Parton collisional effect on the conversion of geometry eccentricities into momentum anisotropies in relativistic heavy-ion collisions
PParton collisional effect on the conversion of geometry eccentricities into momentumanisotropies in relativistic heavy-ion collisions
Long Ma, ∗ Guo-Liang Ma, † and Yu-Gang Ma
1, 2, ‡ Key Laboratory of Nuclear Physics and Ion-beam Application (MOE),Institute of Modern Physics, Fudan University, Shanghai 200433, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
We explore parton collisional effects on the conversion of geometry eccentricities into azimuthalanisotropies in Pb+Pb collisions at √ s NN = 5.02 TeV using a multi-phase transport model. Theinitial eccentricity ε n (n = 2,3) and flow harmonics v n (n = 2,3) are investigated as a function of thenumber of parton collisions ( N coll ) during the source evolution of partonic phase. It is found thatpartonic collisions leads to generate elliptic flow v and triangular flow v in Pb+Pb collisions. Onthe other hand, partonic collisions also result in an evolution of the eccentricity of geometry. Thecollisional effect on the flow conversion efficiency is therefore studied. We find that the partons withlarger N coll show a lower flow conversion efficiency, which reflect differential behaviors with respectto N coll . It provides an additional insight into the dynamics of the space-momentum transformationduring the QGP evolution from a transport model point of view. PACS numbers: 25.75.-q
I. INTRODUCTION
In high-energy heavy-ion collisions, at the extremeconditions of high temperature and high baryon den-sity, strongly-interacting quark gluon plasma (QGP) isexpected to be created. The pressure gradient of theinitial compressed QGP leads to an anisotropic expan-sion and transfers initial-state spatial anisotropy to thefinal-state momentum azimuthal anisotropy, which canbe measured through momentum information of the fi-nal charged hadrons [1–7]. Characterized by the flowcoefficients v n (n = 2,3,4), azimuthal anisotropies of thefinal-state particles are suggested to be sensitive to notonly the early stage partonic dynamics but also prop-erties of the source [8–12]. Experimentally, systematicstudies have been performed for v n in both large heavy-ion collision systems and small collision systems [13–21].Sizable v n observed in experiment indicates that the hotand dense QGP source is like a nearly perfect fluid.Due to the fluid-like behavior observed for QGP, hy-drodynamic models have been widely used to make pre-dictions and are successful in describing flow harmonicsat both RHIC and LHC energies [22–29]. Besides hydro-dynamic models, a multiphase transport model (AMPT)is also employed in studies of anisotropic flow in highenergy collisions. Including both partonic and hadronicinteractions, a multiphase transport model can reason-ably reproduce experimental flow measurements in bothlarge and small collision systems [30–37].In recent years, an escape mechanism was proposedchallenging the commonly believed hydrodynamical ori-gin of the flow anisotropies [38, 39]. It is found ∗ [email protected] † [email protected] ‡ [email protected] that instead of collectivity from partonic interactions,anisotropic parton escape dominates the flow genera-tion in d+Au collision system as well as the semi-centralAu+Au collisions. Though parton escape makes con-siderable contribution, it was also realized that partonicinteraction is essential for generating v n in strong inter-acting systems and v n from partonic interaction becomesdominant in collision systems with large parton-partoninteraction cross-section. Extensive studies have beenperformed on the harmonic flow, dihadron correlationand energy loss induced by partonic collisions [40–46].Theoretically, final flow anisotropy is suggested to bestrongly correlated with the initial geometric anisotropyin relativistic heavy-ion collisions [11, 47–54]. It has beenargued that the magnitude and trend of the partonicparticipant eccentricity ε n (n = 2,3) imply specificallytestable predictions for the final flow harmonics [55–57].For a deeper understanding of the transport, it is essen-tial to investigate the parton collisional effect on the ini-tial geometric anisotropy as well as the conversion fromcoordinate space to momentum space which is expectedto provide important information about the evolution dy-namics of early stage .In this paper, we present a