1. Locating critical point of QCD phase transition by finite-size scaling Chen Lizhu 1, X. S. Chen 2, Wu Yuanfang 1 1 IOPP, Huazhong Normal University, Wuhan, China 2 ITP, Chinese Academy of Sciences, Beijing 100190, China 1. Motivation 2. Finite-size scaling form and how to locate critical point by it how to locate critical point by it 3. Critical behaviour of p t corr. at RHIC 4. Discussions and suggestions 5. Summary Thanks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou Defu

2. 1. Motivation (I) ★ QCD phase transitions Deconfinement Deconfinement Chiral symmetry restoration Chiral symmetry restoration → Open question: Whether they occur at the same T c, or not? Whether they occur at the same T c, or not? ● : crossover : first order Two critical endpoints. Lattice-QCD predict: Karsch F., Lecture Notes Phys. 583, 209(2002); Karsch F., Lutgemeier M., Nucl. Phys. B550, 449(1999).

3. 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? How to locate the critical point from observable? ★ Limited size of formed matter ☞ The effect of finite size is not negligible!

4. 1. Motivation (III) ★ Non-monotonous behavior, and why it is not enough At critical point, ● in infinite system: ● in infinite system: correlation lengthξ → ∞. correlation lengthξ → ∞. ● in finite system: ● in finite system: finite and have a maximum, finite and have a maximum, i.e., non-monotonous behavior i.e., non-monotonous behavior ☞ However, the position of the maximum of non- monotonous behavior of observable changes with monotonous behavior of observable changes with system size and deviates from the true critical point. system size and deviates from the true critical point.

5. 1. Motivation (IV) ☞ The absence of non- monotonous behavior monotonous behavior does not mean no CPOD. does not mean no CPOD. Order parameter in 2D-Ising ☞ Non-monotonous behavior is not always associated is not always associated with CPOD. with CPOD. Specific heat in 1D- Ising

6. ☞ The reliable criterion of critical behavior is finite-size scaling of the observable. finite-size scaling of the observable.

7. 2. Finite-size scaling form (I) : reduced variable, like T, or h in thermal-dynamic system. like T, or h in thermal-dynamic system. : critical exponents : scaling function with scaled variable, : critical exponent of correlation length, 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:

8. ★ Fixed point: At critical point, Scaling function: becomes a constant. becomes a constant. It behaves as a fixed point, where all curves converge to. Scaled variable: is independent of size L. In the plot: Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang. Critical characteristics

9. 2009.4.28STAR--Hangzhou 9 ★ If λ=0, fixed point can be directly obtained. Like Binder cumulant ratios. and fluc. of cluster size. Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang.

10. 2009.4.28STAR--Hangzhou 10 ☞ Reversely, if √s c is unknown, the observable at diff. L can help us to find the position of critical point. can help us to find the position of critical point. Fixed point ★ If λ‡ 0 is a justable parameter is a justable parameter

11. is linear function of ! Taking logarithm in both sides of FSS, At critical point, ★ Straight line behavior: ☞ The critical point can also be found from the system size dependence of the observable. dependence of the observable.

12. 2009.4.28STAR--Hangzhou 12 3. Critical behaviour of p t corr. at RHIC STAR Coll. Au + Au collisions at 4 incident energies: 4 incident energies: 20, 62, 130, 200 GeV 20, 62, 130, 200 GeV and 9 centralities (sizes). and 9 centralities (sizes). If √sc is in the RHIC energy, its scaling form should be: ★ Pt corr. as one of critical related observable H. Heiselberg, Phys. Rept. 351, 161(2001); M. Stephanov, J. of Phys. 27, 144(2005).

13. ★ System size: Initial mean size: Scaled mean size of initial system: System size at transition should be a monotonically increasing function of : It will modifies the scaling exponents, but not the position of critical point. So we take L instead of L’ in the following.

14. 2009.4.28STAR--Hangzhou 14 ★ System size dependence of p t correlation. 1.Change the centrality dependence of p t corr. dependence of p t corr. at diff. incident energies at diff. incident energies to the collision energy to the collision energy dependence at diff. sizes. dependence at diff. sizes. 3. The influence of finite size is obvious. 2. Choose 6 centralities at mid-central and central mid-central and central collisions to do the analysis. collisions to do the analysis.

15. 2009.4.28STAR--Hangzhou 15 ★ Fixed-point behavior of p t correlation. Two fixed-point behavior around: With the ratios of critical exponents :

16. 2009.4.28STAR--Hangzhou 16 ★ Straight-line behavior of p t correlation. A parabola fit for data at give √s, √s(GeV) 20 62 130 200 Parameters of parabola fits ☞ the better straight-line behavior happen to be behavior happen to be at√s =62 and 200 GeV at√s =62 and 200 GeV ☞ the slopes of lines are obtained by the fixed points.

17. 2009.4.28STAR--Hangzhou 17 ★ Same analysis for normalized p t correlation. Two fixed-point behavior around: With the ratios of critical exponents :

18. 2009.4.28STAR--Hangzhou 18 4. Discussions and suggestions. 1.√s c =62, and 200 GeV, are both in the range estimated both in the range estimated by lattice-QCD. They may by lattice-QCD. They may imply that deconfinement imply that deconfinement and chiral symmetry and chiral symmetry restoration occur at diff. Tc. restoration occur at diff. Tc. ☻ Discussions 2. The similar ratios of critical exponents at two critical points is consistent with current theoretical estimation, which shows that consistent with current theoretical estimation, which shows that all critical exponents in 3D-Ising are very close to that of 3D-O(4). 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). 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.

19. 2009.4.28STAR--Hangzhou 19 4. Discussions and suggestions (II). ☻ Suggestions 2. To determine precisely the critical incident energy and critical exponents, additional collisions around critical exponents, additional collisions around √s =62 and 200 GeV are required. √s =62 and 200 GeV are required. 1. More data on : greatly helpful in confirming the results. So, the√s and centrality dependence of those observable are called for. and so on will be

20. 2009.4.28STAR--Hangzhou 20 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. 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 p t 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.