Mean-field potential effects on particle and antiparticle elliptic flows in the beam-energy scan program at RHIC
aa r X i v : . [ nu c l - t h ] A ug Mean-field effects on particle and antiparticle elliptic flows in the beam-energy scanprogram at RHIC
Jun Xu, ∗ Che Ming Ko, Feng Li, Taesoo Song, and He Liu Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Cyclotron Institute and Department of Physics and Astronomy,Texas A & M University, College Station, Texas 77843, USA Frankfurt Institut for Advanced Studies and Institute for Theoretical Physics,Johann Wolfgang Goethe Universitat, Frankfurt am Main, Germany (Dated: June 8, 2018)The elliptic flow splitting between particles and their antiparticles has recently been observed bythe STAR Collaboration in the beam-energy scan program at the Relativistic Heavy Ion Collider.In studies based on transport models, we have found that this splitting can be explained by thedifferent mean-field potentials acting on particles and their antiparticles in the produced baryon-rich matter. In particular, we have shown that the experimentally measured relative elliptic flowdifference can help constrain the vector coupling constant in the Nambu-Jona-Lasinio model usedin describing the partonic stage of heavy-ion collisions. This information is useful for locating thecritical point in the QCD phase diagram and thus understanding the phase structure of QCD.
PACS numbers: 25.75.-q, 12.38.Mh, 24.10.Lx, 24.85.+p
I. INTRODUCTION
The main goal of experiments on relativistic heavy-ion collisions is to study the hadron-quark phase tran-sition or the QCD phase structure. At top energiesof the Relativistic Heavy Ion Collider (RHIC) and theLarge Hadron Collider (LHC), the produced quark-gluonplasma (QGP) is essentially baryon free and the phasetransition is thus a smooth crossover according to re-sults from the lattice QCD calculations [1–3]. On theother hand, studies based on various theoretical modelshave predicted that the hadron-quark phase transitionbecomes a first-order one at large baryon chemical po-tential [4–7]. A critical point is thus expected to existbetween the smooth crossover and the first-order phasetransition. To search for its signature, experiments underthe beam-energy scan (BES) program have recently beencarried out at RHIC. Although there is no definitive con-clusion on the location of the QCD critical point, someinteresting phenomena different from those observed inheavy-ion collisions at much higher energies have beenobserved [8, 9], such as the weakening in the charge sep-aration, the net proton number fluctuation, the fluctua-tion of p/π and
K/π ratios, the monotonic decrease ofthe freeze-out eccentricity with increasing beam energyfrom the Hanbury-Brown Twiss analysis, the larger nu-clear modification factor for high-transverse-momentumparticles than in collisions at higher energies, and thesplitting of the direct flow and elliptic flow between par-ticles and their antiparticles.The observed elliptic flow ( v ) splitting between par-ticles and their antiparticles [10] in the BES programat RHIC, which obviously indicates the break down of ∗ Electronic address: [email protected] the number of constituent quark scaling established athigher collision energies [11], has attracted much atten-tion. Various explanations have been suggested for un-derstanding this phenomenon. It was shown in Ref. [12]that the chiral magnetic wave induced by the strongmagnetic field from non-central heavy-ion collisions couldlead to a charge quadrupole moment in the participant re-gion, which would then result in a larger v for positivelycharged particles than negatively charged ones, especiallyfor charged pions due to their similar final-state interac-tions. Also, the different v between particles and theirantiparticles has been attributed to different v of pro-duced and transported particles [13], different rapiditydistributions for quarks and antiquarks [14], and the con-servation of baryon charge, strangeness, and isospin [15].On the other hand, we have