systematic study of the par-tonic collision effect on the initial eccentricity and flowanisotropy in high energy Pb+Pb collisions at √ s NN =5.02 TeV, from a multi-phase transport model. Of partic-ular interest are central collisions in which the averagedenergy density is relatively higher than in non-central col-lisions. Furthermore, influences of partonic collision onthe transfer of eccentricity anisotropy to flow anisotropyare also investigated. This paper is organized as follows:In Sec. II, a multiphase transport (AMPT) model isbriefly introduced. Results and discussion are presentedIn Sec. III. A summary is given in Sec. IV. a r X i v : . [ nu c l - t h ] F e b II. MODEL SETUP
The multi-phase transport model (AMPT) is widelyused for studying transport dynamics in relativisticheavy-ion collisions. The model consists of four maincomponents: the initial condition, partonic interaction,hadronization (quark coalescence), and hadronic inter-actions [58, 59]. Fluctuating initial conditions includingminijet partons and soft string excitations are generatedfrom the Heavy Ion Jet Interaction Generator (HIJING)model [60]. In the string melting scenario, both excitedstrings and minijet partons are melted into partons, i.e.decomposed into constituent quarks according to theirflavor and spin structures. The following evolution ofpartonic matter is described by a parton cascade model -Zhang’s parton cascade (ZPC) model [61], which includeselastic partonic scatterings at present. Partons stop in-teracting when no parton pairs can be found within theinteraction range of pQCD cross section. The transi-tion from the partonic matter to the hadronic matter isachieved using a simple quark coalescence model whichcombines partons into hadrons. The final-stage hadronicinteractions are modeled by a relativistic transport model(ART) including both elastic and inelastic scattering de-scriptions for baryon-baryon, baryon-meson and meson-meson interactions [62].At the parton cascade stage, the differential parton-parton elastic scattering cross section is formularizedbased on the leading order pQCD gluon-gluon interac-tion: dσdt = 9 πα s µ s ) 1( t − µ ) , (1)where α s is the strong coupling constant, s and t are theusual Mandelstam variables and µ is the Debye screen-ing mass in partonic matter. Previous studies show thatby setting proper parton scattering cross sections, AMPTmodel with string melting scenario has been successful indescribing many experimental results in heavy-ion colli-sions at RHIC and LHC energies [35, 44, 63–68].In this study, we employ the string melting version ofthe AMPT model to focus on the partonic phase only.The parton cross section is set to be 3 mb according toRef. [69] which reasonably reproduces the experimentalresults. Pb+Pb collision events are generated over a widecentrality range at center-of-mass energy of 5.02 TeV.Table. I shows the definition of centrality classes and thecorresponding mean number of participant nucleons. III. RESULTS AND DISCUSSIONA. Parton collisions in Pb+Pb collisions
We trace the collisional history of the initially pro-duced partons during the source evolution in Pb+Pb col-lisions. The total number of parton-parton scatterings
TABLE I: Centrality classes of the AMPT events in Pb+Pbcollisions at √ s NN = 5.02 TeV.Centralitypercentile Impact parameter b(fm) (cid:104) N part (cid:105)
0% - 10% 0.0 - 4.9 362.810% - 20% 4.9 - 7.0 263.520% - 30% 7.0 - 8.6 188.230% - 40% 8.6 - 10.0 131.840% - 50% 10.0 - 11.2 86.150% - 60% 11.2 - 12.3 53.8 suffered by each parton before its freezing out is difinedas N coll .Fig. 1 shows the N coll distributions of the freezeoutpartons for different centrality classes. As expected,partons in central Pb+Pb collisions on average suffermore partonic collisions than non-central collisions beforefreezing out as the energy density is higher in more cen-tral collisions. The probability distribution shows a non-monotonic N coll dependence in central collisions which isdifferent from that of the peripheral collisions. A peakaround N coll ∼ N coll in 0-10% central-ity is found to be roughly three times as large as that in40-60% centrality. It indicates that the fraction of par-tons which never collided with other partons decreasesfrom peripheral to central collision class.We study in particular the spatial evolution of theinitial partons in ultra-relativistic heavy-ion collisions.Figs. 2 and 3 present the two-dimensional