Measurement of the correlation between elliptic flow and higher-order flow harmonics in lead–lead collisions at √

ATLAS-CONF-2014-022 20May2014

ATLAS NOTE

ATLAS-CONF-2014-22

May 18, 2014

Measurement of the correlation between elliptic flow and higher-order flow harmonics in lead–lead collisions at √

s

= 2.76 TeV

The ATLAS Collaboration

Abstract

Correlations between the elliptic flow coefficient,

, and higher-order flow harmonics,

,

and

are measured using 7

of Pb+Pb collision data at

s_NN =

2.76 TeV collected by the ATLAS experiment at the LHC. The

–v

correlations are measured as a function of centrality, and, for events within the same centrality interval, also as a function of event ellipticity. The results are compared to initial-state eccentricities calculated from initial geometry models. The

–v

correlations within a given centrality interval are very di

erent from the

–v

correlations as a function of centrality. For events within the same centrality interval,

is found to be anti-correlated with

and this anti-correlation is compatible with similar anti-correlations between the corresponding eccentricities

and

. On the other hand, the

and

are found to increase strongly with

. The trend and strength of the

–v

correlations for

4 and 5 are found to disagree with

–

correlations predicted by initial geometry models. Instead, these correlations are found to be consistent with the combined e

ects of a linear contribution to

and non-linear term that is a function of

or of

, as predicted by hydrodynamic models. A simple two-component fit is used to separate these two contributions. The magnitudes of the linear and non-linear contributions to the

–v

and

–v

correlations are found to be consistent with previously measured event-plane correlations.

c

Copyright 2014 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 and dense matter that is thought to be composed of strongly coupled quarks and gluons.

This initially produced matter is lumpy and asymmetric in its shape in the transverse plane [1, 2]. The matter expands under large pressure gradients, which transfer the inhomogenous initial condition into azimuthal anisotropy of produced particles in momentum space [3, 4]. Hydrodynamic models are used to understand the space-time evolution of the matter from the measured azimuthal anisotropy [5–7].

The success of these models in describing the anisotropy of particle production in heavy-ion collisions at RHIC and the LHC [8–14] places important constraints on the transport properties (such as shear viscosity to entropy density ratio) and initial conditions of the produced matter [15–20].

When describing the transverse expansion dynamics, it is convenient to parameterize the final state anisotropy of particle production in terms of a Fourier decomposition in azimuthal angle φ:

dN/dφ∝

1

2

∞

X

n=1

v

cos

) , (1)

where v

and

represent the magnitude and the phase (referred to as the event plane or EP) of the

-order harmonic flow. They are often represented as a two-dimensional vector, or in a complex form:

~v

(v

cos

, v

sin

)

v

. (2) The presence of harmonic flow has been related to various moments of shape configurations of the initially produced fireball. These moments are described by the eccentricity vector ~

calculated from the transverse positions (r, φ) of the participating nucleons relative to their centre of mass [4, 16]:

~

(

cos

,

sin

)

hrⁿi

, (3)

where

and angle

(also known as participant-plane (PP)) represent the magnitude and orientation of the eccentricity vector, respectively. The

denotes an average over the transverse position of all participating nucleons. The eccentricity vectors characterize the spatial anisotropy of the initial produced fireball. According to hydrodynamic model calculations, elliptic flow ~v

and triangular flow ~v

are the dominant harmonics, and they are driven mainly by the ellipticity vector ~

and triangularity vector ~

of the initially produced fireball [21, 22]:

v

, v

. (4) The origin of higher-order (n > 3) harmonics is more complicated; they arise from both ~

and final state non-linear mixing of lower-order harmonics [20, 22, 23]. For example, an analytical calculation shows that the v

signal comprises a term proportional to

(linear response term) and a leading non-linear term that is proportional to

[22, 24]:

v

+

...

= c₀e^i4Φ∗4 +c₁

v

+

... , (5)

where the second line of the equation is derived from Eq. 4, and

denotes the linear component

of v

and coe

cients

,

and

are weak functions of centrality. The non-linear contribution from v

is responsible for the strong centrality-dependent event-plane correlation between

and

observed

by the ATLAS Collaboration [14]. In the same manner, the v

signal comprises a linear component proportional to

and a leading non-linear term involving v

and v

[22, 24]:

v

...

