Sergei A. Voloshin Wayne State University, Detroit, Michigan

Winter Workshop on Nuclear Dynamics, San Diego, March 11-18, 2006

page 1 S.A. Voloshin

Sergei A. Voloshin

Wayne State University, Detroit, Michigan

Toward energy and system size dependence of anisotropic flow

Outline:

1. Flow fluctuations and non-flow:Lee-Yang Zeroes, Fourier Transform, Bessel transform, fitting q-distributions

2. Eccentricity fluctuations3. Compare to a model and to data…

Sorry, no new STAR results…



v2/eps at the time of QM2002/NA49 PRC

- uncertainty in the centrality definition- sqrt(s)=130 GeV data: 0.075 < pt < 2.0 GeV/c- sqrt(s)=200 GeV data: 0.15 < pt < 2.0; - the data scaled down by a factor of 1.06 to account for change in (raw) mean pt.- AGS and SPS – no low pt cut- STAR and SPS 160 – 4th order cumulants - no systematic errors indicated

What happened since then?- New data- New methods (e.g. LYZ)- Non-flow and flow fluctuations havebeen much better understood but the problemhas not been resolved… andno new plot yet (note that there is no such a plotin the STAR AuAu 200GeV PRC “flow” paper)

2

1 dN

Sv

dy 2 2S x y

Motivation for the plot:

2 2

2 2

y x

y x

2 cos 2 i RPv



v2{2}, v2{4}, non-flow and flow fluctuations

* 2 21 2 2

* 1/ 22 1 2

* * 4 2 21 2 3 4 2 2

1/ 4* 2 * *2 1 2 1 2 3 4

;

{2}

2 2 2

{4} 2

iu u v u e

v u u

u u u u v v

v u u u u u u

Correct if v is a constant in the event sample

Several reasons for v to fluctuate in a centrality bin:1) Variation in impact parameter in a centrality bin

(taken out in STAR results)2) Real flow fluctuations (due to fluctuations in

initial conditions or in system evolution)

2

* 22 2 2 22 1

2 2vv uu v vv v

22

22 2* * * 2 24 4

2 2 24 2 12

vv

v uu uuu u v vv

Different directions to resolve the problem:- Find method that have direct/different sensitivity to mean v - Estimate flow fluctuations by other means

2 equations, at least 3 unknowns:v, δ, σ

1/ 422 42 2 2; {4} / 2v v v

2 2{LYZ}v v

Subject of this talk

Flow fluctuations and q-distribution method

?



v2 from q-distributions

-- The results are very close to those from 4-particle correlation analysis.-- Difficult to trace the contribution of flow fluctuations.

STAR, PRC 66 (2002) 034904

2{2}v

2{ }distv q



Fourier transform of the distribution in flow vector componenets

Due to symmetry (no acceptanceeffects!) only real part is non-zero

cos( )RPv M shift

General strategy:Let x01 be the first root of equation J0(x0i)=0.x01~=2.045.

Then: v = k01/M, wherek01 is the first zero of f(k)



v2{LYZ} – flow from Lee Yang Zeroes

?

2 2{LYZ}v v How accurate is this statement?



Using Bessel transform

LYZ ==== Fourier transform of distribution in Qx, and/or Qy

== Bessel transform of the distribution in Q== Fitting of Q-distribution !?

/q Q M

k

0 /v k M

0k



Error calculation

.x qxfA.C("out/ds5_AuAu200.root",4)from qx: v=0.0622787+/-0.00047226from qy: v=0.0621035+/-0.000531313

root [6] .x qqfA.C("out/ds5_AuAu200.root",4) v=0.0628762+/-0.000210269

Error on k0 is shown in red.

Good agreement between results from FourierTransforms of qx and qy distribution, fitting q-distributionand Bessel transform of q-distribution.



Differential flow

From the above expression one can get differential flow in different ways. First way:

Alternatively:

?

2 2{LYZ}v v How accurate is this statement?



Simulations, pure flow

4M events, 400 particle in each event,Case 1 : 50% events with v=0.04 and 50% events with v=0.06Case 2: 100% with v=0.06Case 3: 100% with v=0.04

For the case 1, v{2} and v{4} as expected, e.g. v{2}=sqrt(0.04^2+0.06^2).v{BT} is significantly lower than 0.05, close to v{4} !!

Case #



Simulations, + non-flow

Similar to the previous case +”non-flow”: 300 “direct” particles and 100=50*2 - 50 pairs with the same azimuth.

As expected, only v{2} is strongly affected by non-flow.



What is wrong with BT/LYZ?