shown in our recent stud-ies [16–20] that the observed v splitting can be explainedby the different mean-field potentials acting on particlesand their antiparticles. The effect of the mean-field po-tential or the nuclear matter equation of state on theelliptic flow was already known in heavy ion collisions atSIS energies of the order 1 AGeV [21]. At this energy,the expansion of the hot participant matter is blockedby the cold spectator matter, resulting in the emissionof more particles out of the reaction plane and thus anegative v . A more repulsive mean-field potential leadsto a fast expansion of the participant matter, thus astronger blocking effect and an even more negative v .This becomes different in relativistic heavy-ion collisionswhere the blocking effect is absent as the spectator mat-ter quickly moves away from the participant region. As aresult, particles with repulsive potentials are more likelyto leave the system, while those with attractive poten-tials are more likely trapped in the system. For the caseof a positive eccentricity of the participant region in non-central collisions, the v is then enhanced for particleswith repulsive potentials and suppressed for those withattractive potentials. In the baryon-rich matter formedin heavy-ion collisions at √ s NN = 7 . ∼
39 GeV in theBES program at RHIC, particles usually have repulsive orless attractive potentials compared to the strong attrac-tive potentials for their antiparticles, and the differencebecomes smaller with decreasing net baryon density athigher collision energies. This qualitatively explains theobserved decreasing v splitting with increasing beam en-ergy.This paper is organized as follows. In Sec. II, we brieflyreview the physics contents of a multiphase transportmodel based on which our studies were carried out. Thehadronic potentials and the partonic potentials as well astheir effects on v splitting between particles and antipar-ticles are discussed in Sec. III and Sec. IV, respectively.Results with the inclusion of both potentials are shownin Sec. V. Finally, conclusions and outlook are given inSec. VI. II. THE AMPT MODEL
Our study of the mean-field effects on v splitting be-tween particles and their antiparticles were based on amultiphase transport (AMPT) model [22]. To prop-erly describe the relative contributions of the partonicphase and the hadronic phase on the final elliptic flow ofhadrons, we use the string melting version, which con-verts hadrons produced from initial collisions into theirvalence quarks and antiquarks, and was shown to success-fully reproduce the charged particle multiplicity, the col-lective flow, and the dihadron correlations at both RHICand LHC [23].The initial conditions in the AMPT model are obtainedfrom the heavy-ion jet interaction generator (HIJING)model [24], where both soft and hard scatterings are in-cluded by using the Monte Carlo Glauber model withnuclear shadowing effects. In the string melting ver-sion, the interaction in the partonic phase is describedby parton-parton elastic scatterings based on the Zhang’sparton cascade (ZPC) model [25]. After the freeze-out ofpartons, a spatial coalescence model is used to describethe hadronization process with the hadron species deter-mined by the flavor and invariant mass of its constituentquarks and/or antiquarks. The evolution of resultinghadronic phase is described by a relativistic transport(ART) [26], which has been generalized to include bothbaryon-antibaryon annihilations to two-meson states andtheir inverse processes [22]. The cross sections for theformer are determined by the branching ratios for pro-ducing corresponding number of pions according to thephase space considerations, while the cross sections forthe latter are based on the detailed balance relations. Forheavy ion collisions at top energies at RHIC and LHC,the mean-field potentials for particles and their antipar-ticles are not included in either ZPC or ART as they areless important than partonic and hadronic scatterings onthe collision dynamics [23]. In Refs. [16–20], the AMPT model was extended to include mean-field potentials inthe hadronic phase and the partonic phase in order tostudy heavy ion collisions at energies carried out in theBES program. III. EFFECTS OF HADRONIC POTENTIALS