distributionsof the initial partons [plots (a)-(c)] and freezeout partons[plots (d)-(f)] in central and peripheral Pb+Pb collisions,where the initial parton distributions in the collision zoneare compared with the final parton distributions for dif-ferent N coll intervals. We find that partons sufferingsmall N coll tend to distribute in the outer region closeto the source surface whereas partons with large N coll are seen concentrating more in the central area. This isconsistent with the expectation that due to the energydensity distribution of the bulk matter, outgoing partonsfrom the inner source suffers more collisions when pass-ing through the bulk matter than partons close to thesource surface.We further investigate the partonic collision history ofsome selected partons before their freezing out. For aselected parton, we define those partons which collidedwith the selected parton during its evolution as “asso-ciated partons” of the selected parton. The numbersof parton collisions for the selected parton and associ-ated parton are defined as N selcoll and N assoccoll , respectively.Fig. 4 (a) and (b) show the N assoccoll probability distribu-tions for five N selcoll intervals of the selected partons for pe-ripheral and central Pb+Pb collisions, respectively. Onecan find that for a selected freezeout parton which suf-fered N selcoll collisions, the number of collisions that its as-sociated partons suffer ( N assoccoll ) can be distributed overa wide range. For example, for a selected parton with a FIG. 1: (Color online) Probability distributions of N coll for the freezeout partons for Pb+Pb collisions at √ s NN = 5.02 TeVin AMPT simulation. Results are shown for six centrality classes. small N selcoll , a long-tailed N assoccoll distribution can be ob-served suggesting that many large- N assoccoll partons playan important role in the collisional history of small- N selcoll parton. We quantitatively extract the mean value of N assoccoll for each N selcoll intervals and plot the relations asshown in Fig. 4(c). One can find that (cid:104) N assoccoll (cid:105) is largerthan (cid:104) N selcoll (cid:105) at small (cid:104) N selcoll (cid:105) , which indicates the partonswhich suffer a small number of collisions prefer to collidewith the partons which suffer a larger number of colli-sions. But with the increasing of (cid:104) N selcoll (cid:105) , (cid:104) N assoccoll (cid:105) tendsto be close to and then less than (cid:104) N selcoll (cid:105) . It indicatesthat the partons which suffer a large number of collisionsprefer to collide with the partons which suffer a smallnumber of collisions. In this sense, all partons are com-plemented with each other during the whole evolution ofthe partonic phase in Pb+Pb collisions. B. Collisional effect on the flow anisotropy
Azimuthal anisotropy coefficients v n (n = 2,3..) aretypically used to characterize the different orders of har-monic flow of the collision system. In simulation stud-ies, one can calculate v n with respect to the participantplane of the collision event [70]. The n th-order partici-pant plane angle ψ n for a single event is in the form: ψ n { P P } = 1 n (cid:34) arctan (cid:10) r sin( nϕ P P ) (cid:11) (cid:104) r cos( nϕ P P ) (cid:105) + π (cid:35) , (2) where r and ϕ P P are the position and azimuthal angle ofeach parton in the transverse plane in the initial stage ofAMPT and the bracket (cid:104) ... (cid:105) denotes per-event average.Then v n with respect to the participant plane ψ n { P P } is defined as v n { P P } = (cid:104) cos [ n ( φ − ψ n { P P } )] (cid:105) , (3)where φ in this study is the azimuthal angle of partonin the momentum space, and the average (cid:104)· · · (cid:105) denotesevent average. The above method for v n calculation is re-ferred to as participant plane method. Participant planemethod takes into account initial geometric fluctuationeffect, and has been widely used in many studies [50].Besides the participant plane method, the multi-particle cumulant method was also proposed for study-ing flow via particle correlations. It has been successfullyused in both model and experimental studies to quanti-fied the harmonic flow [16, 63, 71]. Usually, the two- andfour-particle cumulants can be written as C n { } = (cid:104)(cid:104) (cid:105)(cid:105) , C n { } = (cid:104)(cid:104) (cid:105)(cid:105) − (cid:104)(cid:104) (cid:105)(cid:105) . (4)The integral flow can be derived