1. 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 , China
Thanks to: Liu Lianshou, Nu Xu, Li Liangsheng, Hou Defu, Li Jiarong
1. Motivation
2. How to locate critical point by
finite-size scaling
3. Critical behaviour of pt corr. at RHIC
4. Discussions and suggestions
5. Summary
arXiv:0904:1040, proceedings of CPOD’09 at BNL

2. 1. Motivation → critical point. ? Open question:
★ QCD phase transitions ● ?
Lattice-QCD predicts:
Deconfinement
Chiral symmetry restoration ●
Critical endpoint
: crossover
: first order
→ critical point.
Open question:
Karsch F., Lecture Notes Phys. 583, 209(2002); Karsch F. , Lutgemeier M., Nucl. Phys. B550, 449(1999). Y. Aoki et al, arXiv: ; A.Bazavov et al, arXiv:

3. 一、关于相图 From Xu Mingmei （许明梅） (pure guesswork)
--- See talk by Pisarski, p.18.
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4. 有效的场论模型： The number of CP: 0, 1, 2, From Xu Mingmei （许明梅） NJL model
Model calculations suggest that there could be none, one or even two critical points depending on the parameters in the QCD Lagrangian.
See talks by B.-J. Schaefer, J. Kapusta, K. Fukushima, Koch, …
有效的场论模型：
NJL model
Linear sigma model
Linear sigma model with quarks or nucleons
PNJL
PNJL with Polyakov loop dynamics
Quark-meson model
Quark-meson model with Polyakov loop dynamics
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5. 1. Motivation RHIC, SPS & FAIR Question:
★ Current status of relativistic heavy ion experiments:
RHIC, SPS & FAIR
critical point
Question:
How to locate the critical point from observable?

6. ★ Finite size of the formed matter
Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca,
Maged Elhajal, and Frederic Mila, PRL91, (2003).
the finite-size effect is negligible!
When
due to critical slowing down ！
Boris Berdnikov and Krishna Rajagopal, PRD61, (2000).
So when
The finite-size effect has to be taken into account at RHIC!
M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, (2000).

7. ★ effects of finite size
2D-Ising
● Infinite system:
at critical point,
correlation length ξ → ∞.
observable
● Finite system:
observable →
finite & has a maximum →
non-monotonous behavior
Li Liangsheng and X.S. Chen; Chen Lizhu,
Li Liangsheng, X.S. Chen and Wu Yuanfang.
M. A. Stephanov, PRL102, (2009);
R. A. Lacey, et.al.,PRL98, (2007).
☞ The position of the maximum changes with size and
deviates from the true critical point !
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8. ☞The finite-size scaling of the observable.
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9. 2. How to locate critical point by finite-size scaling
Finite-size scaling form:
M. E. Fisher, in Critical Phenomena,
(Academic, New York, 1971).
E. Brezin, J. Phys. (Paris) 43, 15 (1982).
X. S. Chen, V. Dohm, and A. L. Talapov,
Physica A232, 375 (1996);
X. S. Chen, V. Dohm, and N. Schultka,
PRL, 77, 3641(1996).
in the vicinity of √sc ,
: reduced variable,
like T, or h in thermal-dynamic system.
: critical exponent of
: scaling function with scaled variable,
: critical exponent of correlation length

10. Critical characteristics of FSS
★ Fixed point:
At critical point ,
Scaled variable:
is independent of size L.
Scaling function:
becomes a constant.
2D-Ising
It behaves as a fixed point in,
Fixed point
Li Liangsheng and X.S. Chen; Chen Lizhu,
Li Liangsheng, X.S. Chen and Wu Yuanfang.

