Study of triangular flow $v_3$ in Au+Au and Cu+Cu collisions with a multiphase transport model

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by  Kai Xiao

version1.4
Study of triangular flow v in Au+Au and Cu+Cu collisions with a multiphase
3
transport model
Kai Xiao,1,2 Na Li,3,∗ Shusu Shi,1,2,† and Feng Liu1,2
1Institute of Particle Physics, Central China Normal University, Wuhan, Hubei, 430079, China
2The Key Laboratory of Quark and Lepton Physics (Central China Normal University),
Ministry of Education, Wuhan, Hubei, 430079, China
3Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
(Dated: January 16, 2012)
We studied the relation between the initial geometry anisotropy and the anisotropic flow in a
2 multiphase transport model (AMPT) for both Au+Au and Cu+Cu collisions at √sNN=200 GeV.
1 It is found that unlike the elliptic flow v2, little centrality dependence of the triangular flow v3 is
0 observed. After removing the initial geometry effect, v3/ε3 increases with the transverse particle
2 density,whichissimilartov2/ε2. Thetransversemomentum(pT)dependenceofv3 from identified
particles is qualitatively similar to thepT dependenceof v2.
n
a
PACSnumbers: 25.75.Ld,25.75.Dw
J

3
1 I. INTRODUCTION However, since the reaction plane can not be directly

measured in the experiment, those anisotropic parame-
]
h ters can not be directly obtained. It is found that differ-
A novel state of matter called quark-gluon plasma
-t (QGP) composed by deconfined partons is believed to entmethodsmaycauseupto20%discrepancyonv2 [16],
cl be created experimentally in heavy ion collisions at thusitshouldalsobecarefullyevaluatedforthev3 study.
Besides,v isdirectlyrelatedwiththeinitialfluctuation,
u RHIC [1]. The discovery of large elliptic flow indicates 3
n that the partonic collectivity is built up during the colli- it is interesting to see its system size dependence.
[ sions,and the number-of-quarkscaling suggeststhat the In this paper, we will study the triangular flow v3
in both Au+Au and Cu+Cu collisions in a multiphase
2 partonic degrees of freedom are active [2].
transport model (AMPT) [12]. The relation between v
v 3
The anisotropic flow is usually described by a Fourier
and ε is studied as a function of number of participants
3 3
decompositionoftheazimuthaldistributionwithrespect
1 and transverse momentum. The paper is organized as
to the reaction plane [3]. The second harmonic coeffi-
2 follows: In Sec. II, the observables and technical meth-
6 cient, v2, so called elliptic flow, has the biggest magni- ods are introduced. A brief description of AMPT model
1. tude at high energy collisions [4]. It is believed that the is given in Sec. III. The results and discussions are pre-
observed anisotropy in the momentum space is caused
1 sented in Sec. IV. Finally, a summary is given in Sec. V.
1 by the anisotropy in the coordinate space in the initial
1 condition. Lotsofattentionhasbeen putonthe relation
v: between v2 and spatial eccentricity to see the hydrody- II. OBSERVABLES
namics behavior of the created system [5–7].
i
X
Recent studies show that the event-by-event fluctua-
In a non-central collisions, the overlap region of two
r tion of the initial geometry [8] may play an important
a nucleiisanalmondshape. Sincethepositionofnucleons
role in the study of collective flow. The triangular shape
may fluctuate event by event, as discussed in Ref [13–
inthe initialgeometrywillbe transferredto the momen-
15], those initial geometric irregularities of the colliding
tumspaceasthesystemexpands,andfinallyleadstothe
system can be described by εn:
none zero value of the third harmonic coefficient, v . It
3
is found that the triangularflow v is responsible for the
ridge and shoulder structures and3 the broad away-side q r2cos(nϕ) 2+ r2sin(nϕ) 2
εn = h i h i , (1)
of two-particle azimuthal correlation [9]. Besides, it is r2
h i
also considered to be a good probe to study the viscous
hydrodynamics behavior of the colliding system [10]. where r and ϕ are the polar coordinate position of par-
ticipating nucleons and is the average over all the
Lots of properties of the triangular flow v have been
3 h···i
participants in an event. n refers to the n-th harmonic,
studied in hydrodynamic and transport models [10, 11].
i.e., ε describes the elliptic shape and ε describes the
2 3
triangular shape.
