6A–25A GeV [41]. Moreover, we tend to see a slighthint of sensitivity in protons and net-protons case beyond25A GeV, however, we cannot make any strong claim atthe moment. In all three EoS cases of hydro mode, theminimum is observed between 10A–80A GeV. However,in case BG EoS, the minimum occurs near 10A–25A GeVand in case of other two EoS, where the minimum is abit shifted and lies between 25A–80A GeV. This shiftin minimum somehow leads to a splitting of slope of di-rected flow between bag model and other two EoS whichlie around 25–30A GeV. A strong increase of slope incase of BG model is observed which could possibly beas a result of the first order phase transition incorpo-rated in the bag model and perhaps, hint towards thepossible onset of deconfinement. In the past, similar in-teresting feature around similar beam energy has beenobserved for strange to non-strange ratio around simi-lar energies [53]. Moreover, the slope of directed flowof protons is compared with the available experimentalmeasurements of E895 [25], NA49 [54] and STAR [55] asdepicted in the middle plot of Fig. 3. It reveals that theresults with hydrid mode overestimate the data beyond c.m. y − − − − v − Charged Hadrons: Cascade1A GeV2A GeV4A GeV6A GeV8A GeV10A GeV 25A GeV30A GeV40A GeV80A GeV120A GeV158A GeV c.m. y − − − − v − Charged Hadrons: Hadron Gas1A GeV2A GeV4A GeV6A GeV8A GeV10A GeV 25A GeV30A GeV40A GeV80A GeV120A GeV158A GeV c.m. y − − − − v − Charged Hadrons: Chiral1A GeV2A GeV4A GeV6A GeV8A GeV10A GeV 25A GeV30A GeV40A GeV80A GeV120A GeV158A GeV c.m. y − − − − v − Charged Hadrons: Bag Model1A GeV2A GeV4A GeV6A GeV8A GeV10A GeV 25A GeV30A GeV40A GeV80A GeV120A GeV158A GeV
FIG. 1: Directed flow of charged hadrons as a function of rapidity at different beam energies for different configurations ofUrQMD.
2A GeV. According to the fluid dynamical calculations,the slope of directed flow of the baryons is expected tochange sign attributed to softening of EoS in the pres-ence of first order phase transition. This was tested withvarious freeze-out scenarios using hydrodynamical simu-lations in Ref. [40]. On the other hand, the results withcascade mode underestimate the measurements below 6AGeV and thereafter, show similar trend with slight over-estimation above 30A GeV.Moving forward, we attempt to look at the net-protonsin more detail by inspecting their p T -integrated directedand elliptic flow at midrapidity (-0.5 < y c.m. < v as its slope in all fourcases studied here. Moreover, feature of splitting at 20–30A GeV in case of hydro case is also observed. While incase of v in the right plot, we witness a similar splittingbetween bag model EoS and other two EoS and further-more, a enhanced broad peak between 10–25A GeV incase of bag model which is appeared as a dip in case of v and its slope. We repeat this exercise of v for kaonsand pions as shown in Fig. 5 and here also, kaons andpions confirm the splitting in hydro case, however, thesplitting is not prominent in case of pions. We now move our focus to look at the higher orderflow harmonic coefficient v which has argued to be gen-erated under the influence of 4 th order moment of fluidflow and the intrinsic elliptic flow, v [34–36]. Under theassumption of ideal fluid dynamics and without of anyfluctuations, v and v are related to each other as, v =0.5( v ) . So one can expect to acquire some informationabout the dynamics of the prevailing medium by esti-mating the ratio v / ( v ) . This ratio has been studied inour previous work [41] within beam energy range 6A–25AGeV for different equations of state except Bag model.Prior to this, some phenomenological study has beenperformed for this observable. In particular, the obser-vations using Parton-Hadron-String Dynamics (PHSD)model [56] at different beam energies with Au–Au colli-sions, have shown the ratio v / ( v ) ≈
2. Moreover, theauthors at Ref. [37] have attempted to investigate theenhancement of v in low energy nuclear collisions usingJAM model. Experimentally, the results at RHIC [57–60] indicated the ratio to be unity. Fig 6 depicts theratio as a function of beam energy (E Lab ) for differentEoS and the values always remain below 2 for all fourcases. The ratio v / ( v ) has been claimed to be in asso-ciation with the phenomenon of incomplete equilibration c.m. y1.5 − − − V − − = 40A GeV lab Pions; ECascadeHadron GasChiralBag ModelPions; Data10 40%; NA49 c.m. y1.5 − − − V − − = 158A GeV lab Pions; ECascadeHadron GasChiralBag ModelPions; Data10 40%; NA49 c.m. y1.5 − − − V − − − = 40A GeV lab Protons; ECascadeHadron GasChiralBag Model Protons; Data10 40%; NA49 c.m. y1.5 − − − V − − − = 158A GeV lab Protons; ECascadeHadron GasChiralBag ModelProtons; Data10 40%; NA49
