Directed flow in relativistic heavy-ion collisions from a multiphase transport model

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by  Chong-Qiang Guo

Directed flow in relativistic heavy-ion collisions from a multiphase transport model
Chong-Qiang Guo,1,2 Chun-Jian Zhang,1,2 and Jun Xu∗1
1Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
2University of Chinese Academy of Sciences, Beijing 100049, China
(Dated: January 10, 2017)
Wehavestudied thedirected flowin 197Au+197Aucollisions at √sNN =200 and 39GeV within
a multiphase transport model. As the partonic phase evolves with time, the slope of the parton
directedflowatmid-rapidityregionchangesfromnegativetopositiveasaresultofthelaterdynamics
at200 GeV,while it remainsnegativeat 39GeV duetotheshorterlife timeof thepartonicphase.
The directed flow splitting for various quark species due to their different initial eccentricities is
observed at 39 GeV, while the splitting is very small at 200 GeV. Effects of the hadronization and
thehadronicrescatteringsonthedirectedflowarefurtherdiscussed. Ourstudyservesasabaseline
7 on the understanding of the directed flow in the absence of the mean-field potentials and other
1 exotic mechanisms.
0
2 PACSnumbers: 25.75.-q,25.75.Ld,24.10.Lx

n
a
J I. INTRODUCTION hadron-quark mixed phase. On the other hand, the ef-
fects ofthe EoScould be understoodonly if the directed
7
The main purpose of relativistic heavy-ion collision flowfromthecascadescenariowithoutmean-fieldpoten-

] tials is well settled. In addition, the effect of hadroniza-
experiments is to study the properties of the quark-
h
tion on the directed flow has not attracted much atten-
t gluonplasma(QGP)[1–4]andtounderstandthehadron-
- tion yet.
l quark phase transition. The anisotropic flow, defined as
c
v = cos[n(φ Ψ )] withφbeingtheparticleazimuthal
u n h − n i
angleinmomentumspacewithrespecttotheeventplane
n
[ Ψn and ... denoting the event average, is an impor- Recently, RHIC-STAR Collaboration have reported
tant obsehrvaible in characterizing how the anisotropy in the directed flow of protons and pions in the beam-
1
the initial coordinate space develops into that in the fi-
v energy-scan program [13]. It has been found that the
nal momentum space, as a result of the strong interac-
5 slope of the net-proton directed flow changes sign twice
tion in the QGP matter created in relativistic heavy-
0 between√sNN =11.5GeVand39GeV,andhasamini-
8 ion collisions. The first-order anisotropic flow is named mumbetween√sNN =11.5GeVand19.6GeV.Besides,
1 as the directed flow (v1) (see Ref. [5] for a recent re- splittings of the directed flow between protons and an-
0 view), and it contains the rapidity-odd component and tiprotons as well as that between π+ and π− were ob-
1. the rapidity-even component. The rapidity-odd compo- served at lower collision energies but become small at
0 nent v1odd(y) = −v1odd(−y), which is traditionally called higher collision energies. Efforts have been made using
7 the sideward flow, is attributed to the collective side-
transport model calculations [14–16] but none of them
1 wards deflection of particles. The rapidity-even compo-
havedescribedtheexperimentaldataofthedirectedflow
v: nent v1even(y) = v1even(−y) was realized recently [6, 7], at various collision energies very satisfactorily so far.
i and it is attributed to the event-by-event fluctuation in
X
the initial state of the colliding nuclei. In the present
r study we only talk about the rapidity-oddcomponent of
a
the directed flow. In this study we investigate the directed flow in rel-
Both hydrodynamic [8] and transport model [9] stud- ativistic heavy-ion collisions within a multiphase trans-
ies have shown that v (y) in the midrapidity region, es- port (AMPT) model [17]. Different from the previous
1
pecially that for baryons, is sensitive to the equation of study [18], we will study the time evolution of the di-
state (EoS) of the produced matter. With the increas- rected flow in the partonic phase, the splitting of the
ing collision energies, these calculations predict that the directedflowbetweendifferentquarkspecies,andthe ef-
slope ofv for chargedparticlesnearmid-rapidity region fectduetohadronization. Thisstudyiscarriedoutbased
1
changesfrompositiveto negative,whichis named inthe on the original AMPT model with only scatterings be-
literature as the ”antiflow” [10], the ”third flow compo- tween particles. This serves as a baseline of the directed
nent” [11], or the ”wiggle” [12]. It is argued that this flow properties that one needs to understand before in-
phenomenon could be a possible signature of the phase corporatingtheEoSorthemean-fieldpotential. Therest
transition,originatingfromthesoftnessoftheEoSinthe of the paper is organized as follows. Section II provides
a brief introduction of the AMPT model. The detailed
analysis and discussions of the directed flow results are
giveninSec.III.Finally,asummaryandoutlookisgiven
∗Correspondingauthor: [email protected] in Sec. IV.
