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description
1July 16th-19th, 2007
McGill University
AM
July 16th-19th, 2007 McGill University, Montréal, Canada
July 2007 Early Time Dynamics Montreal
AM
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for the STAR Collaboration
2July 16th-19th, 2007
McGill University
AM
introductionmotivation for this study
• perfect fluid claims• data and model uncertainties
v2 fluctuations: possible access to initial geometry and reduction of data uncertainties
analysis strategy and correction to QM analysis
new results• non-flow (with comparisons to models and fits to autocorrelations measurements)• v2 and v2
• relatioinship to cumulants v{2}, v{4}, v{6}• v2/v2 (with model comparisons)
relationship to preliminary PHOBOS results
3July 16th-19th, 2007
McGill University
AM
perfect fluid
4July 16th-19th, 2007
McGill University
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why perfect?
ballistic expansion
zero mean-free-path zero mean-free-path limitlimit
STAR Preliminary
5July 16th-19th, 2007
McGill University
AM
why perfect?
small viscosity suggested by: 1) pretty good agreement with ideal hydro and
2) independence of v2 shape on system size
in a hydro model viscosity seems to reduce v2
but large v2 is observed in data
6July 16th-19th, 2007
McGill University
AM
why perfect?
Teaney QM2006
small viscosity suggested by: 1) pretty good agreement with ideal hydro and
2) independence of v2 shape on system size
7July 16th-19th, 2007
McGill University
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model and data uncertaintiestypically the real reaction plane is not detected
inter-particle correlations unrelated to the reaction plane (non-flow) can contribute to the observed v2
different methods will also deviate as a result of event-by-event v2 fluctuations.
ambiguity arises in model calculations ambiguity arises in model calculations from initial conditionsfrom initial conditions
perfect fluid conclusion depends on vperfect fluid conclusion depends on v22 measurement and ambiguous comparison measurement and ambiguous comparison to ideal hydroto ideal hydro
my motivation to measure v2 fluctuations: eliminate source of data uncertaintyfind observable sensitive to initial conditions
8July 16th-19th, 2007
McGill University
AM
flow vector distribution
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 n = cos(n(i- j)) (2-particle corr. nonflow)• v2 fluctuations: broadens
J.-Y. Ollitrault nucl-ex/9711003; A.M. Poskanzer and S.A. Voloshin
nucl-ex/9805001
€
q x = Mvsimulated q distribution
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 + Mδn )
σ n,y2 =
1
2(1− v2n + Mδn )
9July 16th-19th, 2007
McGill University
AM
flow vector distribution
€
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 + Mδ2) and σ Y2 =
1
2(1− v4 + Mδ2)
experimentally x, y directions are unknown: integrate over all and study the length of length of the flow vector |qthe flow vector |q22||
from central limit theorem, q2 distribution is a 2-D Gaussian
fold various assumed v2 distributions (ƒ) with the q2 distribution.