directly from two- andfour-particle cumulants through the following equations v n { } = (cid:112) C n { } , v n { } = (cid:112) − C n { } . (5)and estimation of differential flow is according to FIG. 2: (Color online) Distributions of initial partons (upper panels) and freezeout partons (lower panels) for different N coll intervals in the transverse plane for the most central (b = 0 fm) Pb+Pb collisions at √ s NN = 5.02 TeV.FIG. 3: (Color online) Same as Fig. 2 but for b = 11.5 fm. v (cid:48) n { } = d n { } (cid:112) c n { } , v (cid:48) n { } = d n { }− c n { } / (6)where the d n { } and d n { } are the two- and four- particle differential cumulants as defined in Ref. [71].By extracting the parton information in AMPT simu-lation, we study the collisional effects on the developmentof partonic flow and eccentricity in the early stage of the (a) (b) (c) FIG. 4: (Color online) The N assoccoll probability distributions of the “associated parton” collided with different N selcoll intervals ofselected partons for two centrality classes of 50-60% (a) and 0-10% (b) in Pb+Pb collisions at √ s NN = 5.02 TeV. Panel (c)shows the (cid:104) N assoccoll (cid:105) as functions of (cid:104) N selcoll (cid:105) for the two centrality classes. heavy-ion collisions.Fig. 5 shows the simulation results of the anisotropicflow of freezeout partons as a function of N coll in Pb+Pbcollisions at √ s NN = 5.02 TeV. The second and thirdorder flow harmonics of the freezeout partons are de-fined as v f and v f respectively. Similar to the proba-blity distribution of N coll , v f shows non-monotonic N coll dependence for central collisions. A maximum value of v f around N coll ∼ v f shows a similar decreas-ing trend and is comparably much larger in magnitudethan that of the central collisions in magnitude. It gen-erally shows that partons with larger N coll tends to havesmaller v f indicating that with increasing N coll the mo-mentum azimuthal distribution of freezeout partons tendto be isotropic. This could be because large N coll partonscome mostly from the center where the effective gradientsare small. In other words, large N coll partons are less sen-sitive to the geometry than small N coll partons which arecloser to the surface. The same conclusion also holds for v f but which originates basically from the initial-statefluctuations.Besides the participant plane method, we further stud-ied v fn based on multi-particle cumulant methods. Fig. 6shows the two-particle ( v { } ) and four-particle ( v { } )cumulant flow results. Comparisons are made with theresults from the participant plane method. It is gener-ally found that v f from cumulant methods are similar intrend with the v f results from participant plane method.An ordering of v { } > v { part } > v { } is observed,because v { } involves flow fluctuations but v { } sup-presses non-flow contributions.Towards a more quantitative study, we compare theflow anisotropies of the initial partons ( v in ) and finalfreezeout partons ( v fn ) for n = 2,3. Note that the av-eraged v n for all the initial partons is zero due to theisotropy of initial azimuthal distribution in the AMPTmodel. As can be seen in Fig. 7, showing the change of v n from the initial to the final stage of the partonic evolu-tion, the parton-parton collisions generally make v and v increase for different N coll regions. For the partonswith a larger number of N coll , the change of v and v is smaller, since they are more probably located at thecenter of the source where more collisions may randomizetheir motion.In order to quantitatively study the collisional effecton the flow harmonics, we examine the change of parton v n after N coll collisions, i.e. ∆ v n = v fn - v in . Fig. 8 showsthe ∆ v n (n = 2,3) in Pb+Pb collisions as a function of N coll . Results are compared for different flow methods.Significant centrality dependence can be seen for ∆ v n . Itis interesting to see that in central collisions ∆ v showsnon-monotonic N coll dependence whereas ∆ v presentsmonotonic N coll dependence. As shown in Fig. 7, theinitial intrinsic parton v n is quite tiny, the gain in v n isprimarily due to the partonic scatterings throughout thesource evolution. In general, ∆ v n decreases with increas-ing of N coll indicating that small N coll partons contributeto most of the flow anisotropies. C. Collisional effect on the initial eccentricity
