1 Study of order parameters through fluctuation measurements by the PHENIX detector at RHIC Kensuke...
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Study of order parameters through Study of order parameters through fluctuation measurements by the fluctuation measurements by the
PHENIX detector at RHICPHENIX detector at RHIC
Kensuke Homma for the PHENIX collaboration
Hiroshima University
On Aug 11, 2005 at KromerizXXXV International Symposiumon Multiparticle Dynamics 2005
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MotivationsMotivations
• RHIC experiments probed the state of strongly interacting dense medium with many properties consistent with partonic medium. What about the information on the phase transition?
• Is it the first order or second order transition?• Are there interesting critical phenomena such as tricritical
point?
K. Rajagopal and F. Wilczek, hep-ph/0011333
3
Landau’s treatment Landau’s treatment for 2for 2ndnd order phase transition order phase transition
TSUG
hTuTrTgVTG 42 )()(),(/),(
0)()( aTcTaTrSince order parameter should disappear at T=Tc, assume
Valid in a limit where the fluctuation on order parameter is negligible even at T~Tc
Gibb’s free energy
g(T,)
T>Tc =0
T<Tc =a(T-Tc)/2u
120
0
)122(
urh
G
hh h
hhH T
G
V
T
T
S
V
TC
2
2
||4
1,
||2
1
TcTaTcTa
Susceptibility
Specific heat
and CH show divergence or discontinuity,while T varies around Tc.
T>Tc T<Tc
u
TcTag
4
)( 22
4
Susceptibility and density fluctuationsSusceptibility and density fluctuations
rkik
rkik erehrh
0)()(
hTuTrATgVTG 422 )()()(),(/),(
120
2 )1222( urAk
hk
kk
222
222
1
)(
|)|2(2
1
1
)(
|)|(2
1
k
T
TcTaAk
k
T
TcTaAk
k
k
Susceptibility
2/1
2/1
||2
||
TcTa
A
TcTa
A
)'())'()()(( rrTkrr B
Fluctuation-dissipation theorem
|'|
)/|'|exp())'()()((
rr
rrTkrr B
Ornstein-Zernike behavior
kB
kkkk V
Tk
))((
T>Tc
T<Tc
22
)'( 1)'(
k
rrdVerrk
k
With Fourier transformation
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• Multiplicity fluctuations (density fluctuations ) as a function rapidity gap size with as low pt particle as possible.
Correlation length and singular behavior in correlation function.
• Average pt fluctuations (temperature fluctuations)
Specific heat See PRL. 93 (2004) 092301
• In this talk, I will focus on only multiplicity fluctuation measurements.
Fluctuation measurements by PHENIXFluctuation measurements by PHENIX
7.0
lGeometrical acceptance
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E802: 16O+Cu 16.4AGeV/c at AGSmost central events
[DELPHI collaboration] Z. Phys. C56 (1992) 63[E802 collaboration] Phys. Rev. C52 (1995) 2663
DELPHI: Z0 hadronic Decay at LEP2,3,4-jets events
Universally, hadron multiplicity distributions are well described by NBD.
Charged particle multiplicity distributions and Charged particle multiplicity distributions and negative binomial distribution (NBD)negative binomial distribution (NBD)
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2
2
2
22
2
22
2
2
)(
1
)(
1)(1
)(
1
11
/1
1
/1
/
)()1(
)(
)1/(
n
nnF
Fk
nn
k
kk
k
kn
knP
P
k
n
kn
nnn
Bose-Einstein distributionμ: average multiplicity
F2 : second order normalized factorial moment
NBD correspond to multiple Bose-Einstein distribution and the parameter k corresponds to the multiplicity of those Bose-Einstein emission sources. NBD can be Poisson distribution with the infinite k value.
NBD
Negative binomial distribution (NBD)Negative binomial distribution (NBD)
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δη= 0.09 (1/8) : P(n) x 107 δη= 0.18 (2/8) : P(n) x 106
δη= 0.35 (3/8) : P(n) x 105
δη= 0.26 (4/8) : P(n) x 104
δη= 0.44 (5/8) : P(n) x 103
δη= 0.53 (6/8) : P(n) x 102
δη= 0.61 (7/8) : P(n) x 101
δη= 0.70 (8/8) : P(n)
No magnetic fieldΔη<0.7, Δφ<π/2
PHENIX: Au+Au √sNN=200GeV
Charged particle multiplicity distributions Charged particle multiplicity distributions in different din different d gap gap
| Z | < 5cm
-0.35 < η < 0.35
2.16
< φ
< 3
.73
[rad
]
The effect of dead areas have been corrected.