2
II. THE AMPT MODEL 197Au+197Aucollisionsat√sNN =200and39GeV,cor-
respondingto the top RHIC energyanda typicalenergy
in the beam-energy-scan program. Typically, we focus
The string melting version of the AMPT model [17],
onthetime evolutionofv ,thesplittingofv forvarious
which is used in the present study, mainly consists of 1 1
particle species, and the hadronization effect on v . The
four parts: the initial condition generated by Heavy 1
directedflowiscalculatedwithrespecttothe theoretical
Ion Jet Interaction Generator (HIJING) model [19], the
reaction plane, i.e., v = cos(φ) , in the present study.
partonic evolution described by Zhang’s parton cascade 1
h i
(ZPC) model [20], a coalescence model to describe the
hadronization process, and the hadronic evolution de-
scribedbyarelativistictransport(ART)model[21]. The A. Time evolution of v1 in the partonic phase
HIJING model generates hadrons with proton-proton
scatterings as the building brick together with the nu- Beforeweinvestigatethetimeevolutionofv ,let’sfirst
1
clear shadowing effect and the Glauber geometry for have a global impression on the density evolution in the
the colliding nuclei at relativistic energies. The initial partonic phase. The density contours at time 0.5, 1, 1.5,
phase-space distribution of quarks is generated by melt- 2,4,6,8,and10fm/cinthex-o-zplane,i.e.,thereaction
ing hadrons produced in HIJING, containing reasonable plane with x the direction of the impact parameter and
event-by-eventfluctuations. Thequarkinteractioninthe z thebeamdirection,inmid-centralAu+Aucollisionsat
ZPCmodelisdescribedbythe partonictwo-bodyelastic √sNN = 200 GeV and 39 GeV are displayed in Figs. 1
scatterings with the differential cross section given by and 2, respectively. We found that a large amount of
partons are formed before 0.5 fm/c. The strong inter-
dσ 9πα2
s , (1) action between partons lasts until 4 or 6 fm/c. After
dt ≈ 2(t µ2)2 that the interaction becomes weaker, and the disk-like
−
partonic matter moves away in the z direction. The
wheretisthethestandardMandelstamvariableforfour- density evolutions are similar at √sN±N = 200 GeV and
momentum transfer. In the present study we set the 39 GeV, although a higher quark density is observed at
strong coupling constant αs to be 0.47 and the parton higher collision energies where the life time of the par-
screening mass µ to be 3.2264 fm−1, leading to the to- tonic phase is expected to last longer. The direction of
tal cross section of 3 mb. The evolution of the partonic x z >0 (x z <0) is intuitively labelled as the ”+( )”
phase ends when the distance between quarks is out of di·rection for·the ease of discussion, corresponding to−the
the radius of the cross section. The hadronization is de- positive flow and the antiflow. The relative numbers of
scribed by a spatial coalescence model which allows a particles moving in the ”+” and the ” ” directions de-
pairofnearestquarkandantiquarktoformamesonand termine the net directed flow. −
three nearest quarks or antiquarks to form a baryon or
an antibaryon, with the mass and species of the hadron
-3
determinedbytheinvariantmassandtheflavorsofthese 10 (fm ) 2
constituent partons. In the AMPT version used in the 5 t=0.5-fm/c + t=1fm-/c + t=1.5-fm/c + t=2fm-/c + 10
present study, hadrons are formed from nearestquark in 0 18
coordinate space, while whether they are close in mo- 26
maleesncteunmcespisacperoisceneodtedgubaerfaonrteeebda.ryInonadcodaitlieosnc,enmceesoinntchoe- x (fm)-1-50 t=4fm+-/c -+ (a)t=6fm+-/c +- (b)t=8fm+-/c +- (c)t=10+f-m/c +- (d) 3442
AMPT version used in the present study, which reduces 5 50
the combinations of three quarks that are close in coor- 0 58
dinate space and makes the baryoncoalescenceless real- -5 + - + - + - + - 66
istic. An improvement can be done with mix-event coa- -10 (e) (f) (g) (h) 74
-10 -5 0 5 -10 -5 0 5 -10 -5 0 5 -10 -5 0 5 10
lescence [22] in the future. In order to see the effect of a z (fm)
more realistic coalescence on the directed flow, we have
FIG.1: (Color online) Densitycontoursinthex-o-z planeat
also checked with the dynamical coalescence [23] based differenttimestepsinthepartonicphaseofAu+Aucollisions
ontheWignerfunctioncalculationwhichwillbedetailed at √sNN = 200 GeV and impact parameter b = 5 fm, with
in Sec. IIIB and APPENDIX A. The spatial coalescence different colors indicating the quark number densities. The
in the AMPT model is followed by the ART model that particlesthatmoveinthe”+( )”directioncontributetothe
−
contains various elastic, inelastic, and decay channels to positive flow (antiflow).
