Measurement of the distributions of event-by-event flow harmonics in Pb-Pb collisions at √ s

ATLAS-CONF-2012-114 14August2012

ATLAS NOTE

ATLAS-CONF-2012-114

August 12, 2012

Measurement of the distributions of event-by-event flow harmonics in Pb-Pb collisions at √ s

= 2.76 TeV

The ATLAS Collaboration

Abstract

The event-by-event distributions of harmonic flow coefficients

for n=2–4 are mea- sured in Pb-Pb collisions at

2.76 TeV, using charged particles with p

0.5 GeV and in the pseudorapidity range

2.5. These distributions are obtained by unfolding the raw

distributions with a response function that accounts for the smearing of the

due to finite number of tracks in the detector acceptance. The shape of the unfolded distribu- tions is found to be consistent with a 2-D Gaussian for the underlying flow vector in central collisions for

and over the full centrality range for

and

; When these distributions are rescaled to the same

, the resulting shapes are found to be nearly the same between

1 GeV and p

1 GeV. The

distributions are compared with the eccentricity dis- tributions from two initial geometry models: a Glauber model and a model that includes corrections to the initial geometry due to gluon saturation effects. Both models fail to de- scribe the experimental data consistently across the full centrality range. These results may shed light on the nature of the fluctuation of the created matter in the initial state as well as the subsequent hydrodynamic evolution.

c Copyright 2012 CERN for the benefit of the ATLAS Collaboration.

Reproduction of this article or parts of it is allowed as specified in the CC-BY-3.0 license.

1 Introduction

Heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) create hot, dense matter that is thought to be composed of strongly coupled quarks and gluons. One strik- ing observation that supports this picture is the large momentum anisotropy of particle emission in the transverse plane. This anisotropy is believed to be the result of anisotropic expansion of the created mat- ter driven by the pressure gradients, with more particles emitted in the direction of the largest gradients.

The collective expansion or flow of the matter can be modelled by relativistic viscous hydrodynamic theory. The magnitude of the azimuthal anisotropy is sensitive to transport properties of the matter, such as the ratio of the shear viscosity to the entropy density and the equation of state [1, 2].

The anisotropy of the particle distribution in azimuthal angle φ is described by a Fourier series:

dN

dφ ∝

1

2

v

cos n(φ

) (1)

where v

and

represent the magnitude and phase of the n

-order anisotropy (or flow) at corresponding angular scale (2π/n); they can also be conveniently represented by the per-particle “flow vector” [1]:

⇀

v

(v

cos nΦ

, v

sin nΦ

). The

is referred to as the event plane (EP). The experimentally measured event plane direction

can smear around

due to the finite number of particles in the event.

Due to the almond-like shape of the overlap region in non-central collisions, the anisotropy is dom- inated by the second harmonic term v

. However, significant first-order (n=1) and higher-order (n>2) coefficients have also been observed [3–5]. These coefficients are related to additional shape components arising from the fluctuations of the nucleon positions in the overlap region. The amplitudes ǫ

(or eccen- tricities) and the directions

of these shape components can be estimated via a simple Glauber model from the transverse positions (r, φ) of the participating nucleons relative to their center-of-mass [6]:

ǫ

phrⁿ

cos nφ

sin nφ

hrⁿi

, tan nΦ

sin nφ

hrⁿ

cos nφ

(2)

The large pressure gradients and ensuing hydrodynamic evolution can convert these shape deformations into higher-order azimuthal anisotropy in the momentum space. For small fluctuations, one expects v

ǫ

, and the

to be correlated with the minor-axis direction given by

π/n. However, model calculations show that the values of ǫ

are large, and the alignment between

and

π/n are strongly violated for n > 3 due to non-linear effects in the hydrodynamic evolution [7, 8].