11. ★ If λ=0, fixed point can be directly obtained.
Like Binder cumulant ratios,
2D-Ising
and fluc. of mean cluster size,
2D-Ising
Li Liangsheng and X.S. Chen; Chen Lizhu,
Li Liangsheng, X.S. Chen and Wu Yuanfang.
2D-Ising
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12. ★ If λ ‡ 0 ☞ Therefore, the observable at diff. sizes can be used
2D-Ising
☞ Therefore, the observable at diff. sizes can be used
to locate the position of critical point .
Fixed point
is a parameter
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13. ☞ Fixed-point and best straight-line behavior
Taking logarithm in both sides of FSS,
At critical point ,
is linear function of at given !
☞ Fixed-point and best straight-line behavior

14. Au + Au collisions at 9 L (centralities) for each of 4√s .
3. Critical behaviour of pt corr. at RHIC
★ pt corr. as one of critical related observable
H. Heiselberg, Phys. Rept. 351, 161(2001);
M. Stephanov, J. of Phys. 27, 144(2005).
STAR
Au + Au collisions
at 9 L (centralities)
for each of 4√s .
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15. ★ System size: Initial mean size: Scaled initial mean size :
Number of Participants
Impact Parameter
Initial mean size:
Scaled initial mean size :
critical slowing down
expansion
Initial
Near transition
Transition
System size at transition:
☞Whether L’, or L is taken, the critical exponents will be
different, but the critical point is the same!
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16. ★ √s dependence of pt corr. at different system sizes
FSS is valid for L > 10 fm.
L: 10 to 2 fm for most central and peripheral coll.
L: 10 to 5 fm for 6 more central coll.
So 6 more central collisions are chosen !
B. Klein & J. Braun, arXiv:
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17. ★ Fixed-point behavior of pt correlation.
At given √s, the width of :
Two fixed-point around:
Ratios of critical
exponents :
☞ Precise position of the minimum can be obtained by
additional collisions around 62 and 200 GeV.
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18. ★ Straight-line behavior of pt correlation.
A parabola fit for data at give √s,
Parameters of parabola fits
√s(GeV) 20 62
130
200
☞ better straight-line fit
at√s =62 and 200 GeV
☞ the slopes of lines are ：
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19. ★ Critical behavior of normalized pt correlation.
☞Same fixed-point and best straight-line behavior !
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20. 4. Discussions and suggestions.
☻ Supports
√sc ~ 62, and 200 GeV,
are both within the
range estimated by
lattice-QCD.
M. Stephanov, arXiv: hep-lat/ ; Y. Aoki, Z. Fodor, S.D. Katza, and
K.K. Szabo, Phys. Lett. B643, 46(2006); F. Karsch, PoS CFRNC2007.
* Roy A. Lacey, et al, PRL , 98, (2007).
2. The similar ratios of critical exponents at two critical
points is consistent with current theoretical estimation.
Jorge Garca, Julio A. Gonzalo, Physica A 326,464(2003).
Jens Braun1 and Bertram Klein, Phys. Rev. D77, (2008).
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21. 4. Discussions and suggestions (II).
☻ Uncertainties
Quality of data
Poor statistics for data at 20GeV and 130GeV.
2. Error of the system size
It is absent at moment.
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22. 4. Discussions and suggestions (II).
☻ Confirmation of CEP
1. Boundary of phase diagram
1st-order: the finite size scaling of susceptibility is determined by
the geometric dimension, the height and width are
proportional to volume V and 1/V, respectively.
2nd-order : the singular behavior is given by some power of V,
defined by the critical exponents.
Crossover: There would be no singular behavior and the susceptibility
peak would not get sharper when increasing the volume V;
instead, its height and width will be V independent for large
volumes.
Y.Aoki, G. Endrodi, Z. Fodor, S. D. katz, & K.K. Szabo, Nature 443,05120(2006)
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23. 4. Discussions and suggestions (II).
2. Other measurements
more data on,
and the third order moments of conserved charges.
3. More collisions at RHIC
RHIC energy scan
Additional collisions around √s = 62 and 200 GeV.
It is advantage of RHIC.
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24. 5. Summary.
It is pointed out that critical related observable should
follow the finite-size scaling.
2. The method of finding and locating critical point is
established by critical characteristics of finite-size scaling.
3. As an application, critical behavior in current available data from STAR are demonstrated.
4. Confirmation of CEP from QCD phase diagram, more and better data on other critical related observable at current collision energies, and a few additional collisions are suggested.
Thanks!
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