Event-plane flow analysis without non-flow effects

0.21 MB·English

by  Ante Bilandzic

Event-plane flow analysis without non-flow effects
Ante Bilandzic,1 Naomi van der Kolk,1 Jean-Yves Ollitrault,2 and Raimond Snellings1
1NIKHEF, Kruislaan 409, 1098 SJ Amsterdam,The Netherlands
2Institut de Physique Th´eorique (IPhT), CEA/DSM,
CNRS/MPPU/URA2306, CEA-Saclay, F-91191 Gif-sur-Yvette cedex
(Dated: February 3, 2008)
The event-planemethod, which iswidely used toanalyze anisotropic flowin nucleus-nucleuscol-
lisions,isknowntobebiasedbynonfloweffects,especiallyathighpt. Variousmethods(cumulants,
Lee-Yang zeroes) have been proposed to eliminate nonflow effects, but their implementation is te-
dious, which has limited their application so far. In this paper, we show that the Lee-Yang-zeroes
method can be recast in a form similar to the standard event-plane analysis. Nonflow correlations
areeliminated byusingtheinformation fromthelengthoftheflowvector,inadditiontotheevent-
8
plane angle. This opens the way to improved analyses of elliptic flow and azimuthally-sensitive
0
observables at RHICand LHC.
0
2
PACSnumbers: 25.75.Ld,25.75.Gz,05.70.Fh

n
a
J I. INTRODUCTION • Onemustremoveautocorrelations: theparticleun-

5 der study should not be used in defining the event
2 SinceellipticflowhasbeenseenatRHIC[1,2,3],ithas plane, otherwise there is a trivial correlation be-
becomeacrucialobservableforourunderstandingofthe tween φ and ΨR [8]. This means in practice that
]
x system created in heavy-ion collisions. Anisotropic flow one must keep track of which particles have been
e is most often analyzed using the event-plane method [4]. used in defining the event plane, so as to remove
- These analyses are plagued by systematic errors due to them if necessary.
l
c nonflow effects [5]. Nonflow effects are particularly large
u • Nonflow correlations between the particle under
at high p [6], where they are likely to originate from
n t study and the event plane must be avoided. This
jet-like (hard) correlations;they are expected to be even
[ cannot be done in a systematic way, but rapidity
larger at the LHC. The purpose of this paper is to show
gaps are believed to largely suppress nonflow ef-
1
that nonflow effects can be avoided at the expense of a
v fects [3, 10].
slight modification of the event-plane method.
5
Anisotropicflowofanoutgoingparticleofagiventype, • Event-plane flattening procedures must be imple-
1
9 inagivenphase-spacewindow,isdefinedasitsazimuthal mentedtocorrectforazimuthalasymmetriesofthe
3 correlation with the reaction plane [7] detector acceptance [4].
.
1
v ≡hcos(n(φ−Φ ))i (1) A systematic way of eliminating nonflow effects is to
0 n R
use improved methods such as cumulants [11] or Lee-
8
0 where n is an integer (v1 is directed flow, v2 is elliptic Yang zeroes [12]. Cumulants have been used at SPS [13]
: flow), φ denotes the azimuth ofthe particle under study, and RHIC [6, 14]. Lee-Yang zeroes have been imple-
v
Φ theazimuthofthereactionplane,andangularbrack- mented at SIS [15] and at RHIC [16]. The reason why
i R
X etsdenoteanaverageoverparticlesandevents. SinceΦ these methods have not been more widely used is proba-
R
r is not known experimentally, vn cannot be measured di- bly that they are thought to be tedious.
a rectly. In this paper, we show that the method of flow anal-
Themostcommonlyusedmethodtoestimatev isthe ysis based on Lee-Yang zeroes can be recast in a form
n
event-plane method [4]. In each event, one constructs similar to the event-plane method. Specifically, Eq. (2)
an estimate of the reaction plane Φ , the “event plane” is replaced with
R
Ψ [8]. The anisotropic flow coefficients are then esti-
R
v {LYZ}≡hW cos(n(φ−Ψ ))i, (3)
mated as n R R
1 where ΨR is the same as in Eq. (2), and WR is a new
vn{EP}≡ hcos(n(φ−ΨR))i, (2) quantity, the event weight, which depends on the length
R
oftheflowvector. Theadvantageofthisimprovedevent-
where R = hcos(n(Ψ −Φ ))i is the event-plane reso- plane method over the standard event-plane method is
R R
lution, which corrects for the difference between Ψ and that both autocorrelations and nonflow effects are auto-
R
Φ . Thisresolutionisdeterminedineachcentralityclass matically removed.
