first measurements of event-by-event Elliptic Flow Fluctuations

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STARSTAR

Paul SorensenBrookhaven National Laboratoryfor the STAR Collaborationfor the STAR Collaboration

STAR

first measurements of event-by-eventfirst measurements of event-by-event

Elliptic Flow FluctuationsElliptic Flow Fluctuations



Elliptic Flow Fluctuations — Quark Matter 2006Elliptic Flow Fluctuations — Quark Matter 2006 2STARSTAR

outlineoutline

motivation for this studymotivation for this study

1) increase accuracy of 1) increase accuracy of vv22

2) gain sensitivity to initial2) gain sensitivity to initialconditions (CGC?)conditions (CGC?)

1 and 2 are needed for estimates 1 and 2 are needed for estimates of viscosityof viscosity

• introductionintroduction– vv22 uncertainties and uncertainties and eccentricity eccentricity partpart fluctuations

• analysis method:analysis method:– flow vector distributionflow vector distribution– sanity checkssanity checks

• results: results: the rms and mean (the rms and mean (vv22 and and vv22))




elliptic flowelliptic flow

€

v2 = cos 2 ϕ i, j −ψ j( )[ ]i j

true reaction plane unknown: only estimated

particle correlations unrelated to the reaction plane (non-flow) contribute to the observed v2

n-particle (n>2) correlations reduce the non-flow contributions

but methods also deviate if v2 fluctuates




initial eccentricityinitial eccentricity

monte-carlo models indicate large fluctuations of the initial eccentricity:

v2 fluctuations are also likely to be large

the major axis fluctuates to x’:part is relevant to the observed v2

R. Snellings & M. Miller; nucl-ex/0312008

S. Manly, nucl-ex/0510031; R. Bholerao, J.-Y. Ollitrault,

nucl-th/0607009;S. Voloshin, nucl-th/0606022

S. Manly (PHOBOS), nucl-ex/0510031




analysis method: flow vector analysis method: flow vector dist.dist.

q-vector and v2 related by definition: v2 = cos(2i)=q2,x/√M

sum over particles is a random-walk central-limit-theorem

width depends on• multiplicity: narrows due to failure of CLT at low M• non-flow: broadens gn = cos(n(i- j)) (2-part. correlations)• v2 fluctuations: broadens

J.-Y. Ollitrault nucl-ex/9711003; A.M. Poskanzer and S.A. Voloshin

nucl-ex/9805001

€

q x = Mv

simula

ted q

distrib

utio

n

j

j is observed angle for event j after summing over tracks i

qx

qy

€

qn,x =1

Mcos(nϕ i)

i=1

M

∑

qn,y =1

Msin(nϕ i)

i=1

M

∑

σ n,x2 =

1

2(1+ v2n − 2vn

2 + gn )

σ n,y2 =

1

2(1− v2n + gn )

€

qn,x =1

Mcos(nϕ i)

i=1

M

∑

qn,y =1

Msin(nϕ i)

i=1

M

∑

σ n,x2 =

1

2(1+ v2n − 2vn

2 + gn )

σ n,y2 =

1

2(1− v2n + gn )




analysis method: flow vector analysis method: flow vector lengthlength

€

1

q2

dN

dq2

=1

πσ Xσ Y

e−

1

2

q22 +Mv2

2

σ X2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ 1−

σ X2

σ Y2

⎛

⎝ ⎜

⎞

⎠ ⎟

nq2

v2 M

⎛

⎝ ⎜

⎞

⎠ ⎟

n1

n!Γ n +1 2( )In

q2v2 M

σ X2

⎛

⎝ ⎜

⎞

⎠ ⎟

n= 0,2,4,...

∞

∑

€

1

q2

dN

dq2

=1

π σ Xσ Y

e−

1

2

q22 +Mv2

2

σ X2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ 1−

σ X2

σ Y2

⎛

⎝ ⎜

⎞

⎠ ⎟

nq2

v2 M

⎛

⎝ ⎜

⎞

⎠ ⎟

n1

n!Γ n +1 2( )In

q2v2 M

σ X2

⎛

⎝ ⎜

⎞

⎠ ⎟

n= 0,2,4,...

∞

∑€

1

q

dN

dqd(ΔΦ)=

1

2πσ Xσ Y

e−

1

2

q cos2ΔΦ− M v2( )2

σ X2

+q 2 sin 2 2ΔΦ

σ Y2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

σ X2 =

1

2(1+ v4 − 2v2

2 + g2) and σ Y2 =

1

2(1− v4 + g2)

€

1

q

dN

dqd(ΔΦ)=

1

2πσ Xσ Y

e−

1

2

q cos2ΔΦ− M v2( )2

σ X2

+q 2 sin 2 2ΔΦ

σ Y2

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

σ X2 =

1

2(1+ v4 − 2v2

2 + g2) and σ Y2 =

1

2(1− v4 + g2)

experimentally x, y directions are unknown: integrate over all and study the length of the flow vector |qlength of the flow vector |q22||

from central limit theorem, q2 distribution is a 2-D Gaussian

fold various assumed v2 distributions (ƒ) with the q2 distribution

€

1

q2

d ˜ N

dq2

=1

q2

dv2

dN

dq2

f v2 − v2 ,δv2( )−∞

∞

∫

€

1

q2

d ˜ N

dq2

=1

q2

dv2

dN

dq2

f v2 − v2 ,δv2( )−∞

∞

∫

function now accounts for non-flow, non-flow, vv22, and fluctuations (, and fluctuations (vv22))