For nucleon and antinucleon potentials, they can becalculated from the relativistic mean-field model used inRef. [27] to describe the properties of nuclear matter, i.e., U N, ¯ N ( ρ B , ρ ¯ B ) = Σ s ( ρ B , ρ ¯ B ) ± Σ v ( ρ B , ρ ¯ B ) , (1)where Σ s ( ρ B , ρ ¯ B ) and Σ v ( ρ B , ρ ¯ B ) are the nucleon scalarand vector self-energies in a hadronic matter of baryondensity ρ B and antibaryon density ρ ¯ B , with ”+” for nu-cleons and ” − ” for antinucleons, respectively. Nucleonsand antinucleons contribute both positively to Σ s butpositively and negatively to Σ v , respectively, as a resultof the G -parity invariance. The potentials for strangebaryons and antibaryons are reduced relative to those ofnucleons and antinucleons according to the ratios of theirlight quark numbers.The kaon and antikaon potentials in the nuclearmedium are also taken from Ref. [27] based on the chiraleffective Lagrangian, that is U K, ¯ K = ω K, ¯ K − ω with ω K, ¯ K = q m K + p − a K, ¯ K ρ s + ( b K ρ net B ) ± b K ρ net B (2)and ω = p m K + p , where m K is the kaon mass, andthe values of a K , a ¯ K , and b K can be found in Ref. [28]. Inthe above, ρ s is the scalar density and is calculated self-consistently in terms of the effective quark and antiquarkmasses determined from the same relativistic mean-fieldmodel of Ref. [27] and ρ net B = ρ B − ρ ¯ B is the net baryondensity. The ”+” and ” − ” signs are for kaons and an-tikaons, respectively.The pion potentials are related to their self-energiesΠ ± s according to U π ± = Π ± s / (2 m π ), where m π is thepion mass. Only contributions from the pion-nucleon s -wave interaction to the pion self-energy were includedin our study, and they have been calculated up to thetwo-loop order in the chiral perturbation theory [29]. Inasymmetric nuclear matter of proton density ρ p and neu-tron density ρ n , the resulting π − and π + self-energies aregiven, respectively, byΠ − s ( ρ p , ρ n ) = ρ n [ T − πN − T + πN ] − ρ p [ T − πN + T + πN ]+Π − rel ( ρ p , ρ n ) + Π − cor ( ρ p , ρ n ) , Π + s ( ρ p , ρ n ) = Π − s ( ρ n , ρ p ) . (3)In the above, T + and T − are, respectively, the isospin-even and isospin-odd πN s -wave scattering T -matrices,which are given by the one-loop contribution in chiralperturbation theory; Π − rel is due to the relativistic cor-rection; and Π − cor is the contribution from the two-looporder in the chiral perturbation theory. Their expressionscan be found in Ref. [29].For nucleon and strange baryon resonances in ahadronic matter, we simply extend the above result bytreating them as neutron- or proton-like baryons accord-ing to their isospin structure [26] and light quark num-bers.
10 20 30 4002040 [ v ( P )- v ( P ) ]/ v ( P ) ( % ) s NN1/2 (GeV) solid: without Uopen: with U: p and p: K + and K - : + and - Au+Au at b = 8 fmstring melting AMPT |y| < 1
FIG. 1: (Color online) Relative elliptic flow difference betweenprotons and antiprotons, K + and K − , and π + and π − withand without hadronic mean-field potentials in Au+Au colli-sions at √ s NN = 7 .
7, 11 .
5, and 39 GeV and impact parameterb = 8 fm. Taken from Ref. [16].
The mean-field potentials are included in the ARTmodel by using the test particle method [30]. For the par-ton scattering cross section in ZPC and the ending timeof the partonic phase, they are determined from fittingthe measured charged particle v and freeze-out energydensity calculated from the extracted baryon chemicalpotential and temperature at chemical freeze-out [31].The resulting relative p T -integrated elliptic flow differ-ence between particles and their antiparticles at midra-pidity ( | y | <