describe the hadronic evolution.
Thedirectedflowofquarksinmid-centralAu+Aucol-
lisionsat√sNN =200GeVand39GeVatdifferenttime
III. ANALYSIS AND RESULTS steps is displayed in Fig. 3, with the solid lines from a
cubic fit of v (y) = F y +F y3. The initial v at both
1 1 3 1
In the present study, we employ the AMPT model to collisionenergies are very small as expected. As the sys-
investigate the directed flow in mid-central (b = 5 fm) tem evolves, the slope of the directed flow at √sNN =
3
10 (fm-3) 2 mthoeredinreecgtaetdivfleo(wpoastitdivieff)eraesntthreapsyidstiteymreegvioolnvsesc,haalnthgoeusginh
t=0.5-fm/c ++ t=1fm-/c + t=1.5-fm/c + t=2fm-/c + 5
5 8 a complicated manner as shown in Fig. 3. At √sNN =
0 11 200 GeV the integrated directed flow becomes saturated
x (fm)-1-50 + - (a) + - (b) + - (c) + - (d) 112470 aattaabloautter8tfimm/ec.,Iwnhialdedaitti√ons,NtNhe=m3a9gGnietVudietiosfstahtueriantteed-
5 t=4fm-/c + t=6fm-/c + t=8fm-/c + t=10f-m/c + 23 grated v1 is larger at lower collision energies due to the
0 26 larger initial first-order eccentricity, as will be discussed
29 in Fig. 7.
-5 + - + - + - + - 32
(e) (f) (g) (h) 35
-10
-10 -5 0 5 -10 -5 0 5 -10 -5 0 5 -10 -5 0 5 10
z (fm)
(a) (b)
FIG.2: (Color online) Same asFig. 1but forAu+ Aucolli-
0.2 0.4
sions at √sNN = 39 GeV.
0.1 0.2
200 GeV grows to a maximum negative value at around %) Au+Au@200GeV Au+Au@39GeV
4 fm/c and then gradually becomes positive, while that v(10.0 bp a=r t5o nfm v1 bp a=r t5o nfm v1 0.0
at √sNN = 39 GeV growsto a maximum negative value y > 0 y > 0
ataround6fm/candbecomessaturated. Themaximum -0.1 y < 0 y < 0 -0.2
slope is largerat 39GeV than at 200GeV. Onthe other
hand, the strong scatterings among partons mostly end -0.2 -0.4
around 4 6 fm/c, and the interaction is expected to
∼ 0 10 20 0 5 10
become weaker in the later dynamics, according to the
t (fm/c)
density evolution shown in Figs. 1 and 2. However, it is
FIG. 4: (Color online) The time evolution of quark directed
seenthat the later dynamics reversesthe slope of the di-
flow (v1) at forward and backward rapidities in mid-central
rectedflowat√sNN = 200GeV butis unable toreverse Au+Aucollisions at √sNN = 200 GeV (a) and 39 GeV (b).