Initial geometry anisotropy of the QGP matter is amain source responsible for generating the final flowanisotropy in relativistic heavy-ion collisions. Thus, it isimportant to study the partonic effect on the initial spa-tial anisotropy. In nuclear-nuclear collisions, the spatialanisotropy of the collision zone in the transverse plane(perpendicular to the beam direction) can be character-ized by the eccentricity ε n . It has been argued that themagnitude and trend of the eccentricity imply testablepredictions for final-state hadronic flow [72–75].The definition of the n th-order harmonic eccentricityin the coordinate space of the participant nucleons orpartons for single collision event is in the form: ε n { part } = (cid:113) (cid:104) r n cos( nϕ ) (cid:105) + (cid:104) r n sin( nϕ ) (cid:105) (cid:104) r n (cid:105) , (7) (a) (b) FIG. 5: (Color online) v fn (n = 2,3) of final freezeout partons from participant plane method as a function of N coll for Pb+Pbcollisions at √ s NN = 5.02 TeV in the AMPT model. Results are shown for different centality classes.FIG. 6: (Color online) v f of final freezeout partons from two-particle ( v { } ) and four-particle ( v { } ) cumulant method asa function of N coll for Pb+Pb collisions at √ s NN = 5.02 TeV in the AMPT model. Comparisons are made with participantplane method for different centrality classes. where r and ϕ are position and azimuthal angle of eachnucleon or parton in the transverse plane. ε n { part } characterizes the eccentricity through the distribution ofparticipant nucleons or partons which naturally containsevent-by-event fluctuation. ε n { part } defined in this wayis usually named as “participant eccentricity”.We study the parton collisional effects on the partoniceccentricity in Pb+Pb collisions. Fig. 9 shows the AMPTresults of the second and third order eccentricities calcu-lated with Eq. (7). Eccentricities of the initial and finalfreezeout partons are denoted as ε in { part } and ε fn { part } respectively. Similarly to the flow harmonics, partonicscattering is found to play an important role in the evolu-tion of eccentricities. ε n of the freezeout partons is largerat larger N coll . One can see in the figures that parton col-lisions generally reduce ε n which is consistent with ourexpectation - during the expansion of the QGP source,the transition of the initial pressure gradient from coor-dinate space to the momentum space will significantlydiminish the spatial anisotropy.In addition, we studied ∆ ε n = ε fn { part } - ε in { part } asa function of the number of parton collisions for different FIG. 7: (Color online) v in of initial state partons and v fn of final freezeout partons as a function of impact parameter for different N coll intervals. centrality classes. The results for second and third orderharmonics are shown in Fig. 10 (a) and (b), respectively.We find that ∆ ε and ∆ ε exhibit clear decreasing N coll dependences. Quantitative difference are seen betweenthe results for ∆ ε and ∆ ε , because ε is purely drivenby initial fluctuations but ε is driven by initial geometry. D. Collisional effect on the flow response to theeccentricity
Impressive progress has been made in studying thefinal-state flow response to the initial eccentricity in rela-tivistic heavy-ion collisions [76, 77]. The success of hydro-dynamical models tells us that elliptic flow v and trian-gular flow v are mainly driven by the linear response tothe initial ellipticity and triangularity of the source geom-etry. As space-momentum correlation is also expected tobe built during the partonic evolution stage, quantitativestudy of the partonic flow response in a event-by-eventtransport model is also important for understanding thedevelopment of final flow .Based on the AMPT model simulations, we studied N coll effects on the flow response by looking into the ra-tio ∆ v n /∆ ε n . Since ∆ ε n (n = 2,3) are negative and ∆ v n (n = 2,3) are positive over all the N coll classes, one couldtake the absolute value of ∆ v n /∆ ε n as an estimation ofthe flow conversion efficiency. Fig. 11 and 12 show theresults for the flow conversion efficiency with respect to∆ ε n and ε in as a function of N coll . Considering absolutevalue, results in both figures show similar trend. The ra-tio of ∆ v n /∆ ε n presents obvious N coll dependences for different centrality classes. We observe that for bothelliptic and triangular flow the conversion efficiency isstrongest in the collision class of 