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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.

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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: 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 endpoint ★ QCD phase transitions Lattice-QCD predicts: • Deconfinement • Chiral symmetry restoration : 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:0903.4155; A.Bazavov et al, arXiv:0903.4379.

  3. From Xu Mingmei （许明梅） 一、关于相图 (pure guesswork) --- See talk by Pisarski, p.18. Weihai’09

  4. From Xu Mingmei （许明梅） The number of CP: 0, 1, 2, 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 Weihai’09

  5. 1. Motivation ★ 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 When the finite-size effect is negligible! Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca, Maged Elhajal, and Frederic Mila, PRL91, 177202(2003). due to critical slowing down ！ Boris Berdnikov and Krishna Rajagopal, PRD61, 105017(2000). So when M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, 114028(2000). The finite-size effect has to be taken into account at RHIC!

  7. ● Infinite system: at critical point, correlation length ξ→ ∞. observable ● Finite system: observable → finite & has a maximum → non-monotonous behavior ★ effects of finite size 2D-Ising Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang. M. A. Stephanov, PRL102, 032301(2009); R. A. Lacey, et.al.,PRL98, 092301(2007). ☞ The position of the maximum changes with size and deviates from the true critical point ! 7 Weihai’09

  8. ☞The finite-size scaling of the observable. 8 Weihai’09

  9. 2. How to locate critical point by finite-size scaling : reduced variable, like T, or h in thermal-dynamicsystem. : scaling function with scaled variable, : critical exponent of correlation length : critical exponent of 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 ,

  10. Critical characteristics of FSS Fixed point ★ Fixed point: At critical point , Scaled variable: is independent of size L. Scaling function: becomes a constant. 2D-Ising It behaves as a fixed pointin, 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 11 Weihai’09

  12. Fixed point is a parameter ★ If λ ‡ 0 2D-Ising ☞ Therefore, the observable at diff. sizes can be used to locate the position of critical point . 12 Weihai’09

  13. is linear function of at given ! ★ Straight line behavior: Taking logarithm in both sides of FSS, At critical point, ☞ Fixed-point and best straight-line behavior

  14. 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 . 14 Weihai’09

  15. Number of Participants Impact Parameter ★ System size: 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! 15 Weihai’09

  16. ★ √s dependence of pt corr. at different system sizes FSS is valid forL > 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:0710.1161. 16 Weihai’09

  17. At given √s, the width of : Two fixed-point around: ★ Fixed-point behavior of pt correlation. Ratios of critical exponents : ☞ Precise position of the minimum can be obtained by additional collisions around 62 and 200 GeV. 17 Weihai’09

  18. ☞ the slopes of lines are ： ★ Straight-line behavior of pt correlation. A parabola fit for data at give √s, Parameters of parabola fits ☞ better straight-line fit at√s =62 and 200 GeV 18 Weihai’09

  19. ★ Critical behavior of normalized pt correlation. ☞Same fixed-point and best straight-line behavior ! 19 Weihai’09

  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/0701002; 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, 092301(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, 096008(2008). 20 Weihai’09

  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. 21 Weihai’09

  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) 22 Weihai’09

  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. 23 Weihai’09

  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! 24 Weihai’09

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