Significant v

coefficients for n

6 have been measured at the LHC [3–5]. Strong correlations are also observed for many combinations of two or three

of different order [9]. These findings are consistent with the existence of sizable fluctuations in the initial state and strong non-linear effects in the final state [8, 10]. More information on these fluctuations and non-linear effects can be obtained by measuring the v

on an event-by-event (EbE) basis, which gives the probability distribution for each v

. This is the aim of the analysis presented here. Some properties of the v

distributions were previously inferred using the two- and four-particle cumulant methods, with the measured coefficients denoted as v

2

and v

4

[11, 12], respectively. In the limit of Gaussian distribution for the underlying flow vector, they are related to the mean and the standard deviation of the v

[11]:

v

2

v

σ

n

, v

4

v

σ

n

(3)

The relative width of the distribution, defined as σ

/

v

, was measured [13, 14] to be about 0.3-0.5 for

2 depending on centrality. However, these indirect measurements rely on the assumption that the flow vector follows a 2-D Gaussian distribution and hence are subject to significant systematic uncertainty.

In contrast, these and many additional quantities such as kurtosis and skewness can be obtained directly

from the actual v

distributions. The key step in this measurement is to unfold the observed v

distribution for the smearing due to the effect of finite per-event charged particle multiplicity. This measurement is possible in ATLAS, due to its large detector acceptance and the large particle multiplicities produced in lead-lead collisions at the LHC.

2 ATLAS detector and trigger

The ATLAS detector [15] provides nearly full solid angle coverage of the collision point with track- ing detectors, calorimeters and muon chambers, which are well suited for measurements of azimuthal anisotropies over a large pseudorapidity range

. This analysis primarily uses two subsystems: the inner detector (ID) and the forward calorimeter (FCal). The ID is contained within the 2 T field of a supercon- ducting solenoid magnet, and measures the trajectories of charged particles in the pseudorapidity range

|

η

< 2.5 and over the full azimuth. A charged particle passing through the ID traverses typically three modules of the silicon pixel detector (Pixel), four double-sided silicon strip modules of the semiconduc- tor tracker (SCT), and a transition radiation tracker for

η

< 2. The FCal consists of three longitudinal sampling layers and covers 3.2 <

η

< 4.9. In heavy ion collisions the FCal is used mainly to measure the event centrality and event planes [5, 16].

The minimum-bias Level-1 trigger used for this analysis requires signals in two zero-degree calorime- ters (ZDC), each positioned at 140 m from the collision point, detecting neutrons and photons with

η

>

8.3, or either one of the two minimum-bias trigger scintillator (MBTS) counters, covering 2.1 <

η

< 3.9 on each side of the nominal interaction point. The ZDC Level-1 trigger thresholds on each side are set below the peak corresponding to a single neutron. A Level-2 timing requirement based on signals from each side of the MBTS is imposed to remove beam backgrounds.

3 Event and track selections

This analysis is based on approximately 8 µb

of Pb-Pb data collected in 2010 at the LHC with a nucleon-nucleon center-of-mass energy

2.76 TeV. An offline event selection requires a recon- structed vertex and a time difference

< 3 ns between the MBTS trigger counters on either side of the interaction point to suppress non-collision backgrounds. A coincidence between the ZDCs at forward and backward pseudorapidity is required to reject a variety of background processes, while maintaining high efficiency for non-Coulomb processes. Events satisfying these conditions are required to have a reconstructed primary vertex within

< 150 mm of the nominal center of the ATLAS detector.

The Pb-Pb event centrality [17] is characterized using the total transverse energy (

ET

) deposited in the FCal over the pseudorapidity range 3.2 <

η

< 4.9 at the electromagnetic energy scale [18].

An analysis of this distribution after all trigger and event selections gives an estimate of the fraction of the sampled non-Coulomb inelastic cross-section to be 98

2%. The uncertainty associated with the centrality definition is evaluated by varying the effect of trigger and event selection inefficiencies as well as background rejection requirements in the most peripheral FCal

E_T

interval [17]. The FCal

distribution is divided into a set of 5% percentile bins, together with five 1% bins for the top 5% most central events. A standard Glauber Monte-Carlo analysis [17, 19] is used to estimate the average number of participating nucleons,

, for each centrality interval. These numbers are summarized in Table 1.