that at √sNN = 39 GeV, due to the shorted life time of
the partonic phase at lower collision energies. We note
that a similar non-monotonic behavior of the directed
In order to understand the time evolution of the v
flowwasobservedinRef.[14]butnotdiscussedindetail. 1
slope inFig.3 withsaturatedintegrateddirectedflowat
bothforwardandbackwardrapiditiesasshowninFig.4,
(a) (b) weplotthetimeevolutionofd(N+ N−)/dyasafunction
0.4 Au+Au@200GeV Au+Au@39GeV 0.8 ofrapidityyinFig.5,withN+(N−−)beingthenumberof
b = 5 fm b = 5 fm particles that contribute positively (negatively) to v at
0.2 parton v1 parton v1 0.4 forwardrapiditybutnegatively(positively)tov atba1ck-
1
%) ward rapidity, corresponding to the number of particles
(10.0 0.0 movinginthe”+( )”directionaslabelledinFigs.1and
v −
2,contributingtothepositiveflow(antiflow). Itisseenin
-0.2 0.5fm/c 6fm/c -0.4 Fig.5thatN+ isalmostequaltoN− atthe initialstage,
11f.m5f/mc / c 180fmfm/c/c 01.f5mf/mc / c 46ffmm//cc corresponding to a zero v1. As the system evolves, two
-0.4 24ffmm//cc 1250ffmm//cc 12.f5mf/mc / c 81f0mfm/c/c -0.8 negativepeaks growatforwardandbackwardrapidities,
correspondingtothe picturethatmoreparticlesmoveto
-2 -1 0 1 -2 -1 0 1 2 x direction at forward rapidity and to +x direction at
−
y backward rapidities, i.e., a net antiflow. As these par-
ticles are further accelerated in the later dynamics after
FIG.3: (Coloronline)Directedflow(v1)ofquarksversusra- 4 6 fm/c in the beam direction, the peaks gradually
pidity(y)atdifferenttimestepsinmid-centralAu+Aucolli- ∼
moveto largerrapidity regions. At the later stage of the
sions at √sNN = 200 GeV (a) and 39 GeV (b).
partonic phase at √sNN = 200 GeV, only particles that
contribute to the positive flow stay in the mid-rapidity
region,leadingtoapositiveslopeofmid-rapidityv . So,
1
the time evolution of the v1 slope in Fig. 3 at √sNN =
B. Saturation of v1 200 GeV is due to the transfer of particles among differ-
ent rapidity regions,leading to a positive v slope at the
1
Next, we display the integrated directed flow at for- freeze-out stage. At √sNN = 39 GeV, the saturation of
ward and backward rapidities as a function of time in v takes longer time while the life time of the partonic
1
Fig. 4. It is interesting to see that the integrated v at phase is shorter, leading to a negative v slope at the
1 1
the forward (backward) rapidity monotonically becomes freeze-out stage.
4

as that between u and d quarks are already seen but the
2 Au+Au@200GeV b = 5 fm Au+Au@39GeV b = 5 fm difference is comparable to the statistical error, while at
0 √sNN = 39 GeV it is clearly seen that d¯quarks have a
y more negative directed flow than u and d quarks. The
d
)/ -2 splitting between directed flows of various quark species
N-
N-+-4 couldbepartiallyresponsibleforthev1 splittingbetween
( protons and antiprotons as well as that between π+ and
d
-6 π−, as reported in Ref. [13].
-8 0.5fm/c 6fm/c 0.5fm/c 4fm/c
1fm/c 8fm/c 1fm/c 6fm/c
1.5fm/c 10fm/c 1.5fm/c 8fm/c
-10 24ffmm//cc 1250ffmm//cc (a) 2fm/c 10fm/c (b) 0.2 (a) (b) 0.3
-4 -2 0 2 4 -4 -2 0 2 4
y Au+Au@200GeV Au+Au@39GeV
b = 5 fm b = 5 fm
FIG.5: (Coloronline)Rapiditydependenceofd(N+−N−)/dy initial parton 1 initial parton 1
indicating thecompetition between thepositive flow and the
1
antiflow at different time steps in partonic phase of mid- 0.0 0.0
central Au+Au collisions at √sNN = 200 GeV (a) and 39
GeV (b).