R
through a standard procedure [9]. The paper is organized as follows. In Sec. II, we de-
The analogy between Eq. (2) with Eq. (1) makes the scribe the method for a detector with perfect azimuthal
methodratherintuitive,butitspracticalimplementation symmetry, and we explain why it automatically gets rid
has a few subtleties: of autocorrelations and nonflow correlations, in contrast
2
to the standard event-plane method. Readers interested this modulus has a sharp minimum, compatible with 0
in applying the method should jump to Appendix A, within statistical errors [18]. This is the Lee-Yang zero,
which describes the recommended practical implemen- whichisdeterminedbyfindingthefirstminimumnumer-
tation, taking into account anisotropies in the detector ically. We denote its value by rθ. The estimate of the
acceptance. InSec.III,wepresentresultsofMonte-Carlo integrated flow is then defined by
simulations. Lee-Yang zeroes are compared to 2- and 4-
particle cumulants. Sec. IV concludes with a discussion V = j01, (8)
ofwherethemethodshouldbeapplicable,andofitslim- n rθ
itations.
where j01 ≃ 2.40483 is the first zero of the spherical
Bessel function J0(x).
Please note that the above procedure only makes use
II. DESCRIPTION OF THE METHOD
of the projection of the flow vector onto an arbitrary di-
rection θ. For a perfect detector, azimuthal symmetry
A. The flow vector ensures that rθ is independent of θ, up to statistical er-
rors. In practice, however, it is recommended to repeat
The first step of the flow analysis is to evaluate, for the analysis for several values of θ (see Appendix A).
each event, the flow vector of the event. It is a two-
dimensional vector Q=(Q ,Q ) defined as
x y
C. Event weight
M
Qx =Qcos(nΨR) ≡ wj cos(nφj) Once rθ is determined, the event weight is defined by
j=1
X
M 1
Q =Qsin(nΨ ) ≡ w sin(nφ ), (4) WR ≡ J1(rθQ), (9)
y R j j C
j=1
X
whereJ1(x)isthesphericalBesselfunctionoffirstorder,
where the sum runs over all detected particles, M is the Q is the length of the flow vector, and C is a normal-
observed multiplicity of the event, φj are the azimuthal ization constant which is the same for all events in the
angles of the particles measured with respect to a fixed centralityclass. Thisconstantisdeterminedbyrequiring
directioninthelaboratory. Thecoefficientswj inEq.(4) thattheprojectionoftheflowvectorQontothereaction
areweightsdependingontransversemomentum,particle plane corresponds to the integrated flow V , defined by
n
massandrapidity. Thebestweight,whichminimizesthe Eq. (6). This gives the condition
statistical error (or, equivalently, maximizes the resolu-
tion)isvn itself,wj(pT,y)∝vn(pT,y)[17]. Areasonable hWRQi=Vn. (10)
choice for elliptic flow at RHIC (and probably LHC) is
Inserting Eqs. (8) and (9), we obtain
w =p .
T
Lee-Yang zeroes use the projection of the flow vector
1
ontoafixed,arbitrarydirectionmakingananglenθ with C = rθQJ1(rθQ) (11)
respect to the x-axis. We denote this projection by Qθ: j01
(cid:10) (cid:11)
where angular brackets denote an averageover events.
Qθ ≡Q cosnθ+Q sinnθ =Qcos(n(Ψ −θ)). (5)
x y R Thisnormalizationconstantcanbeshowntoberelated
totheresolutionparameterχusedinthestandardevent-
planeanalysis(seeAppendixA3foraprecisedefinition):
B. Integrated flow
j2
01
The first step of the analysis is to measure the inte- C =exp −4χ2 J1(j01). (12)
grated flow V , defined as the average value of the pro- (cid:18) (cid:19)
n
jection of Q onto the true reaction plane: Fig. 1 displays the weight W as a function of the
R
length of the flow vector Q, together with the proba-
Vn ≡hQcos(n(ΨR−ΦR))i (6) bility distribution of Q, for two values of the resolution
parameter. The normalized distribution of Q is [5]
where angular brackets denote an average over events.