= c₀e^i5Φ∗5+c₁

v

v

... (6) This decomposition of the v

signal explains the measured EP correlation involving

,

and

[14].

Another important feature of heavy ion collisions is that the number of participating nucleons is finite and their positions fluctuate randomly in the transverse plane, leading to strong event-by-event (EbyE) fluctuation of

and

. Calculations based on a Monte-Carlo (MC) Glauber model show that, even for events in a very narrow centrality interval,

can fluctuate from zero to several times its mean value, leading to a very broad probability density distribution

) [13]. These fluctuations also re- sult in non-trivial correlations between eccentricities and PP angles of different order characterized by

,

, ...,

,

, ...) [25, 26]. Consequently, such initial geometry fluctuations can influence the development of harmonic flow and lead to non-trivial correlations between final-state flow harmonics characterized by

, v

, ...,

,

, ...) [20, 21, 26]. Measuring these final-state correlations, and understanding them in terms of both fluctuations of initial-state geometry and final-state non-linear dy- namics is a central focus of current studies of collective flow in heavy ion collisions [7].

ATLAS has recently measured the EbyE distributions of the elliptic and triangular flow,

) and

), in lead–lead (Pb+Pb) collisions [13]. This measurement has shown that events within a narrow centrality interval can have very broad

) and

) distributions. If events with di

erent v

or v

values could be selected cleanly, one would be able to control the relative contributions of linear and non- linear terms to v

and v

in Eqs. 5 and 6, and hence separate these two contributions. An experimental method for selecting on event-shapes has been proposed by several groups [25, 27]. In this method, events in a narrow centrality interval are further classified according to the observed v

signal (m

2 and 3) in a forward rapidity range. This classification gives events with similar multiplicity but with very di

erent ellipticity or triangularity. The values of v

are then measured at mid-rapidity using the standard flow techniques. Since the

) distributions are very broad, the correlation between v

and v

can be explored over a wide v

range. The event-shape selection technique is sensitive to any differential correlation between

and

for events with the same centrality, which would otherwise be washed-out by averaging over events with di

erent initial configurations. One example is the strong anti-correlation between

and

predicted by the MC Glauber model [25, 28]. A recent transport model calculation shows that this correlation survives the collective expansion and appears as a similar anti-correlation between v

and v

[25].

In this note, the correlations between elliptic flow and higher-order flow harmonics are studied using the event-shape selection method. The ellipticity of the events is selected based on the observed v

signal in a forward pseudorapidity range 3.3 <

< 4.8

. The values of v

for

2–5 are then measured at mid-rapidity

< 2.5 using a two-particle correlation method. The extraction of v

in this analysis is identical to a previous ATLAS publication [11], which is also based on the same dataset. The only di

erence is that, in this analysis, the events are classified both by their centrality as in Ref. [11], as well as the v

observed at forward pseudorapidity. Most systematics are the shared by the two analyses.

1ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and thez-axis along the beam pipe. Thex-axis points from the IP to the centre 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).

2 ATLAS detector and trigger

The ATLAS detector [29] provides nearly-full solid angle coverage of the collision point with tracking detectors, calorimeters and muon chambers. All of these 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 full azimuth. A charged particle passing through the ID traverses typically three mod- ules of the silicon pixel detector (Pixel), four double-sided silicon strip modules of the semiconductor tracker (SCT), and a transition radiation tracker for

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

< 4.9. In heavy ion collisions, the FCal is used mainly to measure the event centrality and event planes [11, 30], though in this analysis it is also used to classify the events in terms of v

in the forward rapidity region, as described previously.

The minimum-bias Level-1 trigger used for this analysis requires signals in two zero-degree calorime- ters (ZDC) or either of the two minimum-bias trigger scintillator (MBTS) counters. The ZDCs are posi- tioned at 140 m from the collision point, detecting neutrons and photons with

> 8.3, and the MBTS covers 2.1 <

< 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 selection

This analysis is based on approximately 7 µb

of Pb

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

s_NN =

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 further required to have a reconstructed primary vertex within

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

The Pb+Pb event centrality [31] is characterized using the total transverse energy (

E_T

) deposited in the FCal over the pseudorapidity range 3.2 <

< 4.9 at the electromagnetic energy scale [32].