u u
d d
d d
C. Splitting of v1 for various quark species -0.2 -0.3
-2 0 2 -2 0 2
y
The results discussed in the previous subsections are
FIG. 7: (Color online) Initial first-order eccentricity (ǫ1) of
the average over all quark species. Due to the finite u, d,and d¯quarksversusrapidity (y)in mid-centralAu+Au
chemical potentials reached at lower collision energies,
collisions at √sNN = 200 GeV (a) and 39 GeV (b).
it is always observed that there are splittings of quanti-
ties between particles and antiparticles as well as those
The mechanismhow the initialeccentricity(ǫ )devel-
between particles of different isospin states. The typical n
ops into the final elliptic flow (v ) and triangular flow
examplesaresplittingsoftheellipticflow[24]andthedi- 2
(v ) has been extensively studied (see, e.g., Refs. [25–
rected flow [13] between protons and antiprotons as well 3
asthosebetweenπ+ andπ−. BasedontheAMPTmodel 27]). Generally, a larger initial eccentricity leads to a
larger final anisotropic flow. Since the rapidity-odd di-
withonlyparticlecascade,wewillseehowlargethesplit-
rectedflowinthepresentstudyiscalculatedwithrespect
ting of directed flow between various quark species is, in
to the theoretical reaction plane, we calculate the corre-
the absence of the mean-field potential as well as other
spondingly ǫ according to the particle azimuthal angle
exotic mechanisms. 1
φ in coordinate space as
s

ǫ = cos(φ ) , (2)
0.3 1.5 1 s
h i
Au+Au@200GeV Au+Au@39GeV
0.2 b = 5 fm b = 5 fm 1.0 with ... denotingthe eventaverage. Figure7showsthe
h i
parton v1 parton v1 rapidity distribution of the initial first-order eccentricity
0.1 0.5 for various quark species at both √sNN = 200 GeV and
)
% 39GeV.Itisseenthatǫ hasapositiveslopewithrespect
1
v(10.0 0.0 totherapidity. Duetothe largerinelasticproton-proton
scattering cross section at higher collision energies used
-0.1 -0.5
u u in HIJING [19], the effective overlap region is slightly
-0.2 d d -1.0 largerat√sNN =200GeVcomparedtothatat39GeV,
d d leading to a slightly smaller centrality at higher collision
(a) (b)
-0.3 -1.5 energies even with the same impact parameter b = 5
-2 -1 0 1 -2 -1 0 1 2
fm. This leads to a larger slope of ǫ at lower collision
y 1
energies compared with that at higher collision energies,
FIG. 6: (Color online) Final directed flow (v1) of u, d, and d¯ which is responsible for the larger maximum v slope at
quarksversusrapidity(y)inmid-centralAu+Aucollisionsat 1
about 4 6 fm/c in Fig. 3 and a larger saturated v
√sNN = 200 GeV (a) and 39 GeV (b). ∼ 1
in Fig. 4 at lower collision energies. Moreover, it is seen
that at √sNN = 200 GeV the initial ǫ1 is similar for all
Figure6displaysthedirectedflowofu,d,andd¯quarks the quark species, while at √sNN = 39 GeV d¯quarks
attheir freeze-outstageinmid-centralAu+Aucollisions have a smaller slope of ǫ with respect to the rapidity.
1
at√sNN = 200GeVand 39GeV. At √sNN = 200GeV Since these quarks are melted from hadrons produced
the v splitting between quarks and antiquarks as well in HIJING, the difference is attributed to the different
1
5
production mechanisms of hadrons that have different the spatialcoalescencein AMPT,as wellasthat oftheir
baryon or isospin charges. This explains the different constituentquarksweightedbytheWignerfunction,and
d¯directed flow compared to v of other quark species. that of all quarks at their freeze-out stage. The dif-
1
The ǫ for u and d quarks is the same at higher collision ference between v from the dynamical coalescence and
1 1
energies, and the difference is barely able to observe at thatfromthedefaultspatialcoalescenceinAMPTisob-
lower collision energies. served, consistent with the statement in Ref. [14] that
the directed flow is sensitive to the hadronization treat-
ment. On the other hand, the spatial coalescence in
D. Effect of hadronization on v1 AMPT needs further improvement [22], especially for
baryons as mentioned in Sec. II. It is interesting to see
The directedflow offreeze-outquarksdiscussedinthe that the directed flow of constituent quarks weighted by
previous subsections will be modified in the hadroniza- the Wigner function is quite different from that of all
tion process. In the present study we investigate the quarks at their freeze-out stage. This means that in
hadronization from a dynamical coalescence model [23, the dynamical coalescence scenario only part of partons
28] and the default spatial coalescence model as in closeinphasespacedominatethecontributionofforming
AMPT. In the dynamical coalescence model, the prob- hadrons, while their directed flow is quite different from
ability to form a hadron is proportional to the quark the rest partons. The deep reason for this is worth fur-
Wignerfunctionofthathadron,andtheproportionalco- ther investigation. As a consequence, the directed flow
efficient is the statistical factor by considering the spin, offormedprotonsorpions issimilartothatoftheir con-
flavor,and color degrees of freedom. For detailed formu- stituent partons but different from that of the overall
laeswereferthereadertoAPPENDIX A.Inthedynam- partons. The difference is larger for baryons than for
ical coalescence model, partons that are close in phase mesons, and the v1 slope of protons from the dynamical
spacehavea largerprobability to formhadrons,while in coalescencehasevenchangeditssigncomparedwiththat
the default spatialcoalescencemodelin AMPT,hadrons of the overall partons.