We define the complex-valued function: dN 2χ2Q Q2 2χ2Q
2
dQ = V2 exp −χ V2 +1 I0 V . (13)
Gθ(r)≡ 1 eirQθ (7) n (cid:18) (cid:18) n (cid:19)(cid:19) (cid:18) n (cid:19)
Nevts Forχ≫1,this distributionisanarrowpeakcenteredat
events
X Q=V . The weightdefined by Eqs.(9)and (12) is then
n
Themodulus|Gθ(r)|canbeplottedasafunctionofr for closeto1forallevents. Ifχissmaller,thedistributionof
positive r. In the case where collective flow is present, Qis correspondinglywider,andW is negativeforsome
R
3
v isnon-zeroformostoftheparticles. Next,weassume
n
thattheparticleunderstudydoesnottakepartincollec-
N/dQ 15 Vn =d 0N./0d6Q25, c = 1.5 6 (Q)WR ctiovrereflloawte,di.we.i,thitahafeswvnot=he0r;poanrttihceleost(hfeorrhinasntda,nictemifatyhbeye
d 10 WR belong to the same jet). Such nonflow correlations usu-
MC dN/dQ
4
MC WR allyresultinvn{EP}6=0(generally>0forintra-jetcor-
5 relations). By contrast, our improved estimate v {LYZ}
2 n
vanishes, as we now show.
We separate the flow vector, Eq. (4), into a flow part
0 0
Qf , involving particles which contribute to collective
-2 flow, and a non-flow part Qnf, involving the particle
-5
under interest (autocorrelations) and the few particles
N/dQ 15 Vn =d 0N./0d6Q25, c = 1. 6 (Q)WR wlehctiicvheaflroewc:orrelated to it, but do not contribute to col-
d 10 WR
MMCC dWN/dQ 4 Q=Qf +Qnf. (14)
R
5
2 We further assume that the flow and the nonflow part
are uncorrelated.
0 0 Eqs. (3) and (9) define our estimate of v as
n
-5 -2 vn{LYZ}≡ 1 J1(rθQ)cos(n(φ−ΨR)) . (15)
C
0 0.05 0.1 0.15 0.2 0.25
(cid:10) (cid:11)
|Q| We rewrite the Bessel function as an integral over an-
gles [19]
FofIGQ.,1E:q.(c(o1l3o)r,ownitlihneV)nS=ha0d.e0d62a5re(ase:epSreocb.aIbIiIl)it.yOdpisetnricbiurctlieosn: J1(rθQ)cos(n(φ−ΨR))=−i 2π dθeirθQθcos(n(φ−θ)),
histograms of the distribution of Q obtained in the Monte- 0 2π
Z
Carlo simulation of Sec. III, following the procedure detailed (16)
in Appendix A. Solid curve: weight WR defined by Eqs. (9) withQθ definedbyEq.(5). Sincetheflowvectorappears
and (12). Stars: weights obtained in Sec. III. Top: χ = 1.5, in an exponential, the flow and nonflow contributions to
corresponding to a resolution R=0.86 in the standard anal- Eq.(15) canbe written as a product of two independent
ysis (see Eq. (2)). Bottom: χ = 1, corresponding to a reso- factors:
lution R = 0.71. This is the typical value for a semi-central
Au-Aucollision at RHICanalyzed by theSTAR TPC [6]. v {LYZ}=− i 2π dθ eirθQθf eirθQθnf cos(n(φ−θ) .
n
C 0 2π
Z D ED (17E)
events. These negative weights are required in order to Wethenuseagaintheassumptionthattheflowandnon-
subtractnonfloweffects,aswillbeexplainedinSec.IID. flow part are uncorrelated to write
Ontheotherhand,theyalsosubtractpartoftheflow. In
ordertocompensateforthiseffect,theglobalnormaliza- eirθQθ = eirθQθf eirθQθnf . (18)
tion of the weight increases when χ decreases (compare
D E D ED E
the curves in the top and bottom panels in Fig. 1). This Now, eirθQθ =0bydefinitionoftheLee-Yangzerorθ,
qualitatively explains the χ dependence in Eq. (12).
uptoDstatisticEalfluctuations. Thismeansthatoneofthe
The weight (9) vanishes linearly at Q = 0. This is
two factors in the right-hand side of Eq. (18) vanishes.
physically intuitive. The idea of the flow vector is that
Sinceitistheflowwhichproducesthezero,itmeansthat
by summing over all particles, one increases the relative
weight of collective flow over individual, random motion eirθQθf =0. Inserting into Eq. (17), we find
of the particles. If the flow vector is small in an event, D E
it means that the random motion hides the collective vn{LYZ}=0, (19)
motioninthisparticularevent,whichisthereforeoflittle
up to statistical fluctuations. This completes our proof
use for the flow analysis.
that the method is not biased by autocorrelations and
nonflow effects.