Asthesystemevolves,theanisotropyinthecoordinate
∗Electronicaddress: [e-mail] space is transferred to the anisotropy in the momentum
†Electronicaddress: [e-mail] space due to the pressure gradient. The particle distri-
2
bution then can be written as
dN
1+2 vncos[n(φ Ψn)], (2)
dφ ∝ X − n = 2, Au+Au
n=1
n = 2, Cu+Cu
1.5
where φis the azimuthalangle,andΨn is the n-thevent n = 3, Au+Au
plane angle reconstructed by the final state particles: n = 3, Cu+Cu
sin(nφi) n 1
Ψn = 1 tan−1 Pi . (3) R
n cos(nφi)
 Pi 
0.5
The observed anisotropic flow is defined as the n-th
Flourier coefficient vn:
0
vnobs = cos[n(φ Ψn)] . (4) 0 20 40 60 80
h − i
Most Central (%)
Here is taking the average over all the particles in
h···i
the sample.
This isthe so-calledeventplanemethod ofcalculating
vn. The reconstructedevent plane fluctuates aroundthe FIG.1: Thesecondandthirdharmoniceventplaneresolution
reaction plane. The observed signals need to be revised calculatedbytheparticleswithpseudo-rapidtyregionof η >
| |
by the corresponding resolution [3]: 2 as a function of centrality in both Au+Au and Cu+Cu
collisions at √sNN=200 GeV in AMPT model.
vnobs
vn = R . (5)
n
Due to the finite multiplicity of final state particles, the
resolution
n = 2, Au+Au
Rn = cos[n(Ψn ΨnR)] (6) n = 2, Cu+Cu
h − i
10 n = 3, Au+Au
is usually smaller than 1. Ψ represents the nth real
nR n = 3, Cu+Cu
)
event plane angle. %
(

n
v 5
III. AMPT MODEL
There are four main components in AMPT model:
the initial conditions, parton interactions, hadronization 0
and hadron interactions. The initial conditions are ob- 0 20 40 60 80
tained from the HIJING model [17], which includes the Most Central (%)
spatial and momentum distributions of minijet partons
fromhardprocessesandstringsfromsoftprocesses. The
timeevolutionofpartonsisthentreatedaccordingtothe
ZPC[18]partoncascademodel. Afterpartonsstopinter- FIG. 2: vn as a function of centrality in both Au+Au and
acting, a combined coalescence andstring fragmentation Cu+Cu collisions at √sNN=200 GeV in AMPT model.