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 [31]. The FCal

distribution is divided into a set of 5% percentile bins. A centrality interval refers to a percentile range, starting at 0% relative to the most central collisions. Thus the 0–5% centrality interval corresponds to the most central 5% of the events. A MC Glauber analysis [31, 33] is used to estimate the average number of participating nucleons,

, for each centrality interval. These are summarized in Table 1. Following the tradition of heavy ion analysis, the centrality dependence of the results in this note is presented as a function of

.

The harmonic flow coe

cients, v

, are measured using tracks reconstructed from hits in the ID for

> 0.5 GeV and

< 2.5. At least nine hits in the silicon detectors (out of a typical value of 11) are required for each track, with no missing Pixel hits and not more than one missing SCT hit, taking into account the e

ects 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 [30].

The e

ciency, (

, η), of the track reconstruction and track selection cuts is evaluated using Pb

Pb

Monte Carlo events produced with the HIJING event generator [35]. The generated particles in each

Centrality 0–5% 5–10% 10–15% 15–20% 20–25% 25–30% 30–35%

N_part

382

2 330

3 282

4 240

4 203

4 170

4 142

4 Centrality 35–40% 40–45% 45–50% 50–55% 55–60% 60–65% 65–70%

N_part

117

4 95

4 76

4 60

3 46

3 35

3 25

2

Table 1: The list of centrality intervals and associated

values used in this analysis. The systematic uncertainties are taken from Ref. [34].

event are rotated in azimuthal angle according to the procedure described in Ref. [36] to give harmonic flow consistent with previous ATLAS measurements [11, 37]. The response of the detector is simulated using GEANT4 [38] and the resulting events are reconstructed with the same algorithms applied to the data. The absolute efficiency increases with

by 7% between 0.5 GeV and 0.8 GeV, and varies only weakly for

> 0.8 GeV. However, the e

ciency varies more strongly with η and event multiplicity [39].

For

> 0.8 GeV, it ranges from 72% at η

0 to 57% for

> 2 in peripheral collisions, while it ranges from 72% at η

0 to about 42% for

> 2 in central collisions.

4 Data Analysis

4.1 Event-shape selection

The ellipticity of the events is characterized with the so-called “flow vector” calculated from the trans- verse energy (E

) deposited in the FCal [14, 40]:

*q₂ =

(q

,

)

1

Σ

w

(Σ[w

cos 2φ

],

sin 2φ

])

(7) (8) where the weight w

is the

of the

tower in the FCal. Subtraction of the event-averaged centroid,

, in Eq. 7 removes biases due to detector effects [41]. The magnitude

and the phase

of

are calculated for each event as:

q₂= q

q²_x,2+q²_y,2

, tan 2

q_x,2

. (9)

The angle

is the observed event-plane, which is smeared around the true event plane,

, due to the finite number of particles in an event. A standard technique [42] is used to remove the small residual non- uniformities in the distribution of

. These procedures are identical to those used in several previous flow analyses [11, 13, 14, 41]. To reduce the detector nonuniformities at the edge of the FCal, only the FCal towers whose centroids fall within the interval 3.3 <

< 4.8 are used to calculate the flow vector.

The

defined above is insensitive to the energy scale in the calorimeter. In the limit of infinite

multiplicity it approaches the single-particle flow weighted by w:

(

v

)

w

(

v

)

/

w

. Hence

the EbyE

distribution is expected to follow closely the EbyE v

distribution, except that it is further

smeared due to the finite number of particles. Figure 1 shows the

distributions in two narrow centrality

intervals. These events are first divided into 10 intervals with equal number of events. Since the interval

at the higher (lower) end the distribution covers a much broader

range, it is further divided into four

(two) smaller intervals with equal statistics, resulting in a total of fourteen

intervals (two 5% intervals,

eight 10% intervals and four 2.5% intervals from low end to high end of the

distribution). These

fourteen

intervals are defined separately for each 1% centrality interval, and are then grouped together

into the wider centrality intervals used in this analysis (see Table 1). For example, the first

interval

for the 0–5% centrality interval is the sum of the first

interval in the five centrality intervals, 0–1%,

FCal q2

0 0.02 0.04 0.06

2/dqevtdNevt1/N

10-1

1 10 102

ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb Centrality 0-1%

95-100% 90-95% 80-90% 70-80% 60-70% 50-60% 40-50% 30-40% 20-30% 10-20% 7.5-10% 5-7.5% 2.5-5% 0-2.5%