Ollitrault nucl-ex/9711003;Poskanzer & Voloshin

nucl-ex/9805001




analyzing the multiplicity analyzing the multiplicity dependencedependence

why isn’t it easy: non-flow (g2) broadens the distribution

fluctuations (v2) also broaden the distribution

how do we distinguish between v2 and g2:the width changes have a different dependence on the number of tracks: broadening from v2 scales with √M

analyze the same events sometimes use all tracks, sometimes fewer tracks

fit multiple distributions simultaneously: provides multiple provides multiple equations to solve for the 2 unknownsequations to solve for the 2 unknowns

v2 fixed at 0.0

dN

/dq

2

q2

g2 fixed at 0.0 all parameters free

2/ndf = 1760/651 2/ndf = 705/651 2/ndf = 670/650Au+Au 200 GeV20%-30% central

v2 = (6.05 ± 0.2)%v2 = (2.36 ± 0.3)%g2 = (4 ± 5)%

M=350, 250




systematic studiessystematic studies

Sanity checks:– fit to simulated data: do we get back what we put in? Test passed Test passed for for allall input values input values– fit data from combined centrality bins: does v2 increase as it should? Test passedTest passed– sensitivity to assumed shape of v2 distribution? we used a gaussian we used a gaussian and a function that describes the and a function that describes the partpart distribution: only small distribution: only small deviations in mean or rms <15%deviations in mean or rms <15%– fit using 2, 3, or 5 different multiplicities? some differences on range of allowed g2 values

q

simulated dN/dq



STARSTAR

STAR PreliminarySTAR Preliminary

centrality dependence of r.m.s. and mean ((vv22 and and vv22)) for 200 GeV

Au+Au collisions




width and meanwidth and mean

r.m.s. width of the v2 distribution is 36% of the meanwithin errors independent of centrality




consistency with other consistency with other methodsmethods

For discussion of fluctuations and cumulants see R. Snellings nucl-ex/0312008 and talk by S.

Voloshin

mean v2 lies between 2- and 4-particle cumulant v2






Voloshin

v2 from fit and from v2{2}2-v22 are consistent






Voloshin

v2{4} agrees with q-fit results: non-flow does not have to be invokeddifference between v2{2} and v2{4} apparently dominated by fluctuations




model comparisonsmodel comparisons

For v2/part scaling, see talks by A. Tang and S. Voloshin:

QM2006

v2/part scales with dN/dy1/3: fluctuation of this quantity consistent with datahydro with fluctuating initial conditions suggests sensitivity to EOS




model comparisonsmodel comparisons

standard eccentricity is excluded by data




summarysummary

• using the multiplicity dependence of the |q| distribution we’ve measured v2 fluctuations (v2)

• uncertainty on v2 reduced

v2/v2 (rms/mean) is approximately 36% independent of centrality– agreement with NeXSPheRIO using a QGP EOS– also consistent with phenomenological scaling

v2=0.034*part*(dN/dy)1/3

• new probe for early stages of HI collisions– explore pseudorapidity dependence (v2

1v22 - v2

1v21)

to reveal dynamics beyond geometry– explore sensitivity to CGC– explore sensitivity to EOS (sys. errors on theory?)– explore sensitivity to critical point fluctuations



STARSTAR

supporting materialsupporting material

fits to simulationsderivations

other?













glauber resultsglauber results

0%-5%

not gaussian for central or peripheral

for central it’s well described by:

v*exp[-0.5((v-v0)/v)2]

I use this shape in my fit function




fit resultsfit results

Fit results 0%-5%:

chisquare/ndf = 539.9/574 probability = 84% EDM=0.00159004 STRATEGY= 1 ERROR MATRIX ACCURATE EXT PARAMETER STEP FIRST NO. NAME VALUE ERROR SIZE DERIVATIVE 1 <v2> -4.88725e-08 2.65831e-04 2.65831e-04 -3.44983e+01 2 delta_v2 1.96408e-02 1.20468e-04 0.00000e+00 -6.21924e+00 3 g2 -6.09831e-03 3.23056e-03 -0.00000e+00 2.57024e+01 4 area 5.23554e+03 constant 5 v4 6.91200e-04 constant

new result previousr.m.s. = 0.0128 (0.0151)mean = 0.0246 (0.0235)median = 0.0231mode = 0.0196




vv22 distributions distributionsv2

dN

/dv

2

0%

-5%

5%

-10%

10%

-20%

Positive definite by construction

similar results with the largest deviations in the central bin as expected

dN/dv2 matches dN/dpart distribution

includes impact parameter fluctuations (we should remove that component)

new result 0%-5% previouschange

r.m.s. = 0.0128 0.0151 -15%mean = 0.0246 0.0235 +5%median = 0.0231mode = 0.0196








impact parameter impact parameter fluctuationsfluctuations

• part and v2 should be corrected for impact parameter fluctuations

• part correction comes directly from fixing b from glauber

• v2 corrected using v2corr =b*∂v2/∂b

Documents

first measurements of event-by-event Elliptic Flow Fluctuations