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212111
21212
22
212
212
1
2111
212
2111
212212
)()(
),(1
)(
1
:),(
:),(
:)(
1)()(
),(
)()(
),(),(
dydyyy
dydyyyCKF
k
yyC
yy
y
yy
yy
yy
yyCyyR
inclusive single particle density
inclusive two-particle density
two-particle correlation function
Relation with NBD k
Normalized correlation function
Candidates of function forms with two particle correlation length
beR
eRR
yy
eRR
yy
yy
yy
/||2
/||02
21
/||
02
21
21
21
||
HBT type correlation in E802 : failed to describe data
Empirical two component model with R0=1.0
Relation between k and Relation between k and integrated two particle correlation functionintegrated two particle correlation function
Most general form: many trials failed.
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E802 type function can not describe the dataE802 type function can not describe the data
)]1)(/(1[
2/1)(
/0
/||02
21
eRk
eRR yy
Correlation function used in E802P. Carruthers and Isa Sarcevic,Phys. Rev. Lett. 63 (1989) 1562
NBD k vs. δη at E802
Phys. Rev. C52 (1995) 2663
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Empirical two component fitEmpirical two component fit
PHENIX: Au+Au √sNN=200GeV, Δη<0.7, Δφ<π /2
2
]1/[21
)(
1:
2
/2
2/||
221
beF
kbeR yy
dependent part + independent part with R0 =1
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Participants dependence of Participants dependence of ξξ and and bb
PHENIX: Au+Au √sNN=200GeV
Two particle correlation length Correlation strength of what?
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What is the origin of the two components?What is the origin of the two components?
))'()'()()()(()',(2 rnrnrnrnrrC
2
1 12 )()()'( nrrrrrrC
N
i
N
jji
nVNrnrrrnN
ii
/)(,)()(1
Go back to Ornstein-Zernike’s theory (see Introduction to Phase Transitions and Critical Phenomena by H.E.Stanley)which explains the growth of forward scattering amplitude of lightinteracting with targets at the phase transition temperature.
)'()'()'( 22 rrnrrnrrC
)/|'|exp(
)'(/)'()'(2
rrb
rrnrrrrR
Self interactionrenormalizingsingular part ?
Long range correlation
(r-r’)
r r’
Density of fluid element at r
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ξξ vs. number of participants vs. number of participants
Two particle correlation length
PHENIX: Au+Au √sNN=200GeV
Linear behavior of the correlation length as a function of the number of participants has been obtained in the logarithmic scale.
)log()log(
||
/
3
3
part
cpart
part
N
TTN
TNdydN
One slope fit gives
α = -0.72 ± 0.03
In the case of thermalized ideal gas,
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ConclusionsConclusions• Multiplicity distributions measured in Au+Au collisions at
√SNN=200GeV can be described by the negative binomial distributions.
• Two particle correlation length has been measured based on the empirical two component model from the multiplicity fluctuations, which can fit k vs. d in all centralities remarkably well.
• Extracted correlation length behaves linearly as a function of number of participants in logarithmic scales. Assuming one slope component, the exponent was obtained as -0.72±0.03.
• The interpretation of b parameter is still ambiguous. Any criticize or different view points are more than welcome.
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Backup Slide
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Uncorrected Npart*b vs. NpartUncorrected Npart*b vs. NpartCentrality k-Map in 10% bins
0
50
100
150
200
250
300
350
400
450
50 100 200 500 1000 infinity
intrinsic k
ob
serv
ed k "0-10"
"10-20"
"20-30"
"30-40"
Npart
Npa
rt*
b
Bias on NBD kdue to finite bin size of centrality
Intrinsic k
Obs
erve
d k
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0-5%
15-20%10-15%
0-5%
5-10%
Important HI jargon : Participants (Centrality) Important HI jargon : Participants (Centrality) peripheral central
Relate them to Npart and Nbinary (Ncoll ) using Glauber model.
Straight-line nucleon trajectories
Constant NN=(40 ± 5)mb.
Woods-Saxon nuclear density:
dRr
r o
exp1
1)(
fm
AAR
)03.065.6(
61.119.1 3/13/1
fmd )01.054.0(
b To ZDC
To BBC
Spectator
Participant
Multiplicity distribution Nch
Whether AA is a trivial sum of NN Whether AA is a trivial sum of NN or or something nontrivial ?something nontrivial ?
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Why not observing fluctuations ?Why not observing fluctuations ?• Fluctuation carries
information in early universe in cosmology despite of the only single Big-Bang event.
• Why don’t we use the event-by-event information by getting all phase space information to study evolution of dynamical system in heavy-ion collisions ?
• We can firmly search for interesting fluctuations with more than million times of mini Big-Bangs.
The Microwave Sky image from the WMAP Mission http://map.gsfc.nasa.gov/m_mm.html