FIG. 2: Comparison of directed flow of pions and protons as a function of rapidity with experimental measurements at 40Aand 158A GeV for different configurations of UrQMD with measured directed flow. in the literature [61]. However, the authors have studiedthis observable as a function of K − , number of collisionsper particle. With K being the Knudsen number whichis a dimensionless quantity and a measure of degree ofthermalization, it is a function of system size and beamenergy. The local equilibration is expected to be reachedwhen K − (cid:29)
1. Moreover, the deviations from ideal hy-drodynamics lead to incomplete thermal equilibrium. Asratio shown in Fig. 6 deviate from 0.5, giving the im-pression that the system is not fully equilibrated, thusprevent the use of ideal hydrodynamics in these beamenergy regimes. The results here can be used to makesome robust claims on the degree of thermalization ofthe nuclear fireball after comparison with the data avail-able from future experiments at FAIR and NICA.At last, we attempt to look at the NCQ scaling in theflow coefficients for beam energies examined in this inves-tigation. For this, we specially look at the slope of thedirected flow of various species and their combination un-der the assumption of coalescence sum rule [62, 63] forall four cases of UrQMD and the results are shown inFig 7. First, similar to Ref. [63], we compare the dv /dy values of ¯Λ ( uds ) (black markers) with ( K − (¯ us ) + ¯ p ( uud )) (red markers) where, the same flow for s and ¯ s and similarly, for ¯ u and ¯ d is assumed. Though our re-sults are quantitatively higher than the ones presentedin Ref. [63], however qualitatively, the sum rule seemedto be followed for these two cases at higher beam ener-gies with slight hint of violation below 25A GeV whichat the moment, can not be strongly claimed due to largeuncertainties in all four cases. For same reason, we plotour results above 8A GeV upto 158A GeV. Moreover, wealso look at one more set which is not as simple as ear-lier one. As discussed in Ref. [63], different directed flowfor transported and produced quarks is expected whichare not easy to distinguish in practice. The compari-son of dv /dy of net Λ ( uds ) (blue triangular markers)with the calculation comprising different combinations ofnet p ( uud ), ¯ p ( uud ) and K − (¯ us ) (pink circle and bluesquare markers) is shown in Fig. 7. The combination of K − and ¯ p would give s quark which is assumed to re-place produced u quark in net p in the first coalescencecalculation (pink circle markers). This calculation is ex- quarks transported from the initial nuclei produced in the interactions [GeV] lab E c . m . = y / d y | d v - Charged hadronsCascadeChiralHadron GasBag Model [GeV] lab E c . m . = y / d y | d v - - - – p CascadeChiralHadron GasBag Model [GeV] lab E = c . m . y / d y | d v - ProtonCascadeChiralHadron GasBag Model
Data: E895, NA49, STAR [GeV] lab E c . m . = y / d y | d v - Net-ProtonsCascadeChiralHadron GasBag Model
FIG. 3: Slope of the directed flow of charged hadrons, pions, protons and net-protons as a function of beam energy at midrapidityfor different configurations of UrQMD. [GeV] lab E V V CascadeHadron GasChiralBag Model [GeV] lab E V - V CascadeHadron GasChiralBag Model
FIG. 4: p T integrated directed ( v ) and elliptic ( v ) flow of net-protons as a function of beam energy at midrapidity (0 < y c.m. < pected to hold true at relatively higher energies wheremost of the quarks are produced and may not be validat beam energies considered in this investigation, and itseems to be the case from our observations as shown inFig. 7 for all four cases of UrQMD. Contrary, in the sec-ond calculation where net p is added up with s quark, itis assumed that the transported quarks have dominant contribution in net p , which is quite suitable in the limitof low beam energies, and one of quarks is replaced by s quark. As expected this calculation shows a nice agree-ment with net Λ between 25A–158A GeV which thenbreaks down below 25A GeV in all four cases. This fur-ther may indicate towards possible confinement to decon-finement transition above 25A GeV which has been pre- [GeV] lab E V - - – ; K V CascadeHadron GasChiralBag Model [GeV] lab E V - - – p ; V CascadeHadron GasChiralBag Model
FIG. 5: p T integrated elliptic flow of kaons and pions as a function of beam energy at midrapidity (-0.5 < y c.m. < [GeV] lab E10 ) / ( V V − ) /(V V Chiral [GeV] lab