The v

coefficients are measured with the ID, using reconstructed tracks with p

> 0.5 GeV and

|

η

< 2.5. At least nine hits in the silicon detectors (out of a typical value of 11) are required for each

1ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the z-axis along the beam pipe. The x-axis points from the IP to the center of the LHC ring, and they-axis points upward. Cylindrical coordinates (r, φ) are used in the transverse plane,φbeing the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angleθasη=−ln tan(θ/2).

Centrality 0-1% 1-2% 2-3% 3-4% 4-5%

hN_parti

400.6

1.3 392.6

1.8 383.2

2.1 372.6

2.3 361.8

2.5

Centrality 0-5% 5-10% 10-15% 15-20% 20-25%

hN_parti

382.2

2.0 330.3

3.0 281.9

3.5 239.5

3.8 202.6

3.9

Centrality 25-30% 30-35% 35-40% 40-45% 45-50%

hNparti

170.2

4.0 141.7

3.9 116.8

3.8 95.0

3.7 76.1

3.5

Centrality 50-55% 55-60% 60-65% 65-70%

hNparti

59.9

3.3 46.1

3.0 34.7

2.7 25.4

2.3

Table 1: The relationship between centrality interval and the

estimated from the Glauber model.

track, with no missing Pixel hits and not more than one missing SCT hit, taking into account the effects of known dead modules. In addition, the point of closest approach of the track is required to be within 1 mm of the primary vertex in both the transverse and longitudinal directions [16]. The tracking efficiency is studied by comparing data to Monte Carlo calculations based on the HIJING event generator [20] and a full GEANT4 [21] simulation of the detector. The track reconstruction efficiency in central events is estimated to be about 70% near mid-rapidity and decreases to 50% at forward rapidity [22].

4 Methodology

To illustrate the level of EbE fluctuations in the data, the top panels of Figure 1 show the azimuthal distribution of charged tracks with p

> 0.5 GeV for three central events. The corresponding pair

distributions from the same events are shown in the bottom panels. For each pair of particles two

entries,

φ

φ

and

φ

φ

, are made each with a weight of 1/2, and then folded into [-0.5π,1.5π]

interval. Both distributions are overlaid with the acceptance functions (arbitrary normalization). Rich EbE patterns, beyond the structures seen in the acceptance functions (solid points), are observed. These distributions are the inputs for the EbE v

analyses.

The azimuthal distribution of charged particles in a given event can be expanded into a Fourier series:

dN

dφ ∝

1

2

v

cos n(φ

)

1

2

n=1

v

cos nφ

v

sin nφ)

(4) v

r

v

+

v

, tannΨ

v

v

, v

cos nφ

, v

sin nφ

, (5) where the averages are over all particles in the event for the required η range. The v

is the magnitude of the observed per-particle flow vector:

v

(v

, v

). In the limit of infinite multiplicity and the absence of non-flow effects, it should approach the true flow signal: v

v

. The key issue in measuring the EbE v

is to determine the response function p(v

v

), which is then used to unfold the smearing effect due to finite number of detected particles. Possible non-flow effects from short-range correlations, such as resonance decays, Bose-Einstein correlation and jets, also need to be suppressed.

The rest of this section sets out the steps to obtain the unfolded v

distribution. Section 4.1 explains

how the data are corrected for detector acceptance to obtain

v

, and then how the response function

is obtained from correlation of

v

between two symmetric subevents. Section 4.2 describes how v

can be obtained from two-particle correlation (2PC) as shown in lower panels of Figure 1, and how

the associated response function is obtained. In this analysis the 2PC approach is used primarily as a

consistency check (Section 5). The Bayesian unfolding procedure, applicable to either the single particle

or 2PC data, is described in Section 4.3. The performance of the unfolding and systematics follows in

Section 5.