FCal q2

0 0.05 0.1 0.15

2/dqevtdNevt1/N

10-2

10-1

1 10

ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb Centrality 25-26%

95-100% 90-95% 80-90% 70-80% 60-70% 50-60% 40-50% 30-40% 20-30% 10-20% 7.5-10% 5-7.5% 2.5-5% 0-2.5%

Figure 1: The distributions of the magnitude of the flow vector,

, calculated in the FCal via Eq. 9 in two narrow centrality ranges: 0-1% centrality (left) and 25-26% centrality (right). The vertical lines indicate the boundaries of the fourteen

ranges each containing a certain fraction of events as indicated on the plot.

1–2%,..., 4–5%. The default analysis are obtained in the fourteen non-overlapping

intervals as defined in Fig. 1. But for better precision, sometimes they are re-grouped into five wider

intervals each containing 20% of the statistics, or into two intervals each containing 50% of the statistics.

4.2 Two-particle correlations

The two-particle correlation analysis follows closely a previous ATLAS publication [11] where it is described in detail, so the analysis is only briefly summarized here. For a given event class, the two- particle correlation is measured as a function of relative azimuthal angle

φ

φ

φ

and relative pseudorapidity

η

η

η

. The labels a and b denote the two particles in the pair, which may be selected from di

erent

intervals. The two-particle correlation function is constructed as the ratio of distributions for same-event pairs (or foreground pairs

(

φ,

η)) and mixed-event pairs (or background pairs

φ,

η)):

C(∆

φ,

η)

(∆ φ,

η)

B(∆

φ,

η) . (10)

The mixed-event pair distribution is constructed from track pairs from two separate events with similar centrality and

, such that it properly accounts for the detector ine

ciencies and non-uniformity, but contains no physical correlations. In this analysis, charged particles measured by the ID with a pair acceptance extending up to

η|

5 are used.

This analysis mainly focuses on the shape of the correlation function in

φ. A set of 1-D

φ correla- tion functions is built from the ratio of the foreground distributions to the background distributions, both projected onto

φ.

C(∆

φ)

R S

(∆ φ,

η)d

η

R B(∆

φ,

η)d

η (11) The normalization is fixed by scaling the counts of the mixed-event pairs to be the same as same-event pairs for 2 <

η| < 5, which is then applied for all

η slices.

Figure 2 shows the 1-D correlation functions for 2 <

η| < 5 calculated in the low

region

(0.5 <

< 2 GeV) in three centrality intervals, 0–5%, 25–30% and 55–60%. For each centrality

interval, the correlation functions are also shown for events selected with largest

values and smallest

values. The magnitude of the modulation correlates strongly with the

value reflecting the fact that

[rad]

φ

0 2 ∆ 4

)φ∆C(

0.995 1 1.005

1.01 ATLAS Prelim.

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb

Centrality 0-5%

largest 10% q2

smallest 10% q2

all

<2 GeV

a,b

0.5<pT

|>2 η

∆

|

[rad]

φ

0 2 ∆ 4

)φ∆C(

0.95 1 1.05

ATLAS Prelim.

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb

Centrality 25-30%

largest 10% q2

smallest 10% q2

all

<2 GeV

a,b

0.5<pT

|>2 η

∆

|

[rad]

φ

0 2 ∆ 4

)φ∆C(

0.95 1 1.05

1.1ATLAS Prelim.

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb

Centrality 50-55%

largest 10% q2

smallest 10% q2

all

<2 GeV

a,b

0.5<pT

|>2 η

∆

|

Figure 2: The correlation functions

φ) in three centrality intervals for pairs with

η| > 2 and 0.5 <

< 2 GeV: 0–5% centrality (left), 25–30% centrality (middle) and 50–55% centrality (right).