1) are shown in Fig. 1. It is seen thatwithout hadronic mean-field potentials, the relative v splitting is very small as expected. In the baryon-richand neutron-rich matter formed in Au+Au collisions atBES energies, the hadronic potential is slightly attractivefor nucleons, deeply attractive for antinucleons, slightlyrepulsive for K + , deeply attractive for K − , slightly repul-sive for π − , and slightly attractive for π + . Figure 1 showsthat the sign of the relative v splitting is consistent withthat expected from the different hadronic mean-field po-tentials for particles and their antiparticles. Also, the v difference decreases with increasing beam energy. Theseresults are qualitatively consistent with the experimen-tally measured values of about 63% and 13% at 7 . . v difference between p and ¯ p and between K + and K − , respectively [9]. Similar to the experimen-tal data, the relative v difference between π + and π − isnegative at all energies after including their potentials,although ours have a smaller magnitude. IV. EFFECTS OF PARTONIC POTENTIALS
To include mean-field potentials for u , d , and s quarksand their antiquarks in the partonic phase, we have de-veloped a partonic transport model based on the NJLmodel [17]. The Lagrangian of the 3-flavor NJL model isgiven by [7] L = ¯ ψ ( i ∂ − M ) ψ + G X a =0 (cid:20) ( ¯ ψλ a ψ ) + ( ¯ ψiγ λ a ψ ) (cid:21) + X a =0 (cid:20) G V ψγ µ λ a ψ ) + G A ψγ µ γ λ a ψ ) (cid:21) − K (cid:20) det f (cid:18) ¯ ψ (1 + γ ) ψ (cid:19) + det f (cid:18) ¯ ψ (1 − γ ) ψ (cid:19)(cid:21) , (4)with the quark field ψ = ( ψ u , ψ d , ψ s ) T , the current quarkmass matrix M = diag( m u , m d , m s ), and the Gell-Mannmatrices λ a in SU (3) flavor space. In the case thatthe vector and axial-vector interactions are generatedby the Fierz transformation of the scalar and pseudo-scalar interactions, their coupling strengths are given by G V = G A = G/
2, while G V = 1 . G was used in Ref. [32]to give a better description of the vector meson massspectrum calculated from the NJL model. Other pa-rameters like m u , m d , m s , G , and K are taken fromRefs. [7, 32]. In the mean-field approximation, the aboveLagrangian leads to an attractive scalar mean-field po-tential for both quarks and antiquarks. With a nonva-nishing G V , it further gives rise to a repulsive vectormean-field potential for quarks but an attractive one forantiquarks in a baryon-rich quark matter.For the initial parton distributions, they are taken fromthe HIJING subroutine in the AMPT model. The par-ton scattering cross section is determined by fitting themeasured v of final charged particles. Again the testparticle method is used in the NJL transport model, anddiscrete lattices for space are used to calculate the lo-cal density and potential. The evolution of the partonicphase ends when the energy density of the central celldecreases to about 0 . that is expected for thequark to hadron phase transition. The quark matter isthen converted to hadrons by the coalescence model ofRefs. [33, 34] with the probability for a quark and an an-tiquark to form a meson given by the quark Wigner func-tion of the meson, while the probability for three quarksor antiquarks to coalesce to a baryon or an antibaryongiven by the quark Wigner function of the baryon or an-tibaryon. The relative v difference between resulting p and ¯ p , Λ and ¯Λ, and K + and K − is given in Fig. 2 as K + -K - (Exp.)- (Exp.) p-p ( v - v ) / v ( % ) g V / G p-p (Exp.) - K + -K - FIG. 2: (Color online) Relative elliptic flow difference betweenprotons and antiprotons, K + and K − , and Λ and ¯Λ as afunction of the ratio of the partonic vector coupling constantto the scalar one in Au+Au collisions at √ s NN = 7 . a function of the ratio of the partonic vector couplingconstant to the scalar one in the NJL model, and the ex-perimental results from Ref. [9] are also shown for com-parison. Without vector potential, i.e., g v = G V = 0,there is already slightly v splitting between particlesand their antiparticles, and this is due to the slightlylarger quark than antiquark v as a result of a smallerinitial spatial eccentricity for quarks than for producedantiquarks. With increasing g v , the difference