4.1 Single particle method

In an actual experiment, the azimuthal distribution of particles in Figure 1 needs to be corrected for the non-uniform detector acceptance. This is achieved by dividing the foreground distribution (S) of a given event with the acceptance function (B) obtained from all tracks in all events (see top panels of Figure 1):

dN

dφ ∝C(φ)= S (φ)

B(φ) =

1

2

n=1

v

cos nφ

v

sin nφ) 1

2

n=1

v

cos nφ

v

sin nφ) . (6) v

P

i

(cos nφ

) /ǫ(η

,

)

i

1/ǫ(η

,

)

(cos nφ)/ǫ

h

1/ǫ

, v

(sin nφ)/ǫ

h

1/ǫ

, (7)

where ǫ(η,

) is the tracking efficiency [22] and v

and v

are Fourier coefficients of the acceptance function in azimuth, also weighted by the inverse of the tracking efficiency. Since the azimuthal variation of efficiency corrections are generally small (see Figure 1), the harmonic coefficients are obtained as:

v

v

v

, v

v

v

(8) Figure 2 shows the EbE distribution of per-particle flow vector

v

(v

, v

) and v

obtained for tracks at p

> 0.5 GeV in the 20-25% centrality interval. The azimuthal symmetry in the left panel reflects the random orientation of

v

event to event. Due to finite track multiplicity, the measured flow vector is smeared randomly around the true flow vector by a 2-D response function p(

v

v

).

φ

-2 0 2

Tracks

20 40 60 80

0-5%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

>0.5 GeV pT

Run 169927 Event 400484

φ

-2 0 2

Tracks

40 60

80 0-5%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

>0.5 GeV pT

Run 169927 Event 153906

φ

-2 0 2

Tracks

40 60

80 0-5%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

>0.5 GeV pT

Run 169927 Event 153938

φ

∆

0 2 4

Track Pairs

37000 38000 39000 40000

0-5%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

>0.5 GeV pT

Run 169927 Event 400484

φ

∆

0 2 4

Track Pairs

66000 67000 68000 69000

0-5%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

>0.5 GeV pT

Run 169927 Event 153906

φ

∆

0 2 4

Track Pairs

64500 65000 65500 66000 66500

0-5%

ATLAS Preliminary

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>0.5 GeV pT

Run 169927 Event 153938

Figure 1: Single track φ (top) and track pair

(bottom) distributions for three events (from left to right) in 0-5% centrality interval. The pair distributions have been folded into [-0.5π,1.5π]. The bars indicate the foreground distributions, the solid curves indicate a Fourier parameterization including first six harmonics: dN/dφ

2

i=1cn

cos n(φ

)) for single distributions and dN/d∆φ

2

i=1c_n

cos n(∆φ)) for pair distributions, and the red solid points indicate the acceptance functions.

Tracks with p

> 0.5 GeV are used.

obs

v2,x

-0.2 0 0.2

obs 2,yv

-0.2 0 0.2

0 100 200 300 20-25% ^ATLAS Preliminary

=2.76 TeV sNN

Pb-Pb

b-1

µ

int= 8 L

obs

v2

0 0.1 0.2 0.3

Events

1 10 102

103

104 ATLAS Preliminary

=2.76 TeV sNN

Pb-Pb

b-1

µ

int= 8 L

> 0.5 GeV pT

Figure 2: The distribution of per-particle flow vector

v

(left panel) and the magnitude v

(right panel) for events in 20-25% centrality interval.

)b obs

-(v2,x

)a obs

(v2,x

-0.2 0 0.2

b )obs 2,y-(va )obs 2,y(v

-0.2 0 0.2

0 500 1000 ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

b-1 µ int= 8 L

20-25%

)b obs

-(v2,x

)a obs

(v2,x

-0.2 0 0.2

Events

1 10 102

103

104

/DOF=1.05 χ2

=0.05039 δ2SE

20-25%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

b-1 µ int= 8 L

)b obs

-(v2,y

)a obs

(v2,y

-0.2 0 0.2

Events

1 10 102

103

104

/DOF=1.03 χ2

=0.05018 δ2SE

20-25%

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

b-1 µ int= 8 L

a) b) c)

> 0.5 GeV

T

p

Figure 3: (left) The distribution of the difference between the per-particle flow vectors of the two half- IDs for events in 20-25% centrality interval. (middle) The projection on to the x-axis overlaid with a fit to a Gaussian. (right) The projection on to the y-axis overlaid with a fit to a Gaussian. The width from the fit, δ

, and the quality of the fit, χ

/DOF are shown.