0-10%. It indicates thatmore collisions in more central collisions help transfer ε n into v n , which is a normal concept about the flow con-version efficiency which is an integral effect of all N coll partons. For the differential N coll dependence, ∆ v n /∆ ε n (n = 2,3) presents a smooth increasing trend from smallto large N coll , which indicates that the larger N coll is,the lower the flow conversion efficiency is. The featureseems against common sense, but it can be understoodthrough the above results that parton collisional contri-bution to flow change ∆ v n is more significant at smaller N coll whereas that to eccentricity change ∆ ε n is moresignificant at larger N coll , i.e. changes of ∆ v n and ∆ ε n are not in sync with respect to N coll . But since small- N coll and large- N coll partons are complemented with eachother during the evolution, it is actually hard to fairly saywhich should be given the first credit to the generationof the final flow. In the limit of long evolution time, fi-nal eccentricity ε fn is supposed to approach zero, and weobserve the similar results for ∆ v n / ε in except with theopposite sign, as shown in Fig. 12. IV. SUMMARY
In summary, we studied initial partonic flow anisotropy( v n ) and spatial anisotropy ( ε n ) in Pb+Pb collisions atcenter-of-mass energy of 5.02 TeV using a multi-phasetransport model. By tracing the partonic cascade his-tory in AMPT, the effect of the parton-parton collisions (a) (b) (c) (d) FIG. 8: (Color online) The AMPT results of ∆ v n = v fn − v in (n = 2,3) as a function of N coll , where the upper panels show ∆ v n from participant plane method and the lower panels show ∆ v n from two-particle cumulant methods. (a) (b) (c) (d) (e) (f) FIG. 9: (Color online) ε i,f { part } (upper panels) and ε i,f { part } (lower panels) of the initial state and final freezeout partonsas a function of impact parameter for Pb+Pb collisions at √ S NN = 5.02 TeV. Simulation results are shown for different N coll intervals. (a) (b) FIG. 10: (Color online) The AMPT results of ∆ ε n = ε fn { part } - ε in { part } (n=2,3) as a function of N coll for different centralityclasses. (a) (b) FIG. 11: (Color online) Conversion efficiency ∆ v n /∆ ε n for n=2 (left panel) and n=3 (right panel) as a function of N coll forPb+Pb collisions at 5.02 TeV. (a) (b) FIG. 12: (Color online) ∆ v n / ε in for n=2 (left panel) and n=3 (right panel) as a function of N coll for Pb+Pb collisions at 5.02TeV, where ε in is the initial partonic eccentricity. v n decreaseswith increasing of N coll indicating that small N coll par-tons contribute to most of the flow anisotropies. How-ever, the change of eccentricity is more significant for thelarge- N coll partons. As a result, the partons with larger N coll show a lower flow conversion efficiency, which re-flect the differential behaviors of the flow conversion effi-ciency with respect to N coll . However, since small- N coll and large- N coll partons are always complemented witheach other, it is hard to rank their roles in generatingflow.However, one has to be aware that although the AMPTmodel provides an effective tool to simulate and studyparton-parton collisions in relativistic heavy-ion colli-sions, the initial partonic source configured using con-stituent quarks in the string-melting scenario could in-troduce some intrinsic bias into our study, since the cre-ated QGP should consist of both current quarks and glu-ons. In addition, the approximation of the model treat-ment of the parton interactions is in a way analogous togluon-gluon elastic interaction based on the leading or-der pQCD cross section which could also introduce somebias and lead to an incomplete or improper description, since the QGP evolution involves non-perturbative QCDprocesses. Nevertheless, such a simplified picture of thepartonic evolution in this model is expected to providesome guides to the study of the effect on the conversionrules of the initial eccentricity to the final flow anisotropy.As anisotropic flow of initial partons will transfer tothe final hadrons which are formed from the coalescenceof freeze-out quarks in the transport model, further studyby tracing the hadronization and hadronic evolution ofparticles would be important for fully understanding thecomplete behavior of the anisotropic flow. We postponesuch investigations for our future study. Acknowledgements
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