model are used for the hadronization of partons. The
scattering among the resulting hadrons is described by
IV. RESULTS AND DISCUSSIONS
a relativistic transport (ART) model [19] which includes
baryon-baryon, baryon-meson and meson-meson elastic
and inelastic scattering. Inordertobecomparablewiththeexperimentaldata,
Inourstudy,weanalyzedtheeventsfromAMPTwith the eventplanemethod isusedtocalculatevn. Thepro-
thepartoncrosssectionequalsto3mband10mb. Asall cedure is slightly different between ours and Ref. [11], in
the conclusions are independent on the parton cross sec- which the event plane was reconstructed by initial par-
tion, only the resultsfrom3 mb areshowninthis paper. tons. Charged particles with pT 2 GeV/c, η > 2 are
≤ | |
There are about 8 million events in Au+Au collisions chosen to reconstruct the event plane according to the
and19millioneventsinCu+Cucollisionsat√s =200 Eq. 3. The particles used for the vn measurements are
NN
used. ThestringmeltingAMPTversionisusedsincethe within the η < 1. The η gap used here is to reduce
| |
previousstudyshowsthatthestringmeltingAMPTver- the auto-correlationbetween the particles usedto recon-
sionagreeswiththeexperimentalresultsbetter[12]. The structtheeventplaneandtheparticlesofinterest. Inthe
centrality is defined by the impact parameter. following,the observedvn are all correctedby the corre-
3
0.2 0.2
0.15 0 - 80 % 102 0.15 0 - 80 % 102
0.1 0.1
0.05 10 0.05
v2 0 0.1 v3 0 0.025 10
0.08 0.02
-0.05 v20.06 -0.05 v30.015
-0.1 0.04 1 -0.1 0.01
0.02 0.005 1
-0.15 0 0 0.1 0e.2 0.3 0.4 -0.15 0 0 0.1 0e.2 0.3 0.4
-0.2 2 -0.2 3
0 0.1 0.2 0.3 0.4 e0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 e0.5 0.6 0.7 0.8 0.9 1
2 3
FIG.3: (Color online) v2 asa function ofε2 in Au+Aucolli- FIG.4: (Color online) v3 asafunction of ε3 in Au+Aucolli-
sions at √s =200 GeV in AMPT model. The black points sions at √s =200 GeV in AMPT model. The black points
NN NN
aretheaveragev2 intheselectedε2 bin. Thepadintheright aretheaveragev3 intheselectedε3 bin. Thepadintheright
down corner is theaverage of v2 with smaller scale. down corner is theaverage of v3 with smaller scale.
spondingresolution. Fig.1showstheresolutionofv2and an selected εn bin, and the curves are the connection of
v inbothAu+AuandCu+Cucollisions. Theresolution pointstoguideoureyes. Thepadsintherightdowncor-
3
ofv2 showsa peak inmid-centralcollisionswhichis con- ners are the average values of vn with smaller scale. In
sistentwiththe experimentalresult[20]. This is because Fig. 3, the v increases with ε , which is consistent with
2 2
the resolution of v is affected by both of the v signal theidealhydrodynamiccalculation[21]. While inFig.4,
2 2
and the multiplicity used to reconstructthe event plane. thetriangularflowv firstlyincreaseswithε upto0.17,
3 3
Whiletheresolutionofv onlydependsonthemultiplic- and then decreases. Based on our study, the higher ε
3 3
ity, and keeps decreasing as the multiplicity drops. bin corresponds to the more peripheral collisions. It is
InFig.2,v2 andv3 areshownasfunctionsofcentrality known that v3 is caused by initial geometrical fluctua-
in both Au+Au and Cu+Cu collisions. We can see that tion, and built up by the interactions of constituents.
v2 shows strong centrality dependence since it is mainly The less interactions in the higher ε3 bin may cause the
coming from the elliptic anisotropy in the initial geom- less converting efficiency from ε3 to v3. That could be
etry. Unlike v2, the dependence of v3 on centrality and the reason of decreasing trend of v3 when ε3 is larger
system size are much smaller. The trend of v3 observed than 0.17. Both the trend and the value of v3 show dis-
is the sameas thatin Ref[11]. However,the eventplane crepancytotheidealhydrodynamiccalculations[21]. As
angle Ψn in Ref [11] is obtained from the initial parton discussed in Ref. [10], the viscosity causes the decrease
distribution, which is not observed in the experiment, ofv3, however,the effects to v3 versusε3 is notshownin
while in our study it is from the final state particle dis- the viscous hydrodynamic calculation.