In order to determine p(

v

v

), the tracks in the ID is divided into two subevents with symmetric η range, η > 0 and η < 0, which have nearly the same average track multiplicity (agree within 1%).

The distribution of flow vector differences between the two subevents, p

(

v

)

(

v

)

, is then

obtained in Figure 3a. The physical flow signal cancels in this distribution such that it contains mainly

the effect of statistical smearing. Figure 3b-c show the x and y projections of this residual distribution,

together with fits to a Gaussian function. The fits describe the data across five orders of magnitude

with the same widths, δ

0.050, along both directions with χ

/DOF

1, implying that the smearing

effect is purely statistical. This would be the case if either non-flow effects are small and the smearing is

mostly driven by finite multiplicity, or the number of sources responsible for short-range correlations are

proportional to the multiplicity and they are not correlated between the subevents. The latter could be true

for resonance decays, Bose-Einstein correlation, and jets. Similar behavior is observed for all harmonics

up to centrality interval 50-55%. Beyond that the distributions are more appropriately described by the

Student’s t-distribution, which is a general PDF for the difference between two independent sample-

estimates of the mean. The t-distribution approaches a Gaussian distribution when the number of tracks

is large.

Denoting the width of these 1-D distributions as δ

, then the widths of the 2-D response functions for the half-ID and the full-ID should be δ

/

2 and δ

/2, respectively; they can be obtained by rescaling Figure 3a in both dimensions by a constant factor c: p(

v

v

)

ch

(

v

)

(

v

)

, with c

2 and

2 for the full-ID and half-ID, respectively. A simple shift of this 2-D distribution to

v

(v

, v

), followed by an integration over the azimuthal angle, gives the desired 1-D response function [11, 14, 23]:

p(v^obs_n |

v

)

v

(vobs n )2+v2

n 2δ2 I0

v

v

δ

!

, δ

(

δ

/

2 for half-ID δ

/2 for full-ID

(9) where I

is the modified Bessel function of the first kind and the difference s

v

v

is entirely due to the statistical smearing. This analytical function, commonly referred to as the Bessel-Gaussian func- tion [11], can be used to directly unfold the v

distribution, i.e. the right panel of Figure 2. Alternatively, the actual measured distribution p(v

v

) taking into account sample statistics, can be directly used in the unfolding. This approach is more suitable for peripheral collisions where the analytical description Eq. 9 is not precise.

4.2 Two-particle correlation method

The unfolding based on EbE two-particle correlation method starts with Eq. 6. An EbE pair distribution is obtained by convolving the tracks in the first half-ID with those in the second half-ID:

dN d∆φ ∝









1

2

n

v

cos nφ

v

sin nφ







⊗









1

2

n

v

cos nφ

v

sin nφ









=

1

2

n

h

v

v

v

v

cos n∆φ

v

v

v

v

sin n∆φ

≡

1

2

n

(A

cos n∆φ

sin n∆φ) (10)

where the tracking efficiency re-weighting has been applied for v

and v

for each track similar to Eq. 6. An EbE pair variable v

is subsequently calculated:

v

A²_n+B²_n= r

v

+

v

v

+

v

=

v

v

. (11) The observed flow signal from correlation analysis is defined as:

v

v

v

v

(v

)(v

) (12) where s

v

v

and s

v

v

are independent variables described by the probability distribution in Eq. 9 with δ

δ

/

2. The response function for v

is very different from that for the single particle, but nevertheless can be either calculated analytically via Eq. 9 or generated from the measured distributions in Figure 3. For small v

values, the s

term dominates Eq. 12 and the distribution of v

is broader than v

. For large v

values, s

and s

are approximately Gaussian and hence:

v

v

v

(s

)

v

2

v

/

2

v

(13)

where we have use the fact that the average of two Gaussian random variable is equivalent to a Gaussian with a narrower width, and that the smearing of flow vector for the half-IDs (s

and s

) is

2 broader than that for the full-ID (s). Hence the distribution of v

is expected to approach the v

distribution of the full-ID when v

δ

/

2.