betweenproton and antiproton v also increases because of theincreasing difference of light quark and antiquark v dueto a less attractive potential for quarks than for anti-quarks. Similarly, the relative v difference between Λand ¯Λ increases with increasing g v . On the other hand,the relative v difference between K + and K − becomessmaller or even negative for large values of g v . This isbecause the vector mean field, which acts similarly onlight and strange (anti-)quarks, leads to a smaller antis-trange than strange quark v and also reduces the effectdue to different spatial eccentricities of quarks and anti-quarks. For pions, since the potentials for u and d quarksare the same in the present 3-flavor NJL model due tothe neglect of isovector coupling between quarks and an-tiquarks, there is no v difference between π + and π − . V. EFFECTS OF BOTH HADRONIC ANDPARTONIC POTENTIALS
With only hadronic potentials, Figure 1 shows that ourresults slightly underestimate the v splitting between p and ¯ p and overestimate that between K + and K − in the experimental data. Comparing the results in Fig. 2 ob-tained with only partonic potentials with the experimen-tal data, the relative v difference between protons andantiprotons is still underestimated, while that between K + and K − has a wrong sign. If the effects from themean-field potentials on v in the partonic phase and thehadronic phase are additive, one would expect that therelative v splitting between protons and antiprotons aswell as that between K + and K − can be quantitativelyexplained with both partonic and hadronic potentials fora certain value of the vector coupling constant in the NJLmodel. This is indeed the case as shown in Ref. [18] byreplacing the ZPC model in the AMPT model with theNJL transport model and ending the partonic evolutionwhen the effective mass of light quarks in the central cellincreases to about 200 MeV when the chiral symmetry islargely broken, as well as including the hadronic mean-field potentials in the ART model. -20020406080 STAR data R V = 0.5 R V = 1.1 kaon [ v ( P )- v ( P ) ]/ v ( P ) ( % ) nucleon [email protected] GeV 0-80% | |<1
FIG. 3: (Color online) Relative elliptic flow difference betweenprotons and antiprotons as well as between K + and K − usingtwo different values for the ratio R v of the partonic vector toscalar coupling constants in mini-bias Au+Au collisions at √ s NN = 7 . The relative v difference between protons and antipro-tons as well as that between K + and K − at midpseu-dorapidity ( | η | <
1) in mini-bias Au+Au collisions at √ s NN = 7 . R V = G V /G = 0 . . R V = 0 . R V = 1 . R V the relative v difference be-tween protons and antiprotons increases, while that be-tween K + and K − decreases. It is thus expected thatfurther reducing the value of R V would underestimatethe relative v difference between protons and antipro-tons and overestimate that between K + and K − , whilefurther increasing the value of R V would underestimatethat between K + and K − . To reproduce both the rel-ative v differences between protons and antiprotons aswell as that between K + and K − requires the value of R V to be about 0 . ± . M u ( M e V ) T ( M e V ) q ( M e V ) R V = 0 c h i r a l s u sc ep t. T ( M e V ) q ( M e V ) R V = 0 M u ( M e V ) T ( M e V ) q ( M e V ) R V = 0.5 c h i r a l s u sc ep t. T ( M e V ) q ( M e V ) R V = 0.5 M u ( M e V ) T ( M e V ) q ( M e V ) R V = 1.1 c h i r a l s u sc ep t. T ( M e V ) q ( M e V ) R V = 1.1 FIG. 4: (Color online) The dynamical light quark mass (left)and the corresponding susceptibility (right) in the ( µ q , T )plane from different relative vector coupling constant R V = G V /G . According to Refs. [4–7], the value of the vector cou-pling would affect the location of the critical point inthe QCD phase diagram and thus the QCD phase struc-ture. In Fig. 4 we display the dynamical light quarkmass, which is a function of the quark condensate [17],and the corresponding susceptibility, which is defined asthe derivative of the dynamical quark