D. Autocorrelations and nonflow effects
III. SIMULATIONS
We now explain why the method is insensitive to au-
tocorrelations and nonflow effects. We consider the situ- N =28000eventsweresimulatedwithaMonte-Carlo
ationwherethereis collectiveflowinthesystem,sothat program dubbed GeVSim[20]. In GeVSim the v2 and
4
the particle yield as function of transverse momentum
and pseudorapidity can be set with a user-defined pa-
rameterization. For these simulations events were gen- )
pt c = 1
erated using a linear dependence of v2(pt) in the range (2 0.3 v
v 2
0–2 GeV/c, above 2 GeV/c the v2(pt) was set constant. 0.25 v2{2}
The average elliptic flow is hv2i = 0.0625. We then re- v2{4}
constructed v2(pt) from the simulated events using sev- 0.2 v2{LYZ}
eral methods: the Lee-Yang-zeroes method described in 0.15
AppendixA,2-and4-particlecumulants[11]. Thecorre-
0.1
sponding estimates ofv2 aredenoted by v2{LYZ},v2{2}
0.05
and v2{4}, respectively. v2{2} is generally close to v2
fromthetraditionalevent-planemethod;botharebiased 0
)
bynonfloweffects. Onthe otherhand,v2{4}isexpected pt c =1 and two particle nonflow
to be close to v2{LYZ}, with the bias from nonflow ef- (v2 0.3 v2
v{2}
fects subtracted. The weight w in Eq. (4) was chosen 0.25 2
j v{4}
2
identically1/M forallparticles,withM theeventmulti- 0.2 v2{LYZ}
plicity, so that the integrated flow V defined by Eq. (1)
n 0.15
coincideswiththeaverageelliptic flow,i.e.,V =0.0625.
n
The analysis was repeated twice by varying the multi- 0.1
plicity M used in the flow analysis: the values 256 and 0.05
576 were used, so as to achieve a resolution of χ=1 and
0
1.5. [32]
0 0.5 1 1.5 2 2.5
Fig.2showsthegenerated(input)v2(pt)togetherwith pt [GeV/c]
the reconstructed v2(pt) using cumulants and Lee-Yang
zeroes for χ = 1. The upper panel shows the results in
the case where all correlations are due to flow. In this FIG. 2: Differential elliptic flow v2(pt) reconstructed using
different methods: the line is the input v2. The upper, lower
case, all three methods yield the correct v2(pt) and hv2i panel shows the obtained v2(pt) for the different methods
within statistical uncertainties (see Table I), which are
without and with nonflow present,respectively.
twice larger for v2{4} and v2{LYZ} than for v2{2} (see
Sec. A3).
In the lower panel, simulations are shown which in-
TABLEI:Valueoftheaverageellipticflow v2 reconstructed
clude nonflow effects. Nonflow correlations are intro- h i
using different methods with and without nonflow effects in
duced by using each input track twice, roughly imitat- thesimulated data. The input valueis v2 =0.0625.