are formed by nearest combinations of partons in coor-
dinate space as discussed in Sec. II, and all the partons
are forceto be usedup after hadronization. To speed up E. Effect of hadronic evolution on v1
the program,in the dynamical coalescence only the par-
ton combinations with the relative momentum δp 2.5
≤
GeV/c are allowed to form hadrons, since the probabil-
ityforcombinationswithlargerrelativemomentatoform 6 6
Au+Au@200GeV Au+Au@200GeV
hadrons is rather small. 3 for protons for pions 3
0 0
6 6
(a)Au+Au@200GeV (b) p/ from dynamical coalescence
03 for protons pqa/ llf oq frr odmyn aAmMiPcaTl ccooaalleesscceennccee 03 v(%)1--63 Au+Au@ 39GeV (a) Au+A u@39GeV (b)--63
3 for protons for pions 3
) -3 Au+Au@200GeV -3
% for pions 0 0
(1-6 -6
v (c) Au+Au@39GeV (d)
3 for protons 3 -3 before ART -3
after ART
(c) (d)
-6 -6
0 0 -4 -2 0 2 4 -4 -2 0 2 4
y
-3 Au+Au@39GeV -3
for pions FIG. 9: (Color online) Directed flow (v1) of protons and
-6 -6 charged pions versus rapidity (y) before and after hadronic
-4 -2 0 2 4 -4 -2 0 2 4 rescatteringsinmid-centralAu+Aucollisionsat√sNN =200
y
GeV [(a) and (b)]and 39 GeV [(c) and (d)].
FIG. 8: (Color online) Directed flow (v1) of protons and
charged pions from the dynamical coalescence and from the
The directed flow of hadrons after hadronzation pre-
default spatialcoalescence intheAMPTmodel,aswell asv1
sented in the previous subsection is further modified by
of theirconstituent quarksweighted bytheWigner function,
and v1 of all quarks at their freeze-out stage as a function thehadronicevolutiondescribedbyART,containingvar-
of rapidity in mid-central Au+Au collisions at √sNN = 200 ious elastic, inelastic, and decay channels. To illustrate
GeV [(a) and (b)] and 39 GeV [(c) and (d)]. See text for the effect of hadronic rescatterings on v1 in the AMPT
details. model, we present in Fig. 9 the directed flow of initial
(beforeART)andfinal(afterART)protonsandcharged
Figure 8 displays the directed flow of protons and pions in mid-central Au+Au collision at √sNN = 200
charged pions from the dynamical coalescence and from GeV and 39 GeV. After hadronic scatterings, it is seen
6
thatthe slopeofthe directedflowgenerallybecomesless APPENDIX A. DYNAMICAL COALESCENCE
negative or increases, while the sign of the v slope near FOR HADRONS WITH WIGNER FUNCTION
1
mid-rapidity region is not changed. The effect of the
hadronic evolution on v1 is seen to be larger at lower In the dynamical coalescence model, the probability
collisionenergiescomparedtothatathighercollisionen- for a pair of quark and antiquark to form a meson is
ergies. proportional to the quark Wigner function of the meson
times the statistical factor, i.e.,
IV. SUMMARY AND OUTLOOK ρ2
f (ρ,k ) = 8g exp k2σ2 , (A.3)
M ρ M (cid:18)−σ2 − ρ ρ(cid:19)
ρ
To summarize, we have studied in detail the effects of
partonicscatterings,hadronization,andhadronicrescat- where g =1/36 is the statistical factor for pions, and
M
terings on the directed flow in relativistic heavy-ion col-
lisions within the framework of a multiphase transport 1
ρ = (r r ), (A.4)
model. Duetothe largerproton-protoninelasticscatter- √2 1− 2
ing cross section in HIJING at higher collision energies,
m k m k
whichleadsto a largeroverlapanda smallerinitialfirst- kρ = √2 2 1− 1 2 (A.5)
m +m
ordereccentricityathighercollisionenergies,thedirected 1 2