4.3 Unfolding

In this analysis, the standard Bayesian unfolding procedure [24], as implemented in the RooUnfold framework [25], is used to calculate the unfolded distribution. The true distribution (“cause” c or v

) and the measured distribution (“effect” e or v

or v

) are related via the response function A, which gives the probability that a cause c

is mapped on to a measured value e

:

Ac=e, Aji= p(ej|ci

) (14)

The distribution of c is obtained via an iterative pseudo-inversion procedure: An unfolding matrix ˆ

is determined from the response function and some initial guess of the true distribution ˆc

(called a prior):

ˆc

ˆ

ˆ

ˆc

m,kAmiAjk

ˆc

(15)

The ˆ

is used to calculate the unfolded distribution ˆc

, which is then used to calculate ˆ

and so on.

More iterations reduce the dependence on the prior and gives results closer to the true distributions but with increased statistical fluctuations. So the number of iteration N

needs to be adjusted according to the sample statistics and binning. The prior can be chosen as the v

distribution from the full-ID for the single particle unfolding, and as the v

distribution from the two half-IDs for the 2PC unfolding.

However, more realistic priors can be obtained by rescaling the input distributions by a constant factor R to obtain a distribution with a mean that is closer to that for the true distribution:

v

,

v

v

1 1

q

1

σ

/

v

−

1

! f

,

0, 0.5, 1, 1.5, 2, 2.5 . (16)

where

v

and σ

n

are the mean and the RMS width of the v

distribution, respectively, and v

is measured using the FCal EP method from [5] with the same track selection criteria for the same dataset.

This rescaling is applied to v

on a EbE basis such that the bin size of the resulting distribution remains unchanged. The EP method in general is known to measure a value in between the simple average and the RMS average of the true v

[26, 27]:

h

v

v

h

v

h

v

σ

. (17)

The lower limit is reached when the resolution factor [5] used in the EP method is one, while the upper limit is reached when the resolution factor is close to zero. If the EbE

v

distribution is purely Gaussian, it can be shown that [28]:

q

h

v

2

√

π

v

1.13

v

, σ

v

r

4

π

1

0.523 (18)

Gaussian fluctuations are expected to dominate the v

in the most central collisions, and more generally for the higher-order harmonics. Thus f

0 (default choice) gives a prior that is slightly broader than the true distribution; f

1 gives a distribution that has a mean close to the true distribution; f > 1 typically gives a distribution that is narrower than the true distribution.

The unfolding problem in this analysis has several features that improves its performance:

1. The response function is obtained entirely from the data itself, with no need for Monte Carlo input.

2. Each event provides one entry for the v

distribution and the response function (no efficiency loss), and these distributions can be determined with great precision (about 2.4 million events for each 5% centrality interval).

3. The unfolded v

distribution is always narrower than the v

distribution, hence there is no leakage

problem.

5 Unfolding performance and systematic uncertainties

This section describes the unfolding of the single particle distribution (Eq. 4) and summarizes a series of checks to verify the robustness of the results: a) the number of iterations used, b) comparison with results obtained from a smaller η range, c) variation of the initial prior, d) comparison with the results obtained by unfolding the 2PC distribution, and e) checking for bias due to non-flow, short-range correlations by varing the η gap between the two half-IDs. Only a small subset of the available figures is presented here;

a wider selection can be found in the Appendix.

