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Chen Lizhu 1 , Shao Ming 2 , X.S. Chen 3 , Wu Yuanfang 1

Finite-size behavior of critical related observables at RHIC. Chen Lizhu 1 , Shao Ming 2 , X.S. Chen 3 , Wu Yuanfang 1 1 IOPP, Central China Normal University (CCNU), Wuhan, China 2 University of Science and technology of China, Hefei, Anhui , China

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Chen Lizhu 1 , Shao Ming 2 , X.S. Chen 3 , Wu Yuanfang 1

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  1. Finite-size behavior of critical related observables at RHIC Chen Lizhu1, Shao Ming2, X.S. Chen3, Wu Yuanfang1 1IOPP, Central China Normal University (CCNU), Wuhan, China 2University of Science and technology of China, Hefei, Anhui , China 3ITP, Chinese Academy of Science, Beijing, China 1. Motivation. 2. How to see the finite-size behaviour ? 3. Application to the related observables at RHIC. 4. Summary and outlook. Chen Lizhu, X.S. Chen, Wu Yuanfang, arXiv:0904.1040; 1002:4139; Wu Yuanfang, Chen Lizhu, X. S. Chen, PoS, (CPOD, 2009 )036. CPOD2011, Wuhan, China

  2. 1. Motivation ★ Current status : • 3 possibilities from hadron to QGP! • QGP has been found at RHIC! • Difficulty in determining the boundary. • Possibility to find critical point (CP). • observables: sensitive to ξ • behavior: non-monotonic, or peak, • Expected behavior of CP has not been found! CP Quark Gluon Plasma (QGP) phase C. Blume’ talk at CPOD2011, http://conf.ccnu.edu.cn/~cpod2011/ B. Mohanty’s talk at QM’11, http://qm2011.in2p3.fr/node/629. CPOD2011, Wuhan, China

  3. ★ Possible reasons : ● Expected signal is the case of thermodynamic limits. ● The formed system in relativistic heavy ion collisions are finite in both duration and system size. ● These finite effects may shift, or smear the signals. K. Paech, Eur. Phys. J. C 33 (2004) S627. B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017. L. F. Palhares, E.S. Fraga, and T. Kodama, arXiv 0904.4830. CPOD2011, Wuhan, China

  4. ★ Influence of finite evolution time Due to critical slowing down, the system may: ● not pass the CP; ● pass CP, the correlation length may not be fully developed. 0.5-1fm  2-3fm B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017. So the observables, which are more sensitive to correlation length, are recommended, e.g., the higher cumulants! CPOD2011, Wuhan, China

  5. ★ The influences of finite size : System size L: (1) Infinite system, or very larger: * L→ ∞, at critical point, ξ → ∞. * L>> ξ , ξ → finite maximum, non-monotonic. (Possible for very high energy and central collisions) (2) ξ > L/6 (L/10), finite-size effect is not negligible. (Likely the case of most collisions) (3) Very small, no phase transition. (Possible for low energy, and peripheral collisions.) C. Weber, L. Capriotti, G. Misguich, F. Becca, M. Elhajal, and F. Mila, PRL. 91,177202(2003); Peter Olsson, PRB 55, 3583(1997). CPOD2011, Wuhan, China

  6. ★ Optimistic sides of finite size: ● Finite-size behavior of phase transition is well built up, and served as an identification of transition nature! • An identification of • crossover in Lattice QCD. Y. Aoki, et. Al.,Nature , 443, 675(2006). • Alocation of critical point • in nuclear fragmentation. M. K. Berkenbusch, et. Al., PRL88 (2001) 022701; J. B. Elliott, et. al., PRL88, (2002) 042701. • Size behavior is helpful ! Z. Fodor and T. Hasuda’s talks at preschool, http://conf.ccnu.edu.cn/~cpod2011/ CPOD2011, Wuhan, China

  7. Number of Participants Impact Parameter • ●Relativistic heavy ion collision: • Initial size changes • one magnitude from • peripheral to central coll. • The change is presented by centrality, and is well • measured in experiments. It is possible to test finite-size behavior of QCD phase transition in relativistic heavy ion collisions ! CPOD2011, Wuhan, China

  8. 2. How to see the finite size behavior? ★ Finite-size behavior: 2nd-order: finite-size scaling function, and scaling exponent ( , non-integer), i.e., 1st-order: finite-size scaling function, and scaling exponent is determined by spatial dimension (integer). Crossover: size independent. Z. Fodor and T. Hasuda’s talk at preschool, http://conf.ccnu.edu.cn/~cpod2011/ Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz, K.K. Szabo, Nature ,443, 675(2006); V. Koch, arXiv: 0810.2520. CPOD2011, Wuhan, China