E10 ) / ( V V − lab E10 ) / ( V V − lab E10 ) / ( V V − FIG. 6: V / ( V ) of charged hadrons as a function of beam energy at midrapidity (-0.5 < y c.m. < dicted in prior studies and also in our investigations ear-lier in this section. It is interesting to see the agreementof these sum rule calculations with EoS cases where theunderlying degrees of freedom are not partonic and thisneeds to be understood. However, this is not first timeone has seen the scaling behavior using pure transportUrQMD approach [13]. This also brings up the questionof whether the underlying assumption of coalescence isindeed the source of this agreement. As mentioned ear-lier, the particle production in UrQMD at higher ener- gies is performed in terms of string excitation and subse-quent fragmentations as narrated in Refs. [17, 64]. Asper the string-excitation scheme, the quark-antiquarkor diquark-antidiquark pairs are spontaneously formedin color flux tube between initial quarks and subse-quently, mesons and baryons are produced. The pro-duced hadrons undergo multiple scatterings, however, nostring will be involved after certain energy limit ( √ s < [GeV] lab E c . m . = y / d y | d v - - Cascade L p 31 + - K L net - + Kp 32net p - - + Kp 31 net p - 31net p - [GeV] lab E c . m . = y / d y | d v - - Hadron Gas L p 31 + - K L net - + Kp 32net p - - + Kp 31 net p - 31net p - [GeV] lab E c . m . = y / d y | d v - - Chiral L p 31 + - K L net - + Kp 32net p - - + Kp 31 net p - 31net p - [GeV] lab E c . m . = y / d y | d v - - Bag Model L p 31 + - K L net - + Kp 32net p - - + Kp 31 net p - 31net p - FIG. 7: Comparison of slope of directed flow of net lambda and anti-lambda with various combinations of hadrons under theassumption of coalescence sum rule as a function of beam energies for different configurations of UrQMD. [GeV] lab E - p / - K - - p / - K CascadeHadron GasChiralBag Model ; Data - p / - K 10--20%20--30%30--40% [GeV] lab E + p / + K + p / + K CascadeHadron GasChiralBag Model; Data - p / - K 10--20%20--30%30--40% [GeV] lab E ) - p + + p ) / ( - + K + ( K - p + + p )/( - + K + (K CascadeHadron GasChiralBag Model FIG. 8: K − to π − , K + to π + and (K + + K − )/( π + + π − ) ratio as a function of beam energy for different configurations ofUrQMD and their comparison with experimental measurements. [GeV] lab E - / K + K - /K + K CascadeHadron GasChiralBag Model ; Data - /K + K 10--20%20--30%30--40% [GeV] lab E - p / + p - p / + p CascadeHadron GasChiralBag Model; Data - p / + p [GeV] lab E / PP FIG. 9: K − to K + , π + to π − and anti-proton to proton ratio as a function of beam energy for different configurations ofUrQMD and their comparison with experimental measurements. [GeV] lab E + p P / + p P/ CascadeHadron GasChiralBag Model; Data + p P/ 10--20%20--30%30--40% [GeV] lab E - p / P - - p /P CascadeHadron GasChiralBag Model; Data - p /P 10--20%20--30%30--40% FIG. 10: Proton to π + and anti-proton to π − ratio as a function of beam energy for different configurations of UrQMD andtheir comparison with experimental measurements. model (AQM) is implemented in UrQMD to estimate theunknown hadronic cross-sections [17]. This model as-sumes the existence of very weakly interacting dressedvalance quarks inside the hadrons. B. Particle ratios
In this subsection, we proceed to investigate and un-derstand the effect of different degrees of freedom andphase transition on the particle production. For this,we obtain various particle ratios namely, strange to non-strange, anti-particle to particle and compare them withthe available data. In central collisions, the K + /π + ratio,has been a unique measure of the onset of deconfinementin the literature [53]. It will be interesting to see the be-havior of this observable in case of non-central collisions.We estimate various particle ratios to procure insightsabout the medium properties by studying the impact ofvarious equations of state. In Fig. 8, we show ratio ofK − /π − , K + /π + and (K + + K − ) / ( π + + π − ) as a function of beam energy for different cases of EoS. In left plot,K − /π − shows a monotonic rise for all beam energies ex-cept for bag model which saturates after 20A GeV. Inthe middle plot, K + /π + ratio shows a similar increasingbehavior up to 4A GeV and then start to decrease withhint of stronger drop in case of bag model from 20A–30AGeV. In the right most plot, (K + + K − ) / ( π + + π − ) isobtained as a function of beam energy and similar split-ting seen earlier at 20–30A GeV in case of bag model isobserved. The ratio seems to saturate beyond this rangein other cases. Both K − /π − and K + /π + ratios are com-pared with the experimental data from NA49 experimentat three different centralities as these seem to cover im-pact parameters considered in this work. The chiral andhadron gas EoS are able to reproduce the trend set bydata in both these ratios however the magnitude is over-estimated. Furthermore, we also look at the antiparticleto particle ratio for different EoS. In Fig. 9, K + / K − , π + /π − and ¯p / p ratios are depicted for all four cases ofEoS. K + / K − ratio shows an increase for all beam ener-gies and EoS, however, no sensitivity to the EoS is shown0 c.m. y1 - - d N / d y UrQMD: Cascade1A GeV2A GeV4A GeV6A GeV 8A GeV10A GeV25A GeV30A GeV 40A GeV80A GeV120A GeV158A GeV
