1. Elliptic Flow Fluctuations and Non-flow correlations
Paul Sorensen
Brookhaven National Laboratory
Thank you to the Tata Institute for Fundamental Research and the organizers

2. introduction motivation to measure v2 fluctuations:
ambiguity arises in calculations from uncertainty in initial conditions
perfect fluid conclusion depends on ambiguous comparison to ideal hydro
motivation to measure v2 fluctuations:
find an observable sensitive to initial conditions
analysis of the distribution of the length of the flow vector
non-flow 2 and v2 from the q-distribution
comparison to cumulants v{2}, v{4}
relationship to 2 particle correlations
v2 of events with a “ridge” and/or a “jet”!
See STAR Poster at QM08: Navneet Kumar Pruthi

3. flow vector distributionqx qy
simulated q distribution
j
j is observed angle for event j after summing over tracks i
J.-Y. Ollitrault nucl-ex/ ; A.M. Poskanzer and S.A. Voloshin nucl-ex/
q-vector and v2 related by definition: v2 = cos(2i) = q2,x/√M
sum over particles is a random-walk  central-limit-theorem
width depends on
non-flow: broadens n = cos(n(i- j)) (2-particle corr. nonflow)
v2 fluctuations: broadens

4. flow vector distribution
from central limit theorem, q2 distribution is a 2-D Gaussian
Ollitrault nucl-ex/ ;
Poskanzer & Voloshin nucl-ex/
x, y directions are unknown:  integrate over all  and study the length of the flow vector |q2|
fold v2 distributions ƒ(v2) with the q2 distribution to account for fluctuations v2
correction to QM06 analysis:
those results were an upper limit
dynamic width dominated by non-flow and/or fluctuations  not determined independently

5. See STAR Talk at QM08: Michael Daugherity
non-flow evident
STAR Preliminary
width depends on the track sample
differences are due to more or less non-flow in various samples
smaller 2 for like-sign (charge ordering)
larger 2 for small  (strong short range correlations)
also in 2-D correlations: can be fit with a  independent cos(2) term + non-flow structures
See STAR Talk at QM08: Michael Daugherity
STAR Preliminary

6. dynamic width from dN/dq fit
STAR Preliminary
the well constrained combinations of fit parameters are:
the dynamic width is the difference between the above equations
see Miller, Snellings, nucl-ex/
Caution! relationship of measured 2 from 2 particle correlations and dynamic width is not trivial: depends on ZYAM and 2-component model (see following slides)
includes systematic errors from comparisons to cumulants
minimum 2 derived from differences between subevents [q*q] - [q+*q-]

7. relationship to 2-particle corr.
two-component model and ZYAM*
v2 modulated background
non-flow component
*ZYAM: assumption that there’s zero yield at the minumum of the correlation function
above quantities are related to the width of the q-distribution

8. more about 2-particle correlations
simple case: v2=0.00
2 should be the width of the q-distribution
zyam b makes 2 too small
vary b until
2+2v2 matches q width

9. more about 2-particle correlations
add some fluctuations: v2=0.010
2 should be smaller
zyam b makes 2 too small
vary b until
2+2v2 matches q width

10. more about 2-particle correlations
add some fluctuations: v2=0.015
2 should be smaller
zyam b makes 2 too small
vary b until
2+2v2 matches q width

11. more about 2-particle correlations
add some fluctuations: v2=0.020
2 should be smaller
zyam b makes 2 too small
vary b until
2+2v2 matches q width

12. more about 2-particle correlations
add some fluctuations: v2=0.025
2 should be smaller
zyam b makes 2 too small
vary b until
2+2v2 matches q width

13. more about 2-particle correlations
add huge fluctuations: v2=0.045
2 should be smaller
zyam b makes 2 too small
vary b until
2+2v2 matches q width
v2{2} = 5.154e-02
v2{4} = 4.093e-02
<v2> = 6.083e-02
v2{2} = sqrt(<v2>^2 + sigv^2 + delta) = 5.202e-02
sigv/<v2> = 73.98%

14. more about 2-particle correlations
add huge fluctuations: v2=0.045
2 is now negative
huge v2
and negative 2
the subtracted yield

15. compared to correlation 2D fit
difference between 2-D fit and projection over all correlations
 introduces dependence on uncontrolled assumptions
!! words of caution about inter-analysis comparisons !!
be careful

16. with fit to autocorrelations
v2 and v2
consistency of this picture requires a contribution of non-poissonian fluctuations that breaks ZYAM
with fit to autocorrelations
STAR Preliminary

17. comparison to geometric 
result is not unique: only the width dyn2 = 2+2v22 is uniquely determined
multiple models may find consistency with the data
systematic uncertainties are still under investigation
STAR Preliminary

