Locating critical point of QCD phase transition
by finite-size scaling
Chen Lizhu1, X. S. Chen2 , Wu Yuanfang1
1 IOPP,
Huazhong Normal University, Wuhan, China
2 ITP, Chinese Academy of Sciences, Beijing 100190, China
Thanks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou Defu
1. Motivation
2. Finite-size scaling form and
how to locate critical point by it
3. Critical behaviour of pt corr. at RHIC
4. Discussions and suggestions
5. Summary
1. Motivation (I)
★ QCD phase transitions
Lattice-QCD predict:
• Deconfinement
• Chiral symmetry restoration
Tc  0,
 B  0 : crossover
Tc  0,
B  0
●
: first order
Two critical endpoints.
→ Open question:
Whether they occur at the same Tc, or not?
T  T ,
or
T  8T
Karsch F., Lecture Notes Phys. 583, 209(2002); Karsch
F. , Lutgemeier M., Nucl. Phys. B550, 449(1999).
1. Motivation (II)
★ Current status of relativistic heavy ion experiments:
RHIC at BNL, the SPS at CERN, and future FAIR at GSI
are aimed to find critical point.
Question:
How to locate the critical point from observable?
★ Limited size of formed matter
☞ The effect of finite size is not negligible!
1. Motivation (III)
★ Non-monotonous behavior, and why it is not enough
At critical point,
● in infinite system:
correlation lengthξ → ∞.
●in finite system:
finite and have a maximum,
i.e., non-monotonous behavior
☞However, the position of the maximum of nonmonotonous behavior of observable changes with
system size and deviates from the true critical point.
1. Motivation (IV)
Specific heat in 1D- Ising
Order parameter in 2D-Ising
Tc  2.27
☞Non-monotonous behavior
is not always associated
with CPOD.
☞The absence of nonmonotonous behavior
does not mean no CPOD.
☞The reliable criterion of critical behavior is
finite-size scaling of the observable.
2. Finite-size scaling form (I)
A observable in relativistic heavy ion collision is a function
of incident energy √s and system size L, √s like T, or h.
Finite-size scaling form:


Q ( s , L )  L FQ ( L  )
s 
 
sc

F ( L )

sc

: reduced variable,
like T, or h in thermal-dynamic system.

: scaling function with scaled variable,  L


: critical exponents

: critical exponent of correlation length,    0 
Critical characteristics
★ Fixed point:
  0.
At critical point ,

Scaled variable:
L

 0,
Scaling function: F (0 )  Q (
In the plot:

Q ( s , L) L

vs.
is independent of size L.
s , L)L
L
It behaves as a fixed point,
where all curves converge to.
Li Liangsheng and X.S. Chen; Chen Lizhu,
Li Liangsheng, X.S. Chen and Wu Yuanfang.
 
,
becomes a constant.
★ If λ=0, fixed point can be directly obtained.
Like Binder cumulant ratios.
U 1
M
4
3 M
2
2
 U  0
and fluc. of cluster size.
Li Liangsheng and X.S. Chen; Chen Lizhu,
Li Liangsheng, X.S. Chen and Wu Yuanfang.
2009.4.28
STAR--Hangzhou
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★ If λ‡ 0
☞ Reversely, if √sc is unknown, the observable at diff. L
can help us to find the position of critical point .
a
Q ( s , L)L
Fixed point
a0
Q ( s , L)L
a is a justable parameter
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


 a0
10
★ Straight line behavior:
Taking logarithm in both sides of FSS,
ln Q ( s , L ) 
At critical point



ln L  ln F ( L  )
 0,
ln Q ( s , L ) 


ln L  C
ln Q ( s , L) is linear function of ln L !
☞The critical point can also be found from the system size
dependence of the observable.
3. Critical behaviour of pt corr. at RHIC
★ Pt corr. as one of critical related observable
Nk
1
P( s , L) 
N event
N event