mass with respectto the quark chemical potential. One sees that at smallervalues of quark chemical potential µ q and temperature T ,the dynamical quark mass is large, corresponding to thephase where chiral symmetry is broken. For larger valuesof µ q or T , the dynamical quark mass is similar to thecurrent quark mass, corresponding to the phase wherechiral symmetry is restored. A smooth change from thefirst phase to the second phase represents a cross-over chi-ral phase transition, while a sudden change represents afirst-order phase transition. The critical point is exactlywhere the cross-over transition changes to a first-orderone. One sees from the susceptibility that the criticalpoint moves to lower temperatures with increasing vectorcoupling constant, similar to that observed in Refs. [4– 7]. This is not surprising because the equation of stateof baryon-rich matter becomes stiff with larger values of G V , so that the phase transition is more likely to be acrossover rather than a first-order one. T ( M e V ) q (MeV) R V = 0 R V = 0.5 R V = 1.1 NJL model FIG. 5: (Color online) The chiral phase boundary and thecorresponding critical point (labeled as stars) based on theNambu-Jona-Lasinio model in the ( µ q , T ) plane from differentrelative vector coupling constant R V = G V /G . Defining the phase boundary where the chiral conden-sate is half of the value at ( µ q = 0, T = 0) as the phaseboundary [5], the phase diagram with chiral phase tran-sition with different R V is shown in Fig. 5. One shouldkeep in mind that the phase boundary is only accuratefor the first-order phase transition but is artificially de-fined for a crossover, as shown by the dashed lines inFig. 5. It is thus clearly seen that based on the sameNJL Lagrangian from the transport model study and thephase diagram calculation, the constraint of the vectorcoupling R V = 0 . ∼ . Polyakov = − b · T { e − a/T l ¯ l +ln[1 − l ¯ l − l ¯ l ) +4( l +¯ l )] } , (5)where the parameters a and b are fitted to make thequark condensate and the polyakov loop l cross eachother at about T = 200 MeV at zero quark chemical po-tential [5]. The phase structure with the contribution of T ( M e V ) q (MeV) R V = 0 R V = 0.5 R V = 1.1 pNJL model chiral transition FIG. 6: (Color online) The chiral phase boundary, the corre-sponding critical point (labeled as stars), and the deconfine-ment phase boundary based on the Polyakov-looped Nambu-Jona-Lasinio model in the ( µ q , T ) plane from different relativevector coupling constant R V = G V /G . the polyakov loop is displayed in Fig. 6, where the decon-finement phase boundary is defined as l = 1 / R V doesn’t change even withthe inclusion of the polyakov loop. It would be of greatinterest to included gluon contribution in both the trans-port model study and the phase diagram calculation inthe future study. VI. CONCLUSIONS AND OUTLOOK
We have reviewed our recent studies on the effectsof mean-field potentials on the elliptic flow splitting be-tween particles and their antiparticles in the beam-energyscan program at RHIC. By including the hadronic mean-field potentials in the ART model of a multiphase trans-port model, the v splitting is qualitatively consistentwith the experimental results, while the magnitude ofthe relative v difference between protons and antipro- tons as well as π + and π − are underestimated, and thatbetween K + and K − is overestimated. With only par-tonic potentials from a 3-flavor NJL transport model,the relative v difference between protons and antipro-tons is still underestimated while that between K + and K − has a wrong sign when these hadrons are producedfrom the coalescence of quarks and antiquarks. Includingboth the partonic and hadronic potentials in the AMPTmodel, the relative v difference between protons andantiprotons as well as that between K + and K − canbe quantitatively reproduced, and this further helps toconstrain the R V = G V /G in the NJL model to about R V = 0 . ± .