function accounts for non-flow non-flow , , vv22, and , and fluctuations fluctuations v2v2
€
1
q2
d ˜ N
d q2
=1
q2
dv2
dN
d q2
f v2 − v2 ,σ v2( )−∞
∞
∫
Ollitrault nucl-ex/9711003;Poskanzer & Voloshin
nucl-ex/9805001
note: QM results found with wrong multiplicity dependence for this term:
• forced this fit parameter to zero• forced v2 to it’s maximum value
that data therefore represents upper limit on v2 fluctuations: derived under the accidental approximation of minimal non-flow
€
2 = cos2 ϕ1 −ϕ 2( )nonflow
€
2 ≈1
21+ v4 − 2v2
2 + M δ2 + 2σ v2
2( )( )
10July 16th-19th, 2007
McGill University
AM
flow vector distribution
=-1 =0 =1
{/2} {full}
- --
+++- - -
{like-sign}
The width depends on how the track sample is selected. Differences are due to more or less non-flow in various samples:
• less for like-sign (charge ordering)• more for small (strong short range correlations)
€
2 ≈1
21+ v4 − 2v2
2 + M δ2 + 2σ v2
2( )( )
€
2 = cos2 ϕ1 −ϕ 2( )
STAR Preliminary
11July 16th-19th, 2007
McGill University
AM
non-flow term 2
=-1 =0 =1
{/2} {full}
- --
+++- - -
{like-sign}
differences in the width provide a lower limit on the amount of non-flow in the full eventthe total width provides an upper limit
€
2 = cos2 ϕ1 −ϕ 2( )
STAR Preliminary
€
2 ≈1
21+ v4 − 2v2
2 + M δ2 + 2σ v2
2( )( )
12July 16th-19th, 2007
McGill University
AM
non-flow term 2
€
2 ≈1
21+ v4 − 2v2
2 + M δ2 + 2σ v2
2( )( )
differences in the width provide a lower limit on the amount of non-flow in the full eventthe total width provides an upper limit
€
2 = cos2 ϕ1 −ϕ 2( )
STAR Preliminary
13July 16th-19th, 2007
McGill University
AM
v2 and v2
STAR Preliminary
range of allowed v2 values specifiedupper limit on v2 fluctuations given
14July 16th-19th, 2007
McGill University
AM
comparison to cumulant analysisInformation determined from analysis of cumulants
€
v 2{ }2
= v2
+ σ v2 + δ
v 4{ }2
≈ v2
−σ v2
width ≈ δ + 2σ v2 = v 2{ }
2− v 4{ }
2
centroid ≈ v2
−σ v2 ≈ v 4{ }
from fit to the q-distribution
only values on curves are allowed: all parameters are correlatedonce one is determined, the others are specified
15July 16th-19th, 2007
McGill University
AM
v2 and v2
STAR Preliminary
new level of precision being approachedstill significant fluctuations after including minijets from the autocorrelations
with fit to autocorrelations
16July 16th-19th, 2007
McGill University
AM
comparison to geometric
fluctuations from finite bin widths have not been removed yetlikely to reduce ratio below the model!
STAR Preliminary
17July 16th-19th, 2007
McGill University
AM
comparison to geometric
fluctuations from finite bin widths have not been removed yetlikely to reduce ratio below the model!
STAR Preliminary
systematic uncertainties are still large and under investigation
18July 16th-19th, 2007
McGill University
AM
relationship to PHOBOS results
this is essentially an acceptance corrected q-distribution
the underlying analysis turns out to be quite similar and susceptible to the same uncertainties i.e. the width of this distribution can be explained either by non-flow or fluctuations
PHOBOS STAR Preliminary
19July 16th-19th, 2007
McGill University
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conclusionsnew analysis finds that case of zero v2 fluctuations cannot be excluded using the q-vector distributions the non-flow term needs to be accurately determined (see T. Trainor)
analysis places stringent constraints on , v2, and v2: when one parameter is specified, the others are fixed presents a new challenge to models
measurement challenges standard Glauber models: upper limit coincides with participant eccentricity fluctuations accounting for correlations and finite bin widths will likely exclude most glauber models glauber leaves little room for other sources of fluctuations and correlations
CGC based Monte Carlo may leave room for other fluctuations and correlations
non-flow term and fluctuations may follow expected dependence of CGC: still well below hydro prediction (larger initial eccentricity)? can CGC+QGP+hadronic explain , v2, and v2?
20July 16th-19th, 2007
McGill University
AM
correction to previous analysis
but this should be (M-1)2
the difference: how does the fraction of tracks with a partner depend on subevent multiplicity
the consequences: since the multiplicity dependence of the non-flow term is the same as for fluctuations it becomes difficult to distinguish between the two
fraction of tracks with a partner = (n tracks from pair)/Mis a constant*(M-1)= 2 *(M-1)
2 = 0.00047
g2 = 0.109
21July 16th-19th, 2007
McGill University
AM
correlations and fluctuations