ing the effect of resonance decays or track splitting in h i
a detector. Experiments at RHIC have shown [6] that
Method Flow only Flow+nonflow
nonfloweffectsarelargerathigh-p (probablyduetojet-
t v2 2 0.0626 0.0003 0.0764 0.0004
like correlations), and a realistic simulation of nonflow { } ± ±
v2 4 0.0624 0.0005 0.0627 0.0007
effects should take into accountthis p dependence. Our { } ± ±
t v2 LYZ 0.0626 0.0005 0.0629 0.0007
simplified implementation, which does not, is not realis- { } ± ±
tic. It is merely an illustration of the impact of nonflow
effects on the flow analysis. Fig. 2 shows that due to
nonflow effects, the method based ontwo-particle cumu-
tions [21, 22]. The method presented in this paper
lants(v2{2})overestimatesthe averageelliptic flowhv2i. is an improved event-plane method, which eliminates
The error on the average elliptic flow is larger than 20%
the first source of uncertainty, nonflow effects. It has
(see Table I, right column). The transverse-momentum
been recently argued [23] that cumulants (and there-
dependence of v2(pt) is also not correct, with an excess fore Lee-Yang zeroes, which corresponds to the limit of
atlowp by0.03. Bycontrast,theresultsfrom4-particle
t large-ordercumulants) alsoeliminate eccentricity fluctu-
cumulants (v2{4}) and Lee-Yang zeroes (v2{LYZ}) are, ations [21, 22]. However, a recent detailed study [24]
within statistical uncertainties, in agreement with the
showsthatevenwith cumulants,there mayremainlarge
true generated flow distribution. This shows that the
effects of fluctuations in central collisions and/or small
method presented in this paper is able to get rid of non-
systems. Thisissuedeservesmoredetailedinvestigations.
flow effects.
Letting aside the question of fluctuations, we now dis-
cuss which method of flow analysis should be used, de-
pending on the situation. There are three main classes
IV. DISCUSSION
of methods: the standard event-plane method [4], four-
particlecumulants[11],andtheLee-Yang-zeroesmethod
Two effects limit the accuracy of flow analyses at presented in this paper. When the standard event-plane
high energy: nonflow effects and eccentricity fluctua- method is used, nonflow effects and eccentricity fluctua-
5
tionsaregenerallythemainsourcesofuncertaintyonv , from the event plane; in addition, strong nonflow corre-
n
andtheydominateoverstatisticalerrors. Themagnitude lations are expected within a jet, which would bias the
ofthisuncertaintyisatleast10%atRHICinsemi-central analysis.
collisions;it is largerfor morecentralor moreperipheral
collisions,andalsolargerathighp . Unlessstatisticaler-
t
rorsareofcomparablemagnitudeaserrorsfromnonflow Acknowledgments
effects,cumulantsorLee-Yangzeroesshouldbepreferred
over the standard method. JYO thanks Yiota Foka for a discussion which moti-
The advantages of Lee-Yang zeroes over 4-particle cu- vated this work.
mulants are: 1) They are easier to implement. 2) They
further reduce the error from nonflow effects. 3) Effects
of azimuthal asymmetries in the detector acceptance are APPENDIX A: PRACTICAL IMPLEMENTATION
much smaller. 4) The statistical error is slightly smaller
if the resolution parameter χ > 1. For χ = 0.8, the er- Beforewe describe the implementationof the method,
ror is only 35% larger with Lee-Yang zeroes than with letusmentionthatthereareinfacttwoLee-Yang-zeroes
4-particle cumulants (and 4 times larger than with the methods, depending on how the generating function is
event-plane method). defined: the “sum generating function” makes explicit
Our recommendation is that Lee-Yang zeroes should useofthe flowvector[18],whilethe“productgenerating
be used as soon as χ > 0.8. For small values of χ, typi- function” [30] is constructed using the azimuthal angles
cally χ < 0.6, statistical errors on Lee-Yang zeroes blow of individual particles, and cannot be expressed simply
up exponentially, which rules out the method; the sta- intermsofthe flowvector. Cumulantsalsoexistinboth
tistical error on 4-particle cumulants also increases but versions, the “sum” [17] and the “product” [11]. For
more mildly, and their validity extends down to lower Lee-Yang zeroes, both the sum and the product give es-
values of the resolution if very large event statistics is sentiallythesameresultforthelowestharmonic[15]: the
available. differencebetweenresultsfromthetwomethodsissignif-
A limitation of the present method is that it does not icantlylessthanthestatisticalerror. Ontheotherhand,
apply to mixed harmonics: this means that it cannot the product generating function is significantly better
be used to measure v1 and v4 at RHIC and LHC using than the sum generating function if one analyzes v4 or
the event plane from elliptic flow [25]. Note that v1 can v1 [26] using mixed harmonics. The method described
in principle be measured using Lee-Yang zeroes [26] us- below is strictly equivalent to the sum generating func-
ing the “product” generating function, but this method tion,althoughexpressedindifferentterms. Ontheother
cannot be recast in the form of an improved event-plane hand, the product generating function cannot be recast
method. Higher harmonics such as v4 also have a sen- inaformsimilartotheevent-planemethod,andwillnot
sitivity to autocorrelations and nonflow effects, which be used here.
is significantly reduced by using the product generating The method a priori requires two passes through the
function [11]. data, which are described in Sec. A1 and Sec. A2.