3. Based the NJL Lagrangian used in thetransport model study, we have further calculated thecorresponding phase diagram and found that the result-ing critical point in the chiral phase transition disappearsor is located at very low temperatures based on the ob-tained constraint of R V , and the conclusion remains thesame even if the contribution of polyakov loop to thethermal potential is taken into account.In the previous studies, we have only reproduced therelative v difference in mini-bias Au+Au collisions at √ s NN = 7 . v splitting at higher energies as this would provide thepossibility of studying the net baryon or baryon chemicalpotential dependence of R V . In addition, since the par-tonic phase is more important in mid-central collisionsthan in peripheral collisions, it will also be of interestto study the centrality dependence of the v splitting tofurther constrain the parameters in our model. Sincethe location of the QCD critical point is sensitive to thevalue of R V , further investigations on the v splitting inthe baryon-rich matter can help map out the QCD phasestructure. Acknowledgments
This work was supported by the National Key Ba-sic Research Program of China (973 Program) un-der Contract Nos. 2015CB856904 and 2014CB845401,the ”Shanghai Pujiang Program” under Grant No.13PJ1410600, the ”100-talent plan” of Shanghai Instituteof Applied Physics under Grant No. Y290061011 fromthe Chinese Academy of Sciences, the National NaturalScience Foundation of China under Grant No. 11475243,the US National Science Foundation under Grant No.PHY-1068572, and the Welch Foundation under GrantNo. A-1358. [1] BERNARD C, BURCH T, DETAR C et al. (MILC Col-laboration). Phys Rev D, 2005, : 034504.[2] AOKI Y, ENDRODI G, FODOR Z et al. . Nature, 2006, : 675. [3] BAZAVOV A, BHATTACHARYA T, CHENG M et al. .Phys Rev D, 2012, : 054503.[4] ASAKAWA M and YAZAKI K. Nucl Phys A, 1989, :668. [5] FUKUSHIMA K. Phys Rev D, 2008, : 114028[Erratum-ibid. D, 2008, : 039902].[6] CARIGNANO S, NICKEL D, and BUBALLA M. PhysRev D, 2010, : 054009.[7] BRATOVIC N M, HATSUDA T, and WEISE W. PhysLett B, 2013, : 131.[8] KUMAR L (STAR Collaboration). J Phys G, 2011, :124145.[9] MOHANTY B (STAR Collaboration). J Phys G, 2011, : 124023.[10] STAR Collaboration. Phys Rev Lett, 2013, : 142301.[11] STAR Collaboration. Phys Rev Lett, 2007, : 162301.[12] BURNIER Y, KHARZEEV D E, LIAO J, and YEE HU. Phys Rev Lett, 2011, : 052303.[13] DUNLOP J C, LISA M A, and SORENSEN P. Phys RevC, 2011, : 044914.[14] GRECO V, MITROVSKI M, and TORRIERI G. PhysRev C, 2012, : 044905.[15] STEINHEIMER J, KOCH V, and BLEICHER M. PhysRev C, 2012, : 044903.[16] XU J, CHEN L W, Ko C M, and Lin Z W. Phys Rev C,2012, : 041901.[17] SONG T, PLUMARI S, GRECO V, KO C M, and LI F.Nucl Phys A, arXiv: 1211.5511 [nucl-th].[18] XU J, SONG T, KO C M, and LI F. Phys Rev Lett,2014, : 012301.[19] KO C M, CHEN L W, GRECO V, LI F, LIN Z W,PLUMARI S, SONG T, and XU J. Nucl Sci Tech, 2013, : 050525.[20] KO C M, CHEN L W, GRECO V, LI F, LIN Z W, PLUMARI S, SONG T, and XU J. Acta Phys PolonSupp, 2014, : 183.[21] DANIELEWICZ P, LACEY R, and LYNCH W G. Sci-ence, 2002, : 1592.[22] LIN Z W, KO C M, LI B A, ZHANG B, and PAL S.Phys Rev C, 2005, : 064901.[23] XU J and KO C M. Phys Rev C, 2011, : 021903 (R);2011, : 034904; 2011, : 014903; 2011, : 044907.[24] WANG X N and GYULASSY M. Phys Rev D, 1991, :3501.[25] ZHANG B. Comp Phys Comm, 1998, : 193.[26] LI B A and KO C M. Phys Rev C, 1995, : 2037.[27] LI G Q, KO C M, FANG X S, and ZHENG Y M. PhysRev C, 1994, : 1139.[28] LI G Q, LEE C H, and BROWN G E. Phys Rev Lett,1997, : 5214; Nucl Phys A, 1997, : 372.[29] KAISER N and WEISE W. Phys Lett B, 2001, : 283.[30] WONG C Y. Phys Rev C, 1982, : 1460.[31] ANDRONICA A, BRAUN-MUNZINGERA P, andSTACHEL J. Nucl Phys A, 2010, : 237c.[32] LUTZ M F M, KLIMT S, and WEISE W. Nucl Phys A,1992, , 521.[33] GRECO V, KO C M, and LEVAI P. Phys Rev Lett, 2003, : 202302; Phys Rev C, 2003, : 034904.[34] CHEN L W, KO C M, and LI B A. Phys Rev C, 2003, : 017601; Nucl Phys A, 2003, : 809.[35] FUKUSHIMA K. Phys Lett B, 2004, : 277.[36] FUKUSHIMA K and HATSUDA T. Rep Prog Phys,2011,74