Nk
  p
t ,i
 pt
 p
t, j
 pt

H. Heiselberg, Phys. Rept. 351, 161(2001);
M. Stephanov, J. of Phys. 27, 144(2005).
i 1 j  i , j 1
N k ( N k  1)
N k 1
Au + Au collisions at
4 incident energies:
20, 62, 130, 200 GeV
and 9 centralities (sizes).
STAR Coll.
If √sc is in the RHIC energy,
its scaling form should be:


P ( s , L )  L FP ( L  )
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★ System size:
Number of Participants
Impact Parameter
Initial mean size: 
N part
Scaled mean size of initial system:
N part
L
2N A
System size at transition should be a monotonically
increasing function of L :
'
1
L  cL
,
  0.
It will modifies the scaling exponents, but not the position
of critical point. So we take L instead of L’ in the following.
★ System size dependence of pt correlation.
1. Change the centrality
dependence of pt corr.
at diff. incident energies
to the collision energy
dependence at diff. sizes.
2. Choose 6 centralities at
mid-central and central
collisions to do the analysis.
3. The influence of finite size is obvious.
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★ Fixed-point behavior of pt correlation.
Two fixed-point behavior around:
s  62,
200 GeV
With the ratios of critical exponents : a0,1 , a0,2  2.09, 2.08
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★ Straight-line behavior of pt correlation.
A parabola fit for data at give √s,
c2 (ln L)  c1 ln L  c0
2
Parameters of parabola fits
√s(GeV)
20
62
130
200
c2
1 .8 6  0 .9 3
0 .6  0 .0 9
1 .5 6  0 .4 1
0 .7 7  0 .1
c1
3 .9  0 .8 9
2 .5 9  0 .0 9
3 .4 3  0 .4 1
2 .7 4  0 .1
☞ the better straight-line
behavior happen to be
at√s =62 and 200 GeV
☞ the slopes of lines are
a0,1  2.09, and 2.08 respectively,
obtained by the fixed points.
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★ Same analysis for normalized pt correlation.
N event N k
Rt ( s , L ) 
P( s , L)
pt
,
pt 

p t ,i
k 1 i 1
N event

Nk
k 1
Two fixed-point behavior around:
s  62,
200 GeV
With the ratios of critical exponents : a0,1 , a0,2  0.55
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4. Discussions and suggestions.
☻ Discussions
1. √sc =62, and 200 GeV, are
both in the range estimated
by lattice-QCD. They may
imply that deconfinement
and chiral symmetry
restoration occur at diff. Tc.
M. Stephanov, arXiv: hep-lat/0701002; Y. Aoki, Z. Fodor, S.D. Katza, and
K.K. Szabo, Phys. Lett. B643, 46(2006); F. Karsch, PoS CFRNC2007.
2. The similar ratios of critical exponents at two critical points is
consistent with current theoretical estimation, which shows that
all critical exponents in 3D-Ising are very close to that of 3D-O(4).
Jorge Garca, Julio A. Gonzalo, Physica A 326,464(2003).
Jens Braun1 and Bertram Klein, Phys. Rev. D77, 096008(2008).
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4. Discussions and suggestions (II).
☻ Suggestions
1. More data on :
and so on will be
greatly helpful in confirming the results. So, the√s and
centrality dependence of those observable are called
for.
2. To determine precisely the critical incident energy and
critical exponents, additional collisions around
√s =62 and 200 GeV are required.
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5. Summary.
1. It is pointed out that in relativistic heavy ion collisions, critical
related observable in the vicinity of critical point should follow
the finite-size scaling.
2. The method of finding and locating critical point is established by
finite-size scaling and its critical characteristics, in particular,
fixed point and straight line behavior.
3. As an application, the data of pt correlation from RHIC/STAR are
analyzed. Two fixed-point and straight-line behavior are both
observed around√s =62 and 200 GeV. This demonstrates two
critical points of QCD phase transition at RHIC.
4. To precisely determine the critical endpoints and critical
exponents, more and better data on other critical related
observable at current collision energies, and a few additional
collisions around √s = 62 and 200 GeV are called for.
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