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Locating critical point of QCD phase transition by finite-size scaling

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. Thanks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou Defu.

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Locating critical point of QCD phase transition by finite-size scaling

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  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 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

  2. 1. Motivation (I) ● Two critical endpoints. ★ QCD phase transitions Lattice-QCD predict: • Deconfinement • Chiral symmetry restoration : crossover : first order → Open question: Whether they occur at the same Tc, or not? 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? ★ 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: correlation lengthξ → ∞. ●in finite system: finite and have a maximum, i.e., non-monotonous behavior ☞However, the position of the maximum of non- monotonous behavior of observable changes with system size and deviates from the true critical point.

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

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

  7. 2. Finite-size scaling form (I) : reduced variable, like T, or h in thermal-dynamic system. : scaling function with scaled variable, : critical exponents : 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. Critical characteristics ★ Fixed point: At critical point , Scaled variable: is independent of size L. Scaling function: becomesa constant. In the plot: 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.

  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. STAR--Hangzhou

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

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

  12. 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). 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: STAR--Hangzhou

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

  14. ★ System size dependence of pt correlation. • 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. STAR--Hangzhou

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

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

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

  18. 4. Discussions and suggestions. ☻ Discussions • √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). STAR--Hangzhou

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

  20. 5. Summary. • 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. Thanks! STAR--Hangzhou

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