For each centrality interval, the correlation functions for events with the largest 10% and smallest 10%

q₂

values are also shown. The scale on the y-axis are di

erent for the three panels. The statistical uncertainties are smaller than the symbols.

the global ellipticity can be controlled by

in the forward rapidity. In the 0–5% centrality interval, the correlation function for events with smallest

values shows a double-peak structure on the away-side (

φ

π). This structure reflects the dominant contribution of the v

when the v

signal is suppressed by the

selection. Similar double-peak strucures are also observed in ultra-central Pb+Pb collisions without event-shape selection [11, 43].

The 1-D correlation function in

φ is then expressed as a Fourier series:

C(∆

φ)

R C(∆

φ)d

φ 2π









1

2

n

v

(

,

) cos (n

φ)









(12)

The coe

cients are calculated directly from the correlation function as v

φ)i. The single- particle azimuthal anisotropy coe

cients, v

, are obtained via the factorization relation commonly used for collective flow in heavy ion collisions [11, 12, 44, 45]:

v

(

,

)

v

(

)v

(p

). (13) From this, v

is calculated as:

v

(

)

v

(p

,

)/

v

(p

,

) (14)

where the default transverse momentum range for reference particle,

, is chosen to be 0.5 <

<

2 GeV. The v

values obtained using this method measure, in effect, the root-mean-square (RMS) values of the EbyE v

[44]. A detailed test of the factorization behaviour has been carried out by comparing the v

(

) obtained for di

erent

ranges in Ref. [11, 12], and factorization holds well for

< 4 GeV.

4.3 Systematic uncertainties

Other than the classification of events according to

, the analysis procedure is nearly identical to the

previous ATLAS measurement [11] based on the same dataset. Most systematic uncertainties are the

same, and they are summarized briefly here.

The correlation function relies on the pair acceptance function to reproduce and cancel the detector acceptance e

ects in the foreground distribution. A natural way of quantifying the influence of detector effects on v

and v

is to express the single-particle and pair acceptance functions as Fourier series (as in Eq. 12), and measure the coefficients v

and v

. The resulting coefficients for pair acceptance, v

, are the product of two single-particle acceptances v

and v

. In general, the pair acceptance function in

φ is quite flat: the maximum variation from its average is observed to be less than 0.001, and the corresponding

values are found to be less than 1.5

10

. These v

e

ects are expected to cancel out to a large extent in the correlation function, and only a small fraction contributes to the uncertainties of the pair acceptance function. Three possible residual e

ects for v

are studied in Ref. [11]: 1) the time dependence of the pair acceptance, 2) the e

ect of imperfect centrality matching, and 3) the e

ect of imperfect

matching. In each case, the residual v

values are evaluated by a Fourier expansion of the ratio of the pair acceptances before and after the variation. The systematic uncertainty of the pair acceptance is the quadrature sum of these three estimates, which is δv

< 5

10

for 2 <

η| < 5.

This absolute uncertainty is propagated to the uncertainty in v

, and it is the dominant uncertainty when v

is small, e.g. for v

in central collisions. This uncertainty has been found to be uncorrelated with the

selection, and hence it is assumed not to cancel between di

erent

intervals.

The second type of systematic uncertainty includes the sensitivity of the analysis to track cuts and tracking efficiency, variation of v

between different running periods, and trigger and event selection.

Most systematic uncertainties cancel for the correlation function when dividing the foreground and background distributions. The estimated residual effects are summarized in Table 2. Most of these uncertainties are expected to be correlated between different

intervals.