5.1 Unfolding performance

Figure 4 shows the convergence behavior of single particle unfolding for v

in the 20-25% centrality interval measured with the full-ID. The results converge to within a few percent by N

4 around the peak of the distribution, but the convergence is slower in the tails and there are small, systematic improvements at the level of a few percent for N

8. The refolded distributions, obtained by con- volving the unfolded distributions with the response function, indicate that the convergence is reached for N

8. Figures 5 and 6 shows similar distributions for v

and v

. The unfolding performances are similar to that shown in Figure 4, except that the tails of the unfolded distributions require more iterations to converge. For example, Figure 6 suggests that the bulk region of the distributions, both unfolded and refolded, have converged by N

32, but the tails still exhibit some small changes up to N

64.

The slower convergence for higher-order harmonics simply reflects the fact that the physical v

signal is smaller for larger n, while the values of δ

change only weakly with n. These studies are repeated for all centrality intervals. In general, more iterations are needed for peripheral collisions due to the increase of δ

.

v2

0 0.05 0.1 0.15 0.2

Events

103

104

>0.5 GeV 20-25% pT

f=0 as prior Input

itr=1 N

itr=2 N

itr=4 N

itr=8 N

itr=16 N

itr=32 N

itr=64 N

itr=128 N

ATLAS Preliminary

=2.76 TeV sNN

Pb-Pb b-1

µ

int= 8 L

v2

0 0.05 0.1 0.15 0.2

=128iterRatio to N

0.9 1 1.1

iter=128 Unfolded: ratio to N

obs

v2

0 0.1 0.2

Ratio to input

0.9 1 1.1

Refolded: ratio to input

Figure 4: The performance of unfolding of v

for the 20-25% centrality interval (left panel) for various

, and the ratios of the unfolded distributions to the results after 128 iterations (middle panel). The ratios of the refolded distributions to the input v

are shown in the right panel.

The statistical uncertainties given by the unfolding procedure are verified via a resampling tech- nique [29]. For small N

, the statistical uncertainties as given by diagonal entries of the covariance matrix are much smaller than

N, where N is the number of entries in each bin, indicating the presence

of statistical bias of the prior. But these uncertainties increase with N

, and generally approach

N

for 64

128. In this analysis, the centrality range for each harmonic n is chosen such that the convergence is reached for N

64, defined as the difference between N

64 and N

128 to be less than 5% in any bin when statistics allows. They are 0-70% for v

, 0-60% for v

and 0-45% for v

.

The robustness of the unfolding procedure is checked by comparing the results measured indepen-

dently for the half-ID and the full-ID. The results are shown in Figure 7. Despite the large differences

v3

0 0.05 0.1

Events

103

104

105

>0.5 GeV 20-25% pT

f=0 as prior

Input

itr=1 N

itr=2 N

itr=4 N

itr=8 N

itr=16 N

itr=32 N

itr=64 N

itr=128 N

ATLAS Preliminary

=2.76 TeV sNN

Pb-Pb b-1

µ

int= 8 L

v3

0 0.05 0.1

=128iterRatio to N

0.9 1 1.1

iter=128 Unfolded: ratio to N

obs

v3

0 0.05 0.1 0.15

Ratio to input

0.9 1 1.1

Refolded: ratio to input

Figure 5: The performance of unfolding of v

for the 20-25% centrality interval (left panel) for various

, and the ratios of the unfolded distributions to the results after 128 iterations (middle panel). The ratios of the refolded distributions to the input v

are shown in the right panel.

v4

0 0.02 0.04 0.06

Events

103

104

105

>0.5 GeV 20-25% pT

f=0 as prior

Input

itr=1 N

itr=2 N

itr=4 N

itr=8 N

itr=16 N

itr=32 N

itr=64 N

itr=128 N

ATLAS Preliminary

=2.76 TeV sNN

Pb-Pb b-1

µ

int= 8 L

v4

0 0.02 0.04 0.06

=128iterRatio to N

0.9 1 1.1

iter=128 Unfolded: ratio to N

obs

v4

0 0.05 0.1

Ratio to input

0.9 1 1.1

Refolded: ratio to input

Figure 6: The performance of unfolding of v

for the 20-25% centrality interval (left panel) for various

, and the ratios of the unfolded distributions to the results after 128 iterations (middle panel). The ratios of the refolded distributions to the input v

are shown in the right panel.

in their initial distributions, the final unfolded results agree to within a few percent in the bulk region of the unfolded distribution. Systematic differences are observed in the tails of the distributions for v

, especially in peripheral collisions, with the half-ID results being slightly broader. This behavior reflects mainly the residual non-convergence of the half-ID, since its response function is a factor of

2 broader than that for the full-ID.