  9. ★ Applicability: Thermal equilibrium, or local thermal equilibrium. P. Braun-Munzinger and J. Stachel, arXiv:1101.3167. 2) Transition line is close to freeze-out curve. G. Endrodi et al., JHEP1104, 001 (2011); O. Kaczmarek, F. Karsch et al., Phys.Rev. D83, 014504 (2011); J. Cleymans, K. Redlich, Phys. Rev. Lett.81, 5284 (1998). 3) Survival of critical fluctuations in the final state. M. A. Stephanov, hep-ph/0402115, Int. J. Mod. Phys. A20 (2005) 4387; Ibid., PRL 102 (2009)032301; M. Asakawa, S. Ejiri, M. Kitazawa, PRL. 103, 262301(2009);Y. Hatta et al, PRL 91, 102003 (2003); Cheng et al, PRD79 (2009) 074505; Ibid. Prog.Theor.Phys.Suppl. 186 (2010) 563-566 CPOD2011, Wuhan, China

  10. ► Possible form of finite-size scaling in heavy ion collisions: • If correspondsize: • & temperature: • then write similarly, J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, PRC73, (2006)034905. L T 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). : scaling function with scaled variable, : reduced √s like T, or h in thermodynamicssystem, √sc critical one. : critical exponents of Q, and correlation length. CPOD2011, Wuhan, China

  11. Fixed point ►Fixed point and straight line: At critical energy , Scaling function: 2D-Ising is a constant. It behaves as a fixed pointin, In the case, taking the logarithm, is a straight line. It deviates from line, when: X. S. Chen’s talk at 18th CBM, http://hepd.ep.tsinghua.edu.cn/cbm2011/. CPOD2011, Wuhan, China

  12. ► How to find fixed point from observable? Tune Fixed point Example: 2D-Ising Plot energy dependence of the observable at different system sizes to see if we can find the scaled parameter a0, which makes all size points intersect. CPOD2011, Wuhan, China

  13. ► Quantify the behavior of point-like: The width of all size points at a given √s : An experimental point: 2D-Ising } At CP: For CPOD2011, Wuhan, China

  14. Fixed point ►Energy dependency of the minimum width: size independent Integer a0 Non-integer a0 2D-Ising No fixed point integer a0 1 1st order PT CP Crossover region √s CPOD2011, Wuhan, China

  15. 3. Application to the related observables at RHIC ►Related observables: Higher cumulant ratios of conserved charge (net-baryon, electric charge, strange) Dynamical electric charge fluctuations; Multiplicity fluctuations; pt correlations; …. M. A. Stephanov, K. Rajagopal, and E. Shuyak, PRL 81, 4816(1998); M. A. Stephanov, PRL 102,032301(2009); hep-ph/0402115; M. Asakawa, S. Ejiri, M. Kitazawa, PRL 103, 262301(2009); H. Heiselberg, Phys. Rept. 351, 161(2001); V. Koch, arXiv:0810.2520. CPOD2011, Wuhan, China

  16. ►Centrality dependence of 4th and 6th cumulant ratios of net-proton dis. Lizhu Chen’s talk, and V. Koch’s summary talk at BNL workshop, http://www.bnl.gov/fcrworkshop/. • Both of them are centrality (system size) independent. CPOD2011, Wuhan, China

  17. ►Energy dependence of dynamical electric charge fluc. and pt corr. STAR, Phys. Rev. C79, 024906(2009); STAR,Phys. Rev. C72, 044902(2005). • Weak size dependence at 6 central collisions, and dramatic changes at 3 most peripheral collisions. • Possible reason: volume. • At each energy, let’s multiply the observable at all 9 sizes by La, then • tune a to find minimum width of all 9 size points. CPOD2011, Wuhan, China

  18. ►The width with varying parameter a: Dyn. charge fluc. pt corr. Norm. pt corr. • At each energy, a minimum width for varying parameter a. • a0, corresponding to the minimum width, is close to a • common integer at different energies. CPOD2011, Wuhan, China

  19. ►Energy dependence of minimum width: Dyn. charge fluc. pt corr. Norm. pt corr. • They are equally good point like behavior at 4 measured • incident energies. • After the scale, they are size independent within error. The trivial size effects are absorbed to the power a0. CPOD2011, Wuhan, China

  20. 4. Summary and outlook • Possible finite-size behavior of critical related observables in heavy ion collisions is discussed. • 2.The fixed point method is suggested in searching critical point and nearby phase boundary. • 3. The method has been applied to the related observables at RHIC. No fixed point has been found at measured observables and energies. Scaled observables are size independent within experimental errors. • 4. It should be examined by all coming RHIC/BES data. • Influence of non-thermal effects. • Model investigations. • All is ongoing for a final conclusion. Thanks! CPOD2011, Wuhan, China

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