Although we have only explained how to analyze the
anisotropic flow of individual particles, it is straightfor-
ward to extend the method to azimuthally dependent 1. First pass: locating the zeroes
correlations [27, 28]. The only complication is that the
azimuthaldistributionofparticlepairsgenerallyinvolves Aswithotherflowanalyses,onemustfirstselectevents
sine terms[29],inadditiontothecosine termsofEq.(1). in some centrality class. The whole procedure described
These terms are simply obtained by replacing cos with below must be carried out independently for each cen-
sin in Eq. (3). trality class.
In conclusion, we have presented an improved event- The flow vector (Q ,Q ) is defined by Eq. (4). In
x y
plane method for the flow analysis, which automatically contrastto the standardevent-planemethod, no flatten-
corrects for autocorrelations and nonflow effects. As in ing procedure is required to make the distribution of Q
the standardmethod, eacheventhasits event plane Ψ , isotropic. Corrections for azimuthal anisotropies in the
R
anestimateofthereactionplane,whichisthesameasfor acceptance, which do not vary significantly in the event
the standard method, except for technical details in the sample used, are handled using the procedure described
practicalimplementation. Thetrickwhichremovesauto- below. We do not define the event plane Ψ as the az-
R
correlationsandnonfloweffectsisthatthereisinaddition imuthal angle of the flow vector, as in Eq. (4). The pro-
anevent weight. Anisotropicflowv isthenestimatedas cedurebelowdefinesboththeeventweightandtheevent
n
aweighted averageofcos(n(φ−Ψ )). Astraightforward plane.
R
application of this method would be to measure jet pro- The analysis uses the projection of the flow vector
ductionwithrespecttothe reactionplane atLHC.With onto an arbitrary direction, see Eq. (5). In practice,
the traditionalevent-planemethod, sucha measurement the first pass should be repeated for several equally-
would require to subtract particles belonging to the jet spaced values of nθ between 0 and π. This reduces
6
the statistical error, 5 values of θ suffice, see Eq. A5.
For elliptic flow, for instance, θ takes the values θ =
s
0,π/10,2π/10,3π/10,4π/10. nt104 V = 0.0625
u n
One first computes the modulus |Gθ(r)|, with Gθ de- o c = 1.5
c c = 1.
fined by Eq. (7), as a function of r for positive r. One
determines numerically the first minimum of this func-
tion. This is the Lee-Yang zero. We denote its value by 103
rθ. It must be stored for each θ.
0 0.5 1 1.5 2 2.5 3
2. Second pass: determining the event weight, wR, Y -Y [rad]
EP LYZ
and the event plane, ΨR.
FIG. 3: Distribution of the relative angle between the event
In the second pass, one computes and stores, for each planeΨRdefinedbyEq.(A2),withWR >0,andthestandard
θ, the following complex number: eventplane, for thereconstruction shown in Fig. 2.
Dθ ≡ 1 rθQθeirθQθ, (A1)
one computes the following quantity:
j01Nevts
events
X std std
S ≡W cosnΨ cosnΨ +W sinnΨ sinnΨ ,
R R R R R R
where j01 ≃ 2.40483. Except for statistical fluctuations (A3)
and asymmetries in the detector acceptance, Dθ should where W cosnΨ and W sinnΨ are defined by
R R R R
be purely imaginary. Eq. (A2). The sign of W is then chosen as the sign
R
For each event, the event weight and the event plane ofS,whichensuresthatnΨ −nΨstd lies between−π/2
R R
are defined by and π/2.