Net-protons c.m. y1 - - d N / d y UrQMD: Hadron Gas1A GeV2A GeV4A GeV6A GeV 8A GeV10A GeV25A GeV30A GeV 40A GeV80A GeV120A GeV158A GeV
Net-protons c.m. y1 - - d N / d y UrQMD: Chiral1A GeV2A GeV4A GeV6A GeV 8A GeV10A GeV25A GeV30A GeV 40A GeV80A GeV120A GeV158A GeV
Net-protons c.m. y1 - - d N / d y UrQMD: Bag Model1A GeV2A GeV4A GeV6A GeV 8A GeV10A GeV25A GeV30A GeV 40A GeV80A GeV120A GeV158A GeV
Net-protons
FIG. 11: Rapidity spectra of net-protons at various beam energies for different equations of state. [GeV] lab E = c . m . y / d N / d y | = c . m . y | N / d y d - - CascadeChiralHadron GasBag Model
FIG. 12: Reduced curvature of rapidity spectra of net-protonsas a function of beam energy for different configurations ofUrQMD at midrapidity. in hydro mode. In the middle plot of Fig 9, we see samemagnitude of π + /π − ratio for all EoS at all energies be-yond 4A GeV with decreasing trend as a function of beamenergy. Moreover, data seem to favor hydro mode forboth ratios with slight underestimation by hydro in thecase of K + / K − . Ratio of anti-proton to proton is shownin the right most plot and compared with experimentaldata. Measurements are relatively underestimated by themodel in all cases of EoS. Finally, we study the p /π + and¯p /π − ratios and compare them with the data as shown inFig. 10. In the former case, the ratio is inversely propor-tional to beam energy and show similar magnitude forhydro case with agreement with the experimental mea-surement, however, remain slightly higher in cascade forall beam energies, depicted in the left plot. In the rightplot of Fig. 10, ¯p /π − ratio shows similar trend as dataand sensitivity to all EoS. C. Net-proton rapidity spectra
Understanding the in-medium properties of stoppedprotons by studying their rapidity distributions havebeen a promising observable. In Refs. [42–46], plethoraof studies in this direction has been performed. It hasbeen argued that the irregularities in the stopped protons1may well be the consequence of onset of deconfinementtransition. This occurs due to inherited softest point inthe nuclear equations-of-state in the vicinity of a phasetransition. Such investigations are generally performedin central collisions however, it is also worth to checkthis in non-central collisions. The shape of rapidity spec-tra at midrapidity may contain very crucial informationabout medium and believed to be sensitive to the un-derlying nuclear equations of state. Therefore, we lookat the net-proton rapidity distribution at different beamenergies and equations of state. In Fig. 11, we show ra-pidity distribution of net-protons at mid rapidity for allenergies and EoS considered in this work. Rapidity spec-tra remain flat at high beam energies in case of cascademode in contrast to hydro mode where it shows a veryinteresting feature.As the irregularities in the shape of rapidity spectraat midrapidity can potentially help in explaining the dy-namics of the medium, we quantify the nature of the spec-tra at midrapidity by calculating the double derivative ofthe rapidity spectra at midrapidity i.e. global minima ormaxima as shown in Fig. 12. This quantity is identicalto the one obtained in the Refs. [42–46] and is referred asreduced curvature. For this, the rapidity distributions ofnet-protons are fitted with polynomial at the midrapidityfor all beam energies and EoS. As shown in Fig. 12, thereduced curvature in case of cascade remains constantand zero for all energies. As soon as the hydrodynami-cal evolution is introduced, the corresponding observableshow some sensitivity as a function of beam energy. Wedo not see the so-called “peak-dip-peak-dip” irregularityas seen in the experimental observations and in the cen-tral collisions [42–46]. It is interesting to note that thisobservable has led to show the sensitivity between Chiraland Hadron gas EoS beyond 25A GeV which is the sameenergy at which we have seen some interesting feature forother observables investigated in this work.. The man-gitude and slope of reduced curvature is highest for Bagmodel and decreases for Chiral to Hadron gas beyond25A GeV. On the other hand, it is worth to notice thatthe net-protons are the only species which showed anysensitivity to the underlying degrees of freedom and thatto interestingly for the observable and kinematic vari-able related to longitudinal dynamics such as directedflow and rapidity.