… Nothing really, just the first order approximation mentioned earlier is not good enough. In the graph on the left, the green line shows what one would need to get the correct mean value of v, compared to the black line, what one really gets by transform. One can also track it analyticallyby expanding Bessel function in the vicinity of the first zero.

Summary: BT/LYZ is only slightly better than v2{4} in terms of (in)sensitivity to fluctuations



UrQMD calculations

Fluctuations are too small to see?



Elliptic flow. Initial eccentricity.

Other similar/same quantities:Ollitrault: s

Heiselberg: Sorge: A2

Shuryak: s2

Elliptic flow must vanish if initially the system was created symmetric. Then, at small eccentricities, v2~

“e” -- initializationof energy density;“s” – initialization ofentropy density

2 2

2 2

y x

y x

Not important which one to use, but important to use the same!!!



Eccentricity in the optical Glauber model



Fluctuations in eccentricity fluctuations in v2

2 2

2 2

y x

y x

x,y – coordinates of “wounded” nucleons

v2 ~ fluctuations in flow

Calculations: R. Snellings and M. Miller

One can calculate how cumulants should be affected



Compare to data

Fluctuations in initial geometrycould explain the entire differencebetween v2{2} and v2{4}In fact, using nucleon participants (shown by red line in the plot)generates too much fluctuations,inconsistent with data



UrQMD once more

… the paper was not published for a reason…

Why these fluctuations are not seen in v2{4} compared to real v2?



MC Glauber calculations: “old” and “new”

“New” coordinate system –rotated, shifted

2 2

2 2

y x

y x

2 2

2 2

' ''

' '

y x

y x

Idea known for about a year,“went public” :S. Manly’s talk at QM2005

x

'x

y'y



Eccentricity, fluctuations, Monte-Carlo Glauber, Std vs ‘Participant’

Note:-Relative fluctuations are much smaller.- In general, “apparent” (“participant”) eccentricity values are larger comparedto “standard”.

-In CuCu epsStd{4} failsalmost at all centralities- The fluctuations in apparenteccentricity is much smaller than in standard- The difference betweenstandard and apparent isbigger for CuCu than AuAu

Monte Carlo Glauber nTuples from J. Gonzales (STAR)





Eccentricity, Monte-Carlo Glauber, all four systems.



But should not we use {2}, not eps?It could improve the agreement…

What about v2{4}/{4}?



Does it matter, eps, eps{2} or eps{4}?

This is just an illustration of an effect of using different eccentricity definitions. Centralities for eccentricity calculations are not correct !



Summary

- LYZ method is shown to be ‘identical’ toq-distribution method (and Bessel transform method)

- LYZ/BT is close to v2{4} in terms of sensitivity to flow fluctuations

- Epsparticipant is not only different from Epsstandard, but fluctuates… can/should we use these fluctuations to estimate flow fluctuations?



EXTRA SLIDES



First hydro calculations

2{ }2 2 / 2tpv v

s

In hydro, where the mean free path is by assumption much less than the size of the system,there is no other parameters than the system size (may enter time scales, see below).Then elliptic flow must follow closely the initial eccentricity.

J.-Y. Ollitrault, PRD 46 (1992) 229



Low density limit

d

dN

d

dN

d

dN

0

0( ) ( ) ( )n n n v v v

0 0 0( ) ( ) ( , )n d dt t t v r r r v 0 0( , ) ( )t t d t r v v r v

0 0 0( ) ( ) ( ( ) )n d dt d t v r r v r v v

2 2

2 2exp

2 2

x y

x y

2

2 2 216ji ij i

ij transportjx y i j

dN vv v

R R dy v v

(called “collisionless” in the original paper of Heiselberg and Levy)Below - my own derivation of Heiselberg’s results

Change in the particle flux is proportional to the probability for the particle to interact.

Integrations over: a) particle emission pointb) Over the trajectory of the particle (time) with weight proportional to the density of other particles --“scattering centers”

Particle density at time t assuming free streaming

2

1 dN

Sv

dy 2 2S x y

Heiselberg & Levy, PRC 59 (1999) 2716



v2/ vs particle density, first plot

S.V. & A. Poskanzer, PLB 474 (2000) 27

Uncertainties:Hydro limits: slightly dependon initial conditionsData: no systematic errors,shaded area –uncertainty incentrality determinations.Curves: “hand made”

E877 NA49

“Cold” deconfinement?



“hydro limits” ?

RHIC 160 GeV/A

SPS

SPS 40 GeV/A

b (fm)Suppressed scale!

v 2 /

Minimum in v2/ due to softening of the EoS at phase transition

Q to U. Heinz : Could the solid line in right hand plot be used as HYDRO prediction for v2/eps plot?

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Sergei A. Voloshin Wayne State University, Detroit, Michigan