tribution. Theresultsindicatethatthetriangularflowis The ratio of elliptic flow to eccentricity v /ε gains
2 2
less sensitive to the centrality and system size compared lots of interests by comparing with the hydrodynamic
with the elliptic flow. It could be understood as a result model [14, 22]. Recently, the behavior of triangular flow
of combined effects from initial geometrical fluctuation v inidealhydrodynamicsisalsodiscussed[10]. InFig.5,
3
and collective dynamics which requires the size of bulk we study the vn/εn as a function of transverse parti-
to interact among themselves. cle density. From the plot we can see that vn/εn from
It is commonly assumed that the harmonic flow coef- Au+Au and Cu+Cu are consistent with each other very
ficients vn linearly depends on the εn. This assumption well. As the transverse particle density increases, v3/ε3
is supported by hydrodynamic simulations [10] as long riseswithsmallervaluethanv2/ε2. Itimpliesthatasthe
as one probes deformed initial profiles with only a single particle density increases, the initial geometry asymme-
non-vanishingharmoniceccentricitycoefficient. InFig.3 try transfers to momentum asymmetry more efficiently
and Fig. 4, we investigate the feasibility of this assump- whilethesystemexpands. Besides,thesecondorderhar-
tionfor v2 andv3 respectively. The relationsbetweenvn monic is more efficient than the third order.
and εn are drawn event by event in the two-dimension At last, the transverse momentum dependence of v3
plots. The black points are the average values of vn in for π, K, p and Λ is alsostudied in Au+Au andCu+Cu
4
and Cu+Cu collisions using the AMPT Monte-Carlo
model. We find that the triangular flow v is less sen-
3
sitive to the centrality and system size compared with
40 n = 2, Au+Au the elliptic flow v . The v displays an increasing trend
2 2
n = 2, Cu+Cu
asa functionofε , whichis qualitativelyconsistentwith
2
) 30 n = 3, Au+Au hydrodynamic calculation. We found that v shows an
% n = 3, Cu+Cu 3
increasing trend when ε is less than 0.17, and then de-
( 3
n 20 creases beyond ε3 = 0.17. It may be because of the
e lower converting efficiency from ε to v in the higher
3 3

⁄n ε bin. This decreasing trend is in contrast to the re-
v 10 3
sults of ideal hydrodynamic calculation. Both v /ε and
2 2
v /ε increase with the transverse particle density, and
3 3
0 the second harmonic asymmetry in the initial geometry
seems to transfer to the momentum asymmetry more ef-
0 10 20 30 40 50 60
1 ⁄ S dN ⁄ dy (fm-2) ficientlythanthe thirdharmonic. The triangularflowv3
of identified particles shows a mass ordering in low pT
and meson-baryon splitting at intermediate pT in both
Au+Au and Cu+Cucollisions which is similar to the pT
FIG. 5: vn/εn as a function of transverse particle density in dependence of v2.
both Au+Au and Cu+Cu collisions at √s =200 GeV in
NN
AMPT model.
collisions. In Fig. 6, we can see that v shows quite sim-
3
ilar trend to v2. At low pT, the mass ordering phenom-
ena is observed. The lighter particles are found with
larger v . It indicates that although v is driven by ε ,
3 3 3
its transverse momentum dependence is dominated by
the hydrodynamics behavior of the system. While when
pT 1.5 GeV/c, baryons and mesons are separated into
≥ VI. ACKNOWLEDGMENTS
two groups. The pT dependence of v3 from identified
particles is qualitatively similar to the pT dependence of
v [5,6]. Thev resultsofidentifiedparticlesfromAMPT
2 3
model are similar to the STAR preliminary results [23]. We wish to thank Prof. Fuqiang Wang for useful sug-
gestions, and Dr. Kejun Wu for useful discussions on
the AMPT model. This work was supported in part by
V. SUMMARY
the National Natural Science Foundation of China un-
der grant No. 10775060, 11105060, 11135011, 11147146
Insummary,westudiedtherelationbetweeninitialge- and ‘the Fundamental Research Funds for the Central
ometry parameter εn and anisotropic flow vn in Au+Au Universities’, Grant No. HUST: 2011QN195.
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(a) (b)
20 Au+Au 0 - 80 % 20 Cu+Cu 0 - 60 %
p
k
15 p 15
) L
%
(
10 10
3
v
5 5
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
p (GeV ⁄ c) p (GeV ⁄ c)
T T
FIG.6: v3 asafunctionoftransversemomentumin(a)Au+Auand(b)Cu+Cucollisionsat√sNN=200GeVinAMPTmodel.
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