flow is saturatedata smaller value at√sNN = 200GeV are the relative distance in the coordinate and momen-
compared to that at 39 GeV. As the system evolves, a tum space for the two-particle system, with m , r , and
i i
non-monotonic behavior of the directed flow is observed k being the mass, coordinate, and momentum of the
i
at higher collision energies, and the later dynamics is
ith particle, respectively. The width parameter σ is re-
ρ
able tochangethe slope signofthe directedflow atmid-
latedtotheroot-mean-square(RMS)radiusofthemeson
rapidity region, as a result of the longer life time of the
through the relation
partonicphase,butthisisnotobservedatlowercollision
energies. The splitting of the directed flow for various 3 m2+m2
quark species is observed at lower collision energies due hrM2 i = 2(m1+m2)2σρ2. (A.6)
1 2
totheirdifferentinitialeccentricities,whilesuchsplitting
becomes very small at higher collision energies. The di- Similarly, the probability for three light quarks to form
rected flow is further affected by the hadronization and a baryon is expressed as
the hadronic rescatterings.
In the future study, we will improve the coalescence f (ρ,λ,k ,k )
B ρ λ
andintroducethe mean-fieldpotentialto the multiphase
ρ2 λ2 (A.7)
transport model as in Ref. [22, 29]. It will be of great =82g exp k2σ2 k2σ2 ,
B (cid:18)−σ2 − σ2 − ρ ρ− λ λ(cid:19)
interestto study quantitatively how the results obtained ρ λ
in this work are modified by the mean-field potentials,
where g = 1/108 is the statistical factor for protons,
whichareexpectedtobedifferentfordifferentquarkand B
and
hadronspecies. Ontheotherhand,itisinterestingtosee
that the directed flow of hadrons is mainly determined
2 m r +m r
by only part of partons that are close in phase space λ = 1 1 2 2 r , (A.8)
based on the dynamical coalescence. It is worthy to in- r3(cid:18) m1+m2 − 3(cid:19)
vestigate in more detail the feature of these partons and 3m (k +k ) (m +m )k
3 1 2 1 2 3
have a deeper understanding of the relationbetween the kλ = r2 m +m− +m (A.9)
1 2 3
anisotropic flow and the hadron-quark phase transition.
are the relative distance in the coordinate and momen-
tum space between the third particle and the system
Acknowledgments formed by the first and the second particle. The width
parameterσ is relatedto the RMSradiusofthe baryon
λ
WethankChenZhongformaintainingthehigh-quality through the relation
performance of the computer facility. This work was
supported by the Major State Basic Research Devel- r2 = 3 (m1+m2) σ2
opment Program (973 Program) of China under Con- h Bi 4m m (m +m +m )2 λ
1 2 1 2 3
tract Nos. 2015CB856904 and 2014CB845401, the Na-
[m m (m +m )+m m (m +m )
2 3 2 3 1 3 1 3
tionalNaturalScienceFoundationofChinaunder Grant ×
+ m m (m +m )]. (A.10)
Nos. 11475243 and 11421505, the ”100-talent plan” of 1 2 1 2
Shanghai Institute of Applied Physics under Grant Nos.
The RMS radius of the produced hadron is taken from
Y290061011andY526011011fromtheChineseAcademy
Ref. [30], which is 0.61 fm for π+ and 0.877 fm for pro-
of Sciences, the Shanghai Key Laboratory of Particle
tons, respectively.
Physics and Cosmology under Grant No. 15DZ2272100,
and the ”Shanghai Pujiang Program” under Grant No.
13PJ1410600.
7
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