18. comparison to models upper limit using 2>0 challenges models
confined quark MC:
treats confined constituent quarks as the participants
decreases eccentricity fluctuations
color glass MC:
includes effects of saturation
increases the mean eccentricity
STAR Preliminary
comparison to hydro (NexSPheRio): Hama et.al. arXiv:
eccentricity fluctuations from CGC: Drescher, Nara. Phys.Rev.C76:041903,2007
extraction of Knudsen number: Vogel, Torrieri, Bleicher. nucl-th/
fluctuating initial conditions: Broniowski, Bozek, Rybczynski. Phys.Rev.C76:054905,2007
first disagreement with standard and use of quark MC: Miller, Snellings. nucl-ex/

19. can we eliminate v2 = 0?
is there any evidence for v2 changing event-to-event?
consider events containing two high-pT tracks (pT>2 GeV/c)
is the average v2(pT<2 GeV/c) still the same in this sample?
or when the high pT tracks are correlated at large ? or small ?
or when the tracks are uncorrelated?
large 
small 
STAR Preliminary
Central Au+Au 200 GeV
J. Putschke, QM2006
STAR Preliminary
dN/dq for low pT tracks vs  for the high pT leading hadrons
shape of the q-distribution for underlying event has non-trivial dependence on  and  of the leading and next-to-leading hadron

20. See STAR Poster: Navneet Kumar Pruthi
event classes?
characteristics of the events yielding a “ridge pair” appear to be very different from those yielding a “jet pair”
“jet”
See STAR Poster: Navneet Kumar Pruthi
“ridge”
STAR Preliminary
the “ridge” is calculated by projecting ||>0.7 correlation to ||<0.7
the “jet” is the remaining correlation at ||<0.7 after subtracting the “ridge”
inferred v2 for events associated with “ridge” pairs is large
inferred v2 for events associated with “jet” pairs is small
this conclusion is a direct consequence of the zero-yield at minimum assumption and the 3-component model: (v2 modulated background + ridge + jet)

21. event classes? possible interpretations:
events yielding a “ridge”-like pair have large v2
events yielding a “jet”-like pair have small v2
possible interpretations:
interactions of a jet with the medium and medium response to a jet (radial flow coupling to a jet C.Pruneau, nucl-ex/ )
is this evidence that initial state quantum fluctuations lead to
instabilities and growth of strong color fields M. Strickland, hep-ph/
large q^ and small /s Majumder, Muller, Bass. Phys.Rev.Lett.99:042301,2007
un-quenched jets can preferentially come from events fluctuating towards small q^ and large /s (small flow)?
strong fields lead to the ridge and large v2?
what about jets on the periphery? and tangential jets? momentum conservation effects?

22. summary
we find that the case of zero v2 fluctuations cannot be excluded with dN/dq without knowledge of non-flow, cluster flow, and non-poissonian multiplicity fluctuations
analysis places stringent constraints on 2, v2, and v2:
when one parameter is specified, the others are fixed
measurement challenges standard Monte-Carlo Glauber models:
upper limit is below standard nucleon MC Glauber
upper limit coincides with participating nucleon eccentricity fluctuations
nucleon MC Glauber: leaves no room for other sources of fluctuations & correlations
Is there any evidence that v2 fluctuates? Not from untriggered dN/dq but analysis of high pT triggered events seems to indicate non-zero v2.

23. the following is back-up material

24. mean and width of ƒ(v2)
assuming 2>0; analysis places an upper limit on flow fluctuations
STAR Preliminary
200 GeV Au+Au
STAR Preliminary

25. the statistical decomposition
decomposition into events that yielded:
uncorrelated high pT pairs
correlated and ridge-like high-pT pairs
correlated and jet-like high-pT pairs
total
uncorrelated background
correlated signal
each bin has a different signal to background ratio.
analyze the q-distribution of events contributing to each bin
algebraically solve for the q-distribution of signal and background separately

26. acceptance effects
about 4%
acceptance effects could mimic correlations unrelated to the event plane
easy to quantify using simulations run through a TPC filter
quantified systematics from CLT approximation

27. what’s going on 2 observed in two-component model depends on jet-flow
difference between simulated 2 and observed 2 depends on cluster flow
there are still more variables that can come into play
 highly probable that multiple solutions could match the data

28. comparison to 2 part. correlations
let’s see what happens if we take those values for 2

29. confined quark monte carlo
models of the initial conditions are not trivial

30. See STAR Poster: Navneet Kumar Pruthi
introduction
motivation for this study
use v2 fluctuations to try to access information about initial geometry: distinguish between models of initial conditions
reduce systematic errors on v2
results from 200 GeV Au+Au collisions
analysis procedures and change in QM06 conclusions
non-flow 2 and v2 from the q-distribution
comparison to cumulants v{2}, v{4}
v2 of events with a “ridge” and/or a “jet”!
See STAR Poster: Navneet Kumar Pruthi