Systematic study of particle
spectra in heavy-ion collisions
using Tsallis statistics
Ming Shao, Zebo Tang, Yi Li, Zhangbu Xu
CPPT/USTC

Introduction & Motivation

Why and how to implement Tsallis
statistics in Blast-Wave framework

Results
− strange hadrons vs. light hadrons
− beam energy dependence
−J/y radial flow
 Conclusion
2010/10/18
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Thermalization and Radial flow in HI
STAR whitepaper
Thermalization in heavy-ion collisions ?
－particle ratios agree with thermal
prediction
Matter flows in heavy-ion collisions
– all particles have the same collective
velocity
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Phys. Rev. Lett. 92 (2004) 182301
p T  m ass   T
Teff  T fo  m ass   T
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2
Blast-wave analysis
Multi-strange decouple earlier than light hadrons, with less radial
flow velocity
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Hydrodynamics evolution
π, K, p
Multi-strange W
Hydro parameters:
0 = 0.6 fm/c s0 = 110 fm-3 s0/n0 = 250 Tcrit=Tchem=165 MeV Tdec=100 MeV
Ulrich Heinz, arXiv:0901.4355
Multi-strange particle spectra can be well described by the same
hydrodynamics parameters as light hadrons
in contrast to the Blast-wave results
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Blast-Wave Model
Assumptions:
– Local thermal equilibrium  Boltzmann distribution
– Longitudinal and transverse expansions (1+2)
– Temperature and T are global quantities
boosted
E.Schnedermann, J.Sollfrank, and U.Heinz, Phys. Rev. C48, 2462(1993)
3
E
d N
dp
3

dN
m T dm T
e
(u

p  )/T
fo
random
pd   



R
0
m cosh    p sinh  
T
T
rdrm T K 1 
I




 0 

T fo

  T fo


  tanh
1
r
 r 
 r   S  
R 
  0 .5,1, 2
Extract thermal temperature Tfo and velocity parameter T
BGBW: Boltzmann-Gibbs Blast-Wave
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Limitation of the Blast-wave
• Strong assumption on local
thermal equilibrium
STAR PRC71 (2005) 64902
• Arbitrary choice of pT range of
the spectra
AuAu@200GeV
• Non-zero flow velocity
<T>=0.2 in p+p
• Lack of non-extensive
quantities to describe the
evolution from p+p to central
A+A collisions
pp@200GeV minbias
– mT spectra in p+p collisions
Levy function or mT power-law
– mT spectra in A+A collisions
Boltzmann or mT exponential
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STAR PRL99
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Non-extensive Tsallis statistics
C. Tsallis, H. Stat. Phys. 52, 479 (1988)
http://www.cscs.umich.edu/~crshalizi/notabene/tsallis.html
http://tsallis.cat.cbpf.br/biblio.htm
Wilk and Wlodarzcyk, PRL84, 2770 (2000)
Wilk and Wlodarzcyk, EPJ40, 299 (2009)
Particle pT spectra:
m
exp(  T ) 
T
m
( q  1) mT 1/(q 1)
exp q ( T )  [1 
]
T
T
Exponential  Power law
1/ T
2
 1/ T
1/ T
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2
 q 1
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Tsallis statistics in Blast-wave model
3
E
BGBW:
d N
dp
3
e

dN

p  )/T fo
pd   


mT d mT
  tan h
I0 (z) 
 (u

1
 m T co sh  
rd rm T K 1 
I 0


T
fo


R
0
r
r
 r 
 S  
 R 
 p T sin h  




T
fo



  0 .5,1, 2
2
1

2

ex p [ z co s( )] d  ,
K1 ( z) 
0
 co sh ( y ) ex p [  z co sh ( y )]d y
0
With Tsallis distribution:
exp( 
mT
T
)  exp q ( 
mT
)  [1 
( q  1) m T
T
]
 1 /( q 1 )
T
Tsallis Blast-wave (TBW) equation is:
dN
mT dmT
Y
 mT
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
R
 cosh( y)dy  d  rdr{1 
Y

0
q 1
T
[mT cosh( y ) cosh(  )  pT sinh(  ) cos( )]}
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1/( q 1)
Fit results in Au+Au collisions
Phys. Rev. C 79, 051901 (R) (2009)
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Fit strange hadrons only
All available species
Strangeness, Au+Au 0-10%:
<> = 0.464 +- 0.006
T = 0.150 +- 0.005
q = 1.000 +- 0.002
chi^2/nDof = 51/99
Tstrange>Tlight-hadrons
Strangness decouple from
the system earlier
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Centrality dependence for T and <T>
 Multi-strange hadrons decouple earlier
 Hadron rescattering at hadronic phase doesn’t produce a
collective radial flow, instead, it drives the system off equilibrium
 Partons achieve thermal equilibrium in central collisions
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Beam energy dependence
s  17 . 2 GeV
1. The radial flow velocity at SPS is smaller than that at RHIC.
2. Freeze-out temperatures are similar at RHIC and SPS.
3. The non-equilibrium parameter (q-1) is small in central nucleus-nucleus
collisions at RHIC and SPS except a larger (q -1) value for non-strange
hadrons at RHIC energy
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How about heavy hadrons?
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J/y suppression at RHIC and SPS
quarkonium – gloden probe of QGP
• deconfinement (color screening)
• thermometer
Puzzle!
Grandchamp, Rapp, Brown
PRL 92, 212301 (2004)
nucl-ex/0611020
Regeneration?
Test with J/y flow.
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J/y suppression at RHIC ≈
J/y suppression at SPS
(energy differs by ~10 times)
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J/yElliptic flow
Heavy Flavor decay electron
J/y
Alan Dion, QM2009
Too early to compare with models
Won’t have enough statistics before 2011
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Ermias T. Atomssa, QM2009
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How about radial flow?
Sizeable radial flow for heavy flavor decay electrons
Yifei Zhang, QM2008, STAR, arXiv:0805.0364
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J/y radial flow
<> = 0.06 +- 0.03
T = 0.134 +- 0.006
q =1.0250 +- 0.0014 c2/nDof = 85.03 / 26
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J/y radial flow consistent with 0
Inconsistent with regeneration
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Summary
Identified particle spectra from SPS to RHIC have been
analyzed with Tsallis statistics in Blast-wave description
(light hadrons, multi-strange hadrons, charmonium)
We found in HIC
•Partonic phase
– Partons achieve thermal equilibrium in central heavy-ion collisions
– J/y is not thermalized and disfavors regeneration
•Multi-strange hadrons decouple earlier
•Hadronic phase
– Hadronic rescattering doesn’t produce collective radial flow
– It drives the system off equilibrium
– Radial flow reflects that when the multi-strange decouples
Thank you!
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Back up
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Check— Parameter Correlation
<> = 0.0954 +- 0.0828
T
= 0.1777 +- 0.0328
q
= 1.0106 +- 0.0022
c2/nDof = 151.53 / 37
<> = 0.0000 +- 0.0000
T
= 0.1747 +- 0.1644
q
= 1.0708 +- 0.0435
2
c /nDof = 12.83 / 13
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Check—Strangeness and light hadrons
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Results in p+p collisions
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Temperature fluctuation
Reverse legend
1/ T
2
 1/ T
1/ T
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2
2
 q 1
Wilk and Wlodarzcyk, EPJ40, 299 (2009)
Wilk and Wlodarzcyk, PRL84, 2770 (2000)
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PHENIX Beam Use Request
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