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

Nov.8, 2011Nov.8, 2011 CPOD2011, Wuhan, ChinaCPOD2011, Wuhan, China 11

Chen LizhuChen Lizhu11, Shao Ming, Shao Ming22, X.S. Chen, X.S. Chen33, Wu Yuanfang, Wu Yuanfang11

11IOPP, Central China Normal University (CCNU), Wuhan, ChinaIOPP, Central China Normal University (CCNU), Wuhan, China22University of Science and technology of China, Hefei, Anhui , ChinaUniversity of Science and technology of China, Hefei, Anhui , China

33ITP, Chinese Academy of Science, Beijing, ChinaITP, 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.

1. Motivation 1. Motivation


CCPP

Quark Gluon Plasma(QGP) phase

★ ★ Current status : Current status : ★ ★ Current status : Current status :

C. Blume’ talk at CPOD2011, http://conf.ccnu.edu.cn/~cpod2011/B. Mohanty’s talk at QM’11, http://qm2011.in2p3.fr/node/629.

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!

★ ★ Possible reasons : Possible reasons : ★ ★ Possible reasons : 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.

★ ★ Influence of finite evolution time Influence of finite evolution time ★ ★ Influence of finite evolution time 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.


So the observables, which are more sensitive to correlation length, are recommended, e.g., the higher cumulants!

0.5-1fm 2-3fm

B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017.

★ ★ The influences of finite size : The influences of finite size : ★ ★ The influences of finite size : 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).

★ ★ Optimistic sides of finite size: Optimistic sides of finite size: ★ ★ Optimistic sides of finite size: 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).

A location 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/


It is possible to test finite-size behavior of

QCD phase transition in relativistic heavy ion collisions !

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.


11stst-order: finite-size scaling function, and scaling exponent is -order: finite-size scaling function, and scaling exponent is determined by spatial dimension (determined by spatial dimension (integerinteger).).

22ndnd-order: finite-size scaling function, and scaling exponent -order: finite-size scaling function, and scaling exponent ( ( , non-integer, non-integer), i.e.,), i.e.,

Crossover: size independent.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.

★ ★ Finite-size behavior: Finite-size behavior: ★ ★ Finite-size behavior: Finite-size behavior:

2. How to see the finite size behavior?


P. Braun-Munzinger and J. Stachel, arXiv:1101.3167.

1) Thermal equilibrium, or local thermal equilibrium.

2) Transition line is close to freeze-out curve.

3) Survival of critical fluctuations in the final state.

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).

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

★ ★ Applicability: Applicability: ★ ★ Applicability: Applicability:


► ► Possible form of Possible form of finite-size scaling in heavy ion collisions: finite-size scaling in heavy ion collisions: ► ► Possible form of Possible form of finite-size scaling in heavy ion collisions: finite-size scaling in heavy ion collisions:

1

( , ) ( )QQ s L L F L

: reduced : reduced √s √s likelike T T, or , or hh in in thermodynamicsthermodynamics system, system, √s√scc critical one critical one..

c

c

s s

s

: critical exponents of Q, and correlation length. : critical exponents of Q, and correlation length.

1

( )QF L : scaling function with scaled variable, : scaling function with scaled variable, 1L

J. Cleymans, H. Oeschler, K. Redlich, S. Wheaton, PRC73, (2006)034905.

T

L' 1 , 0.L cL If correspond size:

& temperature:

then write similarly,

M. E. Fisher, in Critical Phenomena, M. E. Fisher, in Critical Phenomena, (Academic, New York, 1971).(Academic, New York, 1971).E. Brezin, J. Phys. (Paris) 43, 15 (1982).E. Brezin, J. Phys. (Paris) 43, 15 (1982).X. S. Chen, V. Dohm, and A. L. Talapov, X. S. Chen, V. Dohm, and A. L. Talapov, Physica A232, 375 (1996).Physica A232, 375 (1996).

M. E. Fisher, in Critical Phenomena, M. E. Fisher, in Critical Phenomena, (Academic, New York, 1971).(Academic, New York, 1971).E. Brezin, J. Phys. (Paris) 43, 15 (1982).E. Brezin, J. Phys. (Paris) 43, 15 (1982).X. S. Chen, V. Dohm, and A. L. Talapov, X. S. Chen, V. Dohm, and A. L. Talapov, Physica A232, 375 (1996).Physica A232, 375 (1996).