The proceduredescribedinthis Appendix differs from
eirθQθ the procedure described in Sec. II only in the case of
W cosnΨ ≡ Re cosnθ
R R Dθ non-uniform acceptance. This agreement can be seen
* ! +
θ in Fig. 1, which displays a comparison between the
eirθQθ
two. The solid line corresponds to the weight defined
W sinnΨ ≡ Re sinnθ , (A2)
R R * Dθ ! + inSec.II(Eqs.(9)and(12)),whilethestarscorresponds
θ
to the weight defined by Eq. (A2), as implemented in
where Re denotes the real part, and angular brackets the Monte-Carlo simulation presented in Sec. III. The
denoteaveragesoverthevaluesofθ definedinsubsection agreementisverygood. Thisagreementcanalsobeseen
A1. Our estimate of v , denoted by v {LYZ}, is then directly on the equations. If the detector has perfect
n n
defined by Eq. (3). azimuthal symmetry, rθ and Dθ in Eq. (A2) are inde-
We nowdiscusshowthe angleΨ definedbyEq.(A2) pendent of θ, up to statistical fluctuations. Neglecting
R
compares with the event-plane from the standard analy- these fluctuations, replacing Qθ with Eq. (5) and inte-
sis. First,wenotethatEqs.(A2)uniquelydeterminethe grating over θ, one easily recovers Eq. (9). If there are
angle nΨ (modulo 2π) only if the signof W is known. azimuthal asymmetries in the detector acceptance, on
R R
The simplest convention is W > 0. In the simpli- the other hand, they are automatically taken care of by
R
fied implementation described in Sec. II, however,where Eq.(A2). The factthatonefirstprojectsthe flowvector
Ψ coincideswith the standardeventplane, W defined onto a fixed direction θ is essential (for a related discus-
R R
by Eq. (9) can be negative, because the Bessel function sion, see [31]).
changes sign (see Fig. 1). The convention W > 0 then
R
leads to a value of nΨ which differs from the standard
R
eventplanebyπ,sincechangingthesignofW amounts 3. Statistical errors
R
to shifting nΨ by π in Eqs. (A2). This is illustrated in
R
Fig. 3, which shows the distribution of the relative angle Thestatisticalerrorstronglydependsontheresolution
between ΨR and the standardeventplane in the simula- parameter [9] χ, which is closely related to the reaction
tion of v2 at LHC describedin Sec.III. The distribution plane resolution in the event-plane analysis. It is given
has two sharp peaks at 0 and π/2. The sign ambiguity by
producesthepeakatπ/2. Thewidthofthepeaksresults
V
from statistical fluctuations. The final result vn{LYZ}, χ= n . (A4)
givenby Eq. (3), does not depend onthe sign chosenfor Q2 +Q2 −hQ i2−hQ i2−V2
x y x y n
W .
R q
Ifonewishestohaveanevent-planeascloseaspossible In this equat(cid:10)ion, V is(cid:11)given by Eq. (8), averagedover θ
n
tothestandardeventplane,onemaychoosethefollowing to minimize the statistical dispersion. The average val-
convention. Denoting by Ψstd the standard event plane, ueshQ i,hQ i, Q2 and Q2 mustbecomputedinthe
R x y x y
(cid:10) (cid:11) (cid:10) (cid:11)
7
2
hfiQrstipvaasnsisthhrfoourgha tshyemdmaettar.icPdleetaescetonro:teththeyatahrQexaiccaenpd- × exp 2jχ012 cos kpπ J0 2j01sin k2πp
y
tance corrections. (cid:20) (cid:18) j2 (cid:18) kπ(cid:19)(cid:19) (cid:18) (cid:18) k(cid:19)π(cid:19)
01
The price to pay for the elimination of nonflow effects −exp −2χ2 cos p J0 2j01cos 2p (A5)
is an increased statistical error. This increase is very (cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21)
modest if χ is larger than 1: If χ=1.5, the error is only
25% larger than with the standard event-plane method.
′
If χ = 1, it is larger by a factor 2. If χ = 0.6, it is 20 whereN denotesthenumberofofobjectsonecorrelates
times larger. This prevents the application of Lee-Yang to the event plane, whatever they are (jets, individual
zeroes in practice for χ smaller than 0.6. particles), and p is the number of equally-spaced values
We now recall the formulas [12] which determine the of θ used in the analysis (see above). The larger p, the
statistical error δvstat on v {LYZ}: smaller the error. The recommended value is p = 5, be-
n n
cause larger values do not significantly reduce the error.
p−1 This equation shows that the statistical error diverges
1 kπ
stat 2
(δv ) = cos exponentially when χ is small.
n 4N′J1(j01)2p p
k=0 (cid:18) (cid:19)
X
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