2010/4/18 1南昌

Correlation between event multiplicity and observed collective

flow

周代梅

华中师范大学粒子物理研究所

2010/4/18 2南昌

Contents• Introduction

• Collective flow and higher harmonicsand higher harmonics

• Results & discussions

• Conclusion

2010/4/18 3南昌

I. IntroductionThe charge particle emitted from the fireball created in relativistic nucleus-nucleus collision exhibit transverse collective flow

Represented by the elliptic flow parameter (v_2) and other harmonic parameters (v_n, n=0,1,3,4,…)

The expected phase transition to QGP should have a dramatic effect on those harmonic parameters

The study of collective flow can provide information on the initial state dynamics and the degree of equilibration attained in heavy-ion collisions

v2 play crucial role in the evidence of QGP

2010/4/18 4南昌

A, Present status

In basic paper of PR,C58(1990)1671,start from triple differential distribution

3 2

31

1[1 2 cos( ( ))]

2 n rnT T

d N d NE v n

d p p dydp

and define that

, where <> indicates an average over all particle in all events.

cos[ ( )]n rv n

II. Collective flow and higher harmonicsand higher harmonics

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0v : isotropic flow

1v : first harmonic, directed flow

2v : second harmonic, elliptic flow (elliptic component）

nv (n=3,4,…): higher harmonics

For the particle number distribution, the coefficient

；221 yxT

T

x pppp

pv

22

2

T

y

T

x

p

p

p

pv

That is widely accepted in theoretical calculations

2010/4/18 6南昌

r is the azimuthal angle of the reaction plane

In theory, if the beam direction zp

Impact parameter vector xp

reaction plane is zx pp

between reaction plane and axisxp

0rparticle azimuthal angle is just the angle spanned by relative to axis

Tp

xp

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B, What we do

Doubts about the method of averaging ?●

3 3

3T T

d N d NE

d p p dydp d

Edpdy z 1zpTransfer into cylindrical system, substitute by y (rapidity), and use

The normalized particle multiplicity distribution is

●

● We first make a transparent derivation starting from invariant particle multiplicity distribution

1ln( )2

z

z

E py

E p

ddydpp

Nd

N TTev

31

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ddydpp

NdddppdyN

TTTTev

3

the event multiplicity and the integrals are taken overthe event multiplicity and the integrals are taken over entire ranges of the variablesentire ranges of the variables

This normalized particle multiplicity distribution is a three dimensional distribution. The dimension can be reduced by integrating over a certain variable

● If one studies , one should take integral over , the three dimensional distribution function reduces to a two dimensional distr. function

yvn

Tp

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ddydpp

Nddpp

N TTTT

ev

31

fdpydp

Nddppyd

Nd

d

ddpydp

Nddppyd

N

pTT

TTevp

TTTTp

evp

2

3

1

1

In numerical calculation a rapidity interval, , is used Instead of a single y value, the corresponding normalizedparticle multiplicity distribution becomes

y

Here the normalization should be

p

yy

yy

TTTTpevp

ydyd

ddpydp

NdddppydN

2

2

3

2010/4/18 10南昌

●

Since is a periodic and even function, it can be expanded by a Fourier series

( )f 2

1

cos2

11)(

nn nvf

● f is the normalized distributions density function of

fNevp is the number of particles emitted into at d

azimuthal angle without constraint on but is

constrained in

Tp yy

,...2,1cos

cos

nn

nfdvn

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where is the average of over

particles in a single event)cos( n )cos( n

As the event multiplicity fluctuates event-by-event, one always has to generate multiple events and to take average over events generated

,...2,1cos nnv eve

n ev

means an average over all events

event-wise average

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Applying recursion formula of cosine function

2coscos1cos2 nnncon

and cos , sin yx

T T

pp

p p

,][,][2

22

21 evT

yxeev

T

xe

p

ppv

p

pv

is an average of first over particles in an event and then average over events (event-wise average)

env ncos

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rather than an average over all particles in all event (particle-wise average)

evevpevevppn NNnv /)cos(

Only if the is independent of constrained event

multiplicity the reduces to

ncos

evpN envp

nv

In fact , the and correlate witheach other (negatively correlate)

)cos( n evpN

2010/4/18 14南昌

Elliptic flow and higher harmonics should be event-wise average rather than particle-wise average. The latter one should be corrected by event multiplicity.

The parton and hadron cascade model AMPT was used to study the correlation between event multiplicity and observed collective flow.

III. Results & discussions

2010/4/18 15南昌The correlate between and )2cos( evpN

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AMPT calculations for 0－ 40% of top multiplicity Au+Au events at RHIC energy: event-wise av. Is nearly 20% larger than particle-wise.

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The correlation between impact parameter and charged multiplicity1 fm change of impact parameter make a change of ～ 100 charged particles in multiplicity. An about 20% increase in multiplicity corresponds to about 2 fm decrease in impact parameter

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fmb 2~ )02.001.0(~2 v

)%3015~(max)/ 22 （vv 7.0max2 v20%mutiplicity change

V_2 is strongly correlated to impact parameter bV_2 is strongly correlated to impact parameter b

2010/4/18 19南昌

IV. Conclusion

(A) Elliptic flow and higher harmonics should be event-wise average rather than particle-wise average. The latter one should be corrected by event multiplicity

(B) Only if the is independent of constrained

event multiplicity the reduces to

ncos

evpN envp

nv

(C) In fact , the and correlate witheach other (negatively correlate)

)cos( n evpN

(D) AMPT model shown that charged multiplicityand v_2 are strongly correlated with impact parameter b

2010/4/18 20南昌

(E) The change of v_2 caused by the different impact parameters in the sample is ~(15-40)% of the maximum! This is a very significant change, which should not be overlooked!

(F) This correlation is important to take into consideration unless very narrow multiplicity bins are discussed.

2010/4/18 21南昌

Thank you !