A wide range of priors are tried in this analysis, including the measured v

distribution and the six rescaled distributions defined by Eq. 16. Figure 8 compares the convergence behavior of these priors for v

in the 20-25% centrality interval. Despite the significantly different initial distributions, the unfolded distributions all converge to the same answer for N

64 within their respective statistical errors.

However, the convergence is faster with an appropriate choice of priors. Furthermore, choosing a prior that is narrower than the unfolded distribution leads to convergence in one direction, and a broader prior converges from the other direction. Taken together, this allows a quantitative evaluation of the residual non-convergence in the unfolded distributions.

Figure 9 compares the convergence behavior between unfolding of single particle v

and unfolding

of 2PC v

in 20-25% centrality interval with f

0 prior. Despite very different response functions

and initial distributions, the unfolded results agree with each other to within a few percent in the bulk

region of the unfolded distribution. The systematic deviations in the tails of the v

distribution (bottom-

v

Events

0 20000 40000 60000

full-ID Input half-ID Input full-ID Unfolded half-ID Unfolded

f=0 as prior ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

b-1 µ int= 8 L iter=128 N

>0.5 GeV pT

20-25%

v2

0 v

50000 100000

20-25%

v3

0 v

50000 100000 150000 200000

20-25%

v4

v2

0 0.05 0.1 0.15 0.2

full-IDhalf-IDUnfolded ^0.9

1 1.1

v3

0 0.05 0.1

0.9 1 1.1

v4

0 0.02 0.04 0.06

0.9 1 1.1

Figure 7: Comparison of the input distributions (solid symbols) and unfolded distributions for N

128 (open symbols) between the half-ID and full-ID in 20-25% centrality interval. The ratios between the unfolded results for half-ID and full-ID are shown in the bottom panels. The results are shown for v

(left), v

(middle) and v

(right).

v

Events

102

103

104

105

Input f=0 f=0.5 f=1 f=1.5 f=2 f=2.5 Final result

Prior 20-25%

>0.5 GeV pT

v

iter=8 N

Single particle unfolding

ATLAS Preliminary

=2.76 TeV sNN Pb-Pb

b-1 µ int= 8 L

v

iter=32 N

v

iter=64 N

v

iter=128 N

v4

0 0.02 0.04 0.06

Ratio

0.9 1 1.1

v4

0 0.02 0.04 0.06

v4

0 0.02 0.04 0.06

v4

0 0.02 0.04 0.06

v4

0 0.02 0.04 0.06

Figure 8: Convergence behavior of v

in 20-25% central collisions for seven choices of priors for different

from left to right. The top panels shows the distributions after certain number of iterations and bottom panels show the ratio to the result for N

128. A common reference, shown by the solid lines in top panels, are calculated by averaging the results for f

0, 0.5 with N

128.

right panel) are mainly due to the remaining non-convergence in the 2PC method, which has a broader response function than the single particle method.

One important issue in EbE v

study is the extent to which the unfolded results are biased by non- flow short-range correlations, which may influence both the v

distributions and the response functions.

However, their influences are expected to be relatively small for low p

particles, which dominate the

measured v

signal; their contributions are also expected to be similar for the v

distributions and

response functions obtained from (

v

)

(

v

)

(Figure 3), and hence are partially cancelled out by

the unfolding procedure. This is supported by the consistency between the single particle and the 2PC

methods (Figure 9). Furthermore, both unfolding methods have been repeated requiring a minimum η

gap between the two subevents used to obtain the input distributions and the response functions. Six