Finally, due to the anisotropy of particle emission, the detector occupancy is expected to be signif- icantly larger in the direction of the event plane where the particle density is larger. This azimuthal angle dependence of the tracking efficiency may lead to a small angle-dependent efficiency variation, which may slightly reduce the measured v

coe

cients. The magnitude of such an occupancy-dependent variation in tracking efficiency has been evaluated using the HIJING simulation with flow imposed on the generated particles [13]. The reconstructed v

values are compared to the generated v

signal. The di

erences are taken as an estimate of the systematics. These di

erences are found to be 1% or less, and are included in Table 2. Since this effect is proportional to the flow signal, it is expected not to cancel between different

ranges.

v

v

v

v

-dependent

Track selection [%] 0.3 0.3 1.0 2.0 yes

Tracking e

ciency[%] 0.1-1.0 0.2-1.5 0.2-2.0 0.3-2.5 yes

Running periods[%] 0.3-1.0 0.7-2.1 1.2-3.1 2.3 no

Trigger [%] 0.5-1.0 0.5-1.0 0.5-1 1.0 yes

MC closure and occupancy e

ects[%] 1.0 1.5 2.0 3.5 yes

Sum of above[%] 1.2-2.0 1.8-3.2 2.6-4.4 4.7-5.4

Table 2: Relative systematic uncertainties for v

, in percent, on the measured v

due to tracking cuts,

tracking e

ciency, variation between di

erent running periods, centrality variation, consistency between

truth and reconstructed v

in HIJING simulation, and the quadrature sum of individual terms. Most of

these uncertainties are correlated between different

ranges.

2v

0 0.1 0.2 0.3

0-10% of q2

20-30% of q2

all q2

70-80% of q2

90-100% of q2

3v

0 0.05 0.1

ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb Centrality 20-30%

<2 GeV

ref

0.5<pT

|>2 η

∆

|

[GeV]

pT

5 10

Ratio

0.5 1 1.5

[GeV]

pT

2 4

Ratio

0.9 1 1.1

4v

0 0.02 0.04 0.06 0.08

ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb

5v

0 0.02 0.04

0.06 ^ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

Pb+Pb

[GeV]

pT

2 4

Ratio

0.6 0.8 1 1.2

[GeV]

pT

2 4

Ratio

0.6 0.8 1 1.2

Figure 3: The v

(

) in 20–30% centrality interval for

2 (top-left),

3 (top-right),

4 (bottom- left) and

5 (bottom-right). They are calculated for reference

of 0.5 <

< 2 GeV. In the top panel of each column, the v

(p

) are shown for events in the 0–10%, 20–30%, 70–80% and 90–100% of the

ranges (open symbols) as well as for inclusive events without

selection (solid symbols). The bottom panel of each column shows the ratios of the v

(

) for

-selected events to those obtained for all events. Only statistical uncertainties are shown.

5 Results

5.1 Fourier coe ffi cients, v

, and their correlations with q

Figure 3 shows the v

(

) for

2–5 extracted from a Fourier decomposition of the one-dimensional (1-D) correlation function in

φ, using the relation in Eq. 14, for events in the 20–30% centrality interval.

The results are observed to correlate strongly with the

selection. The v

values are largest for events selected with the largest

, and smallest for events selected with the smallest

, with a total change of more than a factor of two (see ratios in top-left panel). A similar dependence on the

selection is also seen for v

(

) and v

(

) (two bottom panels). In contrast to this, the extracted v

(

) values are anti-correlated with the

selection; the overall change in v

(

) is also significantly smaller (< 20%

across the

range). The relative changes of v

(p

) between different

ranges are nearly independent

2 FCal q

0 0.05 0.1

2v

0 0.05 0.1 0.15

ATLAS Preliminary

=2.76 TeV sNN

b-1

µ = 7 Lint

< 2 GeV 0.5 < pT

|>2 η

∆

|

Pb+Pb

0-5%

10-15%

25-30%

40-45%

2 FCal q

0 0.05 0.1

3v

0 0.02 0.04 0.06

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10-15%

25-30%

40-45%

2 FCal q

0 0.05 0.1

4v

0 0.01 0.02 0.03

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10-15%

25-30%

40-45%

2 FCal q

0 0.05 0.1

5v

0 0.005 0.01

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|>2 η

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0-5%

10-15%

25-30%

40-45%

Figure 4: The correlations between v

and

in four centrality intervals with

2–5, where v

is calculated in 0.5 <

< 2 GeV. Each panel shows results for one harmonic. Only statistical uncertainties are shown.

of

for

2 and 3, with a possible weak dependence for

4 and 5.