IV Summary
In this article, we have dedicated our efforts to under-stand the impact of various nuclear equations of state onthe several observables of the nuclear matter producedin the low energy collisions of heavy-ions in wide rangeof beam energies, 1A–158A GeV. The UrQMD modelwith intermediate hydrodynamical evolution was em-ployed with different nuclear equations-of-state such asHadron gas, Chiral + deconfinement and Bag model. Westarted with examining the anisotropic flow coefficients of charged and identified hadrons in above-mentioned beamenergy range. A unique feature at 25–30A GeV in theenergy dependence of slope of directed flow of chargedhadrons, protons and net-protons was observed. Theslope using bag model, showed a splitting, leading toa sharp rise compared to other two equations-of-state.This may be attributed to the incorporated first orderphase transition in the former case. Similar feature wasobserved in case of directed flow as well. Apart from thesplitting, the dip within certain energy range for theseequations-of-state hint towards possible onset of decofine-ment. Moreover, we noticed that study of net-protonswith respect to longitudinal component certainly bringsout the sensitivity to underlying degrees of freedom inchiral and hadron gas EoS beyond 20–30A GeV beam en-ergy, however, more evidence in this direction is requiredto make any robust claim. Along with this, efforts havebeen made to study the effect of different EoS on ellipticflow ( v ) of identified hadrons as a function of the beamenergy (E Lab ). As quadrangular flow, v is believed tobe originated from v and 4 th order moment of the fluidflow, the ratio v / ( v ) was examined for wide range ofbeam energies and different EoS. The ratio was alwaysfound to be below 2 for all four cases of EoS and can betested against the data from the future experiments atNICA [8] and FAIR [9, 10].In addition, we made effort to verify NCQ scaling interms of coalescence sum rule for slope of directed flow of¯Λ and net Λ. For this purpose, we used different calcula-tions used in Ref. [63] and compared them with dv /dy of ¯Λ and net Λ. Expected agreement and disagreementbetween them is observed at relevant beam energies forall four cases of equations-of-state. This study may alsohint towards possible onset of deconfinement at certainbeam energy above 25A GeV. However, it was interest-ing to notice similar nature of the results even for puretransport and hadrons gas EoS cases where quarks andgluons are not underlying degrees of freedom.Various particle ratios were calculated for all EoS andstudied as a function of beam energy. Though UrQMDeven with hydro is not able to reproduce the measuredstrange to non-strange ratio, it showed some interestingfeatures in response to various EoS beyond 25A GeV.The ratios found to be sensitive to first order phase tran-sition and exhibited strong behavior in comparison toother cases.We wrapped up by studying the rapidity spectra ofnet-protons for different EoS at various beam energies.The shape of these spectra at mid-rapidity, quantified asa reduced curvature was seen to be sensitive for under-lying EoS and showed larger value in case of bag modelbeyond 25A GeV. It also revealed the sensitivity to theunderlying degrees of freedom beyond 25A GeV. Theseinvestigations provide an opportunity to understand thebehavior of the various observables under different nu-clear equations-of-state and to compare the results withthe outcomes from future experiments.2 Acknowledgements
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