(0) ( , ) ,Q cF Q s L L

, 0.cs s At critical energy At critical energy ,,

Scaling functionScaling function::

is a constant. It behaves as a is a constant. It behaves as a

fixed pointfixed point in,in, ( , ) .Q s L L vs s

2D-Ising2D-Ising2D-Ising2D-Ising

Fixed pointFixed pointFixed pointFixed point

1

FSS: ( , ) ( )QQ s L L F L ►► Fixed point and straight line: Fixed point and straight line: ►► Fixed point and straight line: Fixed point and straight line:


In the case, taking the logarithm, In the case, taking the logarithm,

is a is a straight linestraight line. It deviates from line, . It deviates from line,

when:when:

X. S. Chen’s talk at 18th CBM, http://hepd.ep.tsinghua.edu.cn/cbm2011/.

2D-Ising2D-Ising2D-Ising2D-IsingExample:


Fixed point

( , ) aQ s L L TuneTune TuneTune a

Plot energy dependence of the Plot energy dependence of the observable at different observable at different system sizes to see if we can find the scaled parameter asystem sizes to see if we can find the scaled parameter a0,0,

which makes all size points intersect.which makes all size points intersect.

►► How to find How to find fixed point from observable? fixed point from observable? ►► How to find How to find fixed point from observable? fixed point from observable?


The width of all size points at a given √s :

An experimental point:

}

2D-Ising

►► Quantify the behavior of Quantify the behavior of point-like: point-like: ►► Quantify the behavior of Quantify the behavior of point-like: point-like:

At CP:

For


►►Energy dependency of the minimum widthEnergy dependency of the minimum width ::►►Energy dependency of the minimum widthEnergy dependency of the minimum width ::

1

√s

No fixed pointinteger a0

Non-integer a0Integer a0

Crossover regionCP1st order PT

size independent

2D-Ising2D-Ising2D-Ising2D-Ising

Fixed pointFixed pointFixed pointFixed point

:L( , )Q s L L

s


►►Related observables:Related observables:►►Related observables:Related observables:

3. Application to the related observables at RHIC

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.


►►Centrality dependence of 4Centrality dependence of 4thth and 6th cumulant ratios of net-proton dis. and 6th cumulant ratios of net-proton dis. ►►Centrality dependence of 4Centrality dependence of 4thth and 6th cumulant ratios of net-proton dis. 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.

http://www.bnl.gov/fcrworkshop/


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.

STAR, Phys. Rev. C79, 024906(2009); STAR,Phys. Rev. C72, 044902(2005).

►►Energy dependence of dynamical electric charge fluc. and pEnergy dependence of dynamical electric charge fluc. and p tt corr. corr.►►Energy dependence of dynamical electric charge fluc. and pEnergy dependence of dynamical electric charge fluc. and p tt corr. corr.


►►The width with varying parameter a:The width with varying parameter a:►►The width with varying parameter a:The width with varying parameter a:

Dyn. charge fluc. Norm. pt corr.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.


►►Energy dependence of minimum width:Energy dependence of minimum width:►►Energy dependence of minimum width:Energy dependence of minimum width:

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.

Dyn. charge fluc. Norm. pt corr.pt corr.

Nov.8, 2011Nov.8, 2011 CPOD2011, Wuhan, ChinaCPOD2011, Wuhan, China

4. Summary and outlook4. Summary and outlook

1.1. Possible finite-size behavior of critical related observables in Possible finite-size behavior of critical related observables in heavy ion collisions is discussed.heavy ion collisions is discussed.

2.The fixed point method is suggested in searching critical 2.The fixed point method is suggested in searching critical point and nearby phase boundary. point and nearby phase boundary.

3. The method has been applied to the related observables at 3. The method has been applied to the related observables at RHIC. No fixed point has been found at measured RHIC. No fixed point has been found at measured observables and energies. Scaled observables are size observables and energies. Scaled observables are size independent within experimental errors. independent within experimental errors.

4. It should be examined by all coming RHIC/BES data. 4. It should be examined by all coming RHIC/BES data. Influence of non-thermal effects. Influence of non-thermal effects. Model investigations.Model investigations.

All is ongoing for a final conclusion.All is ongoing for a final conclusion.

2020

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

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