Figure 4 shows the correlation between v

at low

(0.5 <

< 2 GeV) and

for several centrality intervals. Since the relative change of v

to the

selection has a weak

dependence, this plot captures the essential features of the correlation between v

and

selection shown in Fig. 3. Due to the finite number of particles in a event, the measured

value fluctuates relative to the true value producing a weakening of the

–v

correlations. However, for each

interval, the elliptic flow can be measured at mid-rapidity and the mapping between FCal

and mean v

can be used to correct the measured

–v

correlations in Fig. 4 to proper v

–v

correlations. Studying the direct correlation between v

in two different

ranges, and the correlation between v

and higher-order flow harmonics are the main subject of this analysis. These correlations provide insights on fluctuations in the collision geometry in the initial state, as well as non-linear dynamics in the final-state collective expansion [21, 22, 25]. Results from these correlations are discussed in detail in the following sections.

5.2 Correlation between the high-p

v

and the low- p

v

Figure 5 shows the correlation of v

between two

ranges for various centrality intervals. The

axis represents v

values in 0.5–2 GeV range, while the y-axis represents v

values from a higher

range. Each data point corresponds to a 5% centrality interval within the overall centrality range of 0–

70%. Going from central collisions (left end of the data points) to the peripheral collisions (right end

of the data points), the v

first increases and then decreases along both axes, reflecting the characteristic

centrality dependence of v

, well known from pervious flow analyses [10,11]. The rate of decrease of v

is

}

< 2 GeV 0.5 < pT

2 { v

0 0.05 0.1 0.15

} < 3 GeV T2 < p {2v

0 0.1 0.2 0.3

ATLAS Preliminary

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Central

Peripheral

}

< 2 GeV 0.5 < pT

2 { v

0 0.05 0.1 0.15

} < 4 GeV T3 < p { 2v

0 0.1 0.2 0.3

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Central

Peripheral

}

< 2 GeV 0.5 < pT

2 { v

0 0.05 0.1 0.15

} < 6 GeV T4 < p {2v

0 0.1 0.2 0.3

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Central

Peripheral

}

< 2 GeV 0.5 < pT

2 { v

0 0.05 0.1 0.15

} < 8 GeV T6 < p { 2v

0 0.1 0.2 0.3

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Central

Peripheral

Figure 5: The correlation of the v

in 0.5 <

< 2 GeV (x-axis) with v

in four higher

ranges (y-axis), one for each panel. In each panel, the v

values are calculated in fourteen 5% centrality intervals in the centrality range 0–70% without

selection, and the data-points for the most central and most peripheral centrality intervals are indicated explicitly. The error bars and shaded boxes represents the statistical and systematic uncertainties, respectively. These uncertainties are often smaller than the symbol size.

larger at intermediate

(2 <

< 8 GeV), resulting in the boomerang-like structure in the correlation.

The stronger centrality dependence of v

at higher

is consistent with larger viscous-damping effects expected from hydrodynamic calculations [46].

The events in each centrality interval are then further sub-divided into

intervals, as described above. With this further sub-division each data point in Fig. 5 turns into a group of data points, which may follow a different correlation pattern. Figure 6 shows the v

correlation between 0.5–2 GeV and 3–

4 GeV for di

erent

event classes (results for other

ranges are similar). For clarity, the results for the

fourteen different centrality intervals are divided into two different panels, but in each panel the overall

centrality dependence prior to the

selection from Fig. 5 (the “boomerang”) is also shown. Unlike

the centrality dependence, the v

correlation within a given centrality interval follows approximately a

straight line pointing back very close to the origin. This linear relation is quantified by fitting the data to

a linear function in each centrality interval, and the resulting slopes and intercepts are shown in the two

panels of Fig. 7, together with similar results for other

ranges. The small non-zero intercepts can be

attributed to a residual centrality dependence of the v

correlation, due to the finite centrality interval used

for each

-dependence. This approximately linear correlation suggests that, once the event centrality

or the overall event multiplicity is fixed, the viscous-damping e

ects on v

changes very little with the

}

< 2 GeV

T

0.5 < p

2 { v

0 0.05 0.1 0.15

} < 4 GeV T3 < p {2v

0 0.1 0.2

0.3 ATLAS Preliminary

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|>2 η

∆

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selection Centrality 0-70%, no q2

Centrality intervals selection:

with q2

0-5%

10-15%

20-25%

30-35%

40-45%

50-55%

60-65%

}

< 2 GeV

T

0.5 < p

2 { v

0 0.05 0.1 0.15

} < 4 GeV T3 < p {2v

0 0.1 0.2

0.3 ATLAS Preliminary

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|>2 η

∆

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selection Centrality 0-70%, no q2

Centrality intervals selection:

with q2

5-10%

15-20%

25-30%

35-40%

45-50%

55-60%

65-70%

Figure 6: The correlation of the v

in 0.5–2 GeV (x-axis) with v

in 3–4 GeV range (y-axis) in various centrality intervals. The centrality intervals are staggered into the two panels for clarity. The data points in each centrality interval correspond to the fourteen default

intervals. The data are overlaid with the centrality dependence without

selection as shown in top-right panel of Fig. 5. The thin solid straight lines represent a linear fit of the data in each centrality, and error bars represent the statistical uncertainties.

Npart

0 100 200 300 400

Slope: K

1 1.5 2 2.5

3 ATLAS Preliminary

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a

0.5 < pT

<3 GeV

b

2 < pT

<4 GeV

b

3 < pT

<6 GeV

b

4 < pT

Npart

0 100 200 300 400

0Intercept: Y

-0.05 0

0.05 |∆η|>2 ATLAS Preliminary

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Pb+Pb

+ Y0 a} {pT

= K v2 b} {pT

Fit func.: v2

Figure 7: The slope and intercept of the linear fit of the v

correlations between low-p

range 0.5 <

< 2 GeV and four higher

ranges (see text). Only statistical uncertainties are shown.

Examples of the fits are shown in Fig. 6.

variation of the event ellipticity via

selection. The influence of viscous corrections on v

is mainly

controlled by the event centrality (or the overall system size).

2 v

0 0.05 0.1 0.15

3v

0 0.02 0.04

ATLAS Preliminary

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Central Peripheral

2 v

0 0.1 0.2 0.3

3v

0 0.05 0.1

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Centrality 0-70%

Central

Peripheral

2 v

0 0.1 0.2 0.3

3v

0 0.05 0.1

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Centrality 0-70%

Central

Peripheral

2 v

0 0.1 0.2 0.3

3v

0 0.05 0.1

ATLAS Preliminary

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Centrality 0-70%

Central

Peripheral

Figure 8: The correlation of v

(x-axis) with v

(y-axis) both measured in the same

for four

ranges, one for each panel. In each panel, the v

and v

values are calculated in fourteen 5% centrality intervals in the centrality range 0–70%. The data points corresponding to the most central and most peripheral centrality intervals are indicated explicitly. The error bars and shaded boxes represents the statistical and systematic uncertainties, respectively. In many cases, these uncertainties are smaller than the symbol size.

5.3 v

–v

correlation

Figure 8 shows the correlation between v

and v

measured in various fixed

ranges prior to the

selection. The boomerang-like structures in this case reflect mostly the fact that v

has a much weaker centrality dependence than v

[11]. However at higher

, the decrease of v

for peripheral collisions is much more significant than that for v

, and the turn-over point of the boomerang changes from 40–

50% centrality range at low

, and decreases to 30–40% centrality range at high

. This behaviour is consistent with stronger viscous-damping e

ects for v

than for v

for

< 3 GeV or for 0-50%

centrality interval, as predicted by hydrodynamic models [47, 48].

Figure 9 shows the v

–v

correlation in 0.5 <

< 2 GeV for different

event classes (coloured

symbols) overlaid with the centrality dependence taken from the top-left panel of Fig. 8 (thick solid

line). The correlation within a fixed centrality interval follows a very di

erent path than the centrality

dependence: the v

and v

are always anti-correlated with each other within a given centrality, whereas

they are positively correlated as a function of centrality. Since the v

and v

are driven by the initial

eccentricities: v

and v

, one may expect similar anti-correlation between

and

. Indeed,

recent calculations based on a multi-phase transport model [49] show that such anti-correlations exist