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

Locating critical point of QCD phase transitionLocating critical point of QCD phase transitionby finite-size scalingby finite-size scaling

Chen LizhuChen Lizhu11, X.S. Chen, X.S. Chen22, , Wu YuanfangWu Yuanfang11

1 1 IOPP, Huazhong Normal University, Wuhan, ChinaIOPP, Huazhong Normal University, Wuhan, China2 2 ITP, Chinese Academy of Sciences, Beijing 100190, ChinaITP, Chinese Academy of Sciences, Beijing 100190, China

1. Motivation1. Motivation2. How to locate critical point by 2. How to locate critical point by finite-size scaling finite-size scaling 3. Critical behaviour of p3. Critical behaviour of ptt corr. at corr. at

RHICRHIC 4. Discussions and suggestions4. Discussions and suggestions5. Summary5. Summary

Thanks to: Liu Lianshou, Nu Xu, Li Liangsheng, Hou Defu, Li Thanks to: Liu Lianshou, Nu Xu, Li Liangsheng, Hou Defu, Li JiarongJiarong

Thanks to: Liu Lianshou, Nu Xu, Li Liangsheng, Hou Defu, Li Thanks to: Liu Lianshou, Nu Xu, Li Liangsheng, Hou Defu, Li JiarongJiarong

arXiv:0904:1040, proceedings of CPOD’09 at BNL

1. Motivation 1. Motivation

★ ★ QCD phase transitionsQCD phase transitions ★ ★ QCD phase transitionsQCD phase transitions

• DeconfinementDeconfinement• Chiral symmetry restorationChiral symmetry restoration

Open question:Open question: ,T T or T T

0, 0BT

0, 0BT : crossover: crossover

: first order: first order

Lattice-QCD predicts:Lattice-QCD predicts:

Karsch F., Lecture Notes Phys. 583, 209(2002); Karsch F. , Lutgemeier M., 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, Nucl. Phys. B550, 449(1999). Y. Aoki et al, arXiv:0903.4155; A.Bazavov et al, arXiv:0903.4379.arXiv:0903.4379.

→ → critical point.critical point.

T●

Critical endpointCritical endpointCritical endpointCritical endpoint

●●????

22

2009.8.112009.8.11 Weihai’09Weihai’09 33

--- See talk by Pisarski, p.18.

(pure guesswork)

一、关于相图一、关于相图

From Xu Mingmei From Xu Mingmei （许明（许明梅）梅）


2009.8.112009.8.11 Weihai’09Weihai’09 44

The number of CP: 0, 1, 2,The number of CP: 0, 1, 2,

Model calculations suggest that there could be Model calculations suggest that there could be none, one or even twonone, one or even two critical critical points depending on the parameters in the QCD Lagrangian.points depending on the parameters in the QCD Lagrangian.



See talks by B.-J. Schaefer, J. Kapusta, K. Fukushima, Koch, …See talks by B.-J. Schaefer, J. Kapusta, K. Fukushima, Koch, …

有效的场论模型：1. NJL model2. Linear sigma model Linear sigma model with quarks or nucleons3. PNJL PNJL with Polyakov loop dynamics4. Quark-meson model Quark-meson model with Polyakov loop dynamics

1. Motivation 1. Motivation

★ ★ Current status of relativistic heavy ion experiments:Current status of relativistic heavy ion experiments: ★ ★ Current status of relativistic heavy ion experiments:Current status of relativistic heavy ion experiments:

RHIC, SPS & FAIRRHIC, SPS & FAIR

Question:Question:

How to locate the critical point from observable?How to locate the critical point from observable?

critical pointcritical point critical pointcritical point

55

★ ★ Finite size of the formed matter Finite size of the formed matter ★ ★ Finite size of the formed matter Finite size of the formed matter

2009.6.102009.6.10 66

Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca, Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca, Maged Elhajal, and Frederic Mila, PRL91, 177202(2003).Maged Elhajal, and Frederic Mila, PRL91, 177202(2003).

Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca, Cedric Weber, Luca Capriotti, Gregoire Misguich, Federico Becca, Maged Elhajal, and Frederic Mila, PRL91, 177202(2003).Maged Elhajal, and Frederic Mila, PRL91, 177202(2003).

the finite-size effect is negligible!the finite-size effect is negligible! the finite-size effect is negligible!the finite-size effect is negligible! ,6

L WhenWhenWhenWhen

fm3~2: due to critical slowing down due to critical slowing down ！！ due to critical slowing down due to critical slowing down ！！

Boris Berdnikov and Krishna Rajagopal, PRD61, 105017(2000).Boris Berdnikov and Krishna Rajagopal, PRD61, 105017(2000).Boris Berdnikov and Krishna Rajagopal, PRD61, 105017(2000).Boris Berdnikov and Krishna Rajagopal, PRD61, 105017(2000).

fmL 18 ( : ~ 12 )HBT L fmSo whenSo when So whenSo when

The finite-size effect has to be taken into account at RHIC!The finite-size effect has to be taken into account at RHIC! The finite-size effect has to be taken into account at RHIC!The finite-size effect has to be taken into account at RHIC!

M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, 114028(2000).M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, 114028(2000).M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, 114028(2000).M. Stephanov, K. Rajagopal, E. Shuyak, PRD60, 114028(2000).

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★ ★ effects of finite sizeeffects of finite size ★ ★ effects of finite sizeeffects of finite size

☞☞ The position of the maximum The position of the maximum changeschanges with size and with size and deviatesdeviates from the true critical point ! from the true critical point !

Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang.

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).

● ● Infinite system: Infinite system: at critical point,at critical point, correlation length correlation length ξξ →→ ∞. ∞. observable observable ● ● Finite system: Finite system:

observable observable →→ finite finite && has a maximum has a maximum →→ non-monotonous behaviornon-monotonous behavior

2D-Ising2D-Ising2D-Ising2D-Ising

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☞☞The finite-size scaling of the observable.The finite-size scaling of the observable.

2. How to locate critical point by finite-size scaling 2. How to locate critical point by finite-size scaling

1

( , ) ( )QQ s L L F L

Finite-size scaling form: Finite-size scaling form: Finite-size scaling form: Finite-size scaling form: 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. X. S. Chen, V. Dohm, and A. L. Talapov, Talapov, Physica A232, 375 (1996);Physica A232, 375 (1996);X. S. Chen, V. Dohm, and N. Schultka, X. S. Chen, V. Dohm, and N. Schultka, PRL, 77, 3641(1996).PRL, 77, 3641(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. X. S. Chen, V. Dohm, and A. L. Talapov, Talapov, Physica A232, 375 (1996);Physica A232, 375 (1996);X. S. Chen, V. Dohm, and N. Schultka, X. S. Chen, V. Dohm, and N. Schultka, PRL, 77, 3641(1996).PRL, 77, 3641(1996).

: reduced variable, : reduced variable, likelike T T, or , or hh in thermal-dynamic in thermal-dynamic system.system.

: critical exponent of : critical exponent of

1

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

c

c

s s

s

: critical exponent of correlation length : critical exponent of correlation length 0

1L

( , )Q s L

in the vicinity of √sin the vicinity of √sc c ,,in the vicinity of √sin the vicinity of √sc c ,,

99

★ ★ Fixed point:Fixed point: ★ ★ Fixed point:Fixed point:

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

, 0.cs s At critical At critical point ,point ,

Scaling Scaling function:function:

becomes a becomes a constant.constant.

It behaves as a It behaves as a fixed pointfixed point in,in,

( , ) .Q s L L vs s

10,L Scaled variable:Scaled variable: is independent of size L.is independent of size L.

Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng and X.S. Chen; Chen Lizhu, Li Liangsheng, X.S. Chen and Wu Yuanfang.Li Liangsheng, X.S. Chen and Wu Yuanfang.


Critical characteristics of FSSCritical characteristics of FSS1

FSS: ( , ) ( )QQ s L L F L


Fixed pointFixed pointFixed pointFixed point

1010

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★ ★ If If λλ=0, fixed point can be directly =0, fixed point can be directly obtained.obtained.

★ ★ If If λλ=0, fixed point can be directly =0, fixed point can be directly obtained.obtained.

4

221

3

MU

M 0,U

Like Binder cumulant ratios,Like Binder cumulant ratios,

and fluc. of mean cluster and fluc. of mean cluster size,size,



4

4 223

mean

mean

S SF

S S


2D-Ising2D-Ising2D-Ising2D-Ising2D-Ising2D-Ising2D-Ising2D-Ising

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☞ ☞ Therefore, Therefore, the observable at diff. sizes can be usedthe observable at diff. sizes can be used to locate the position of critical point .to locate the position of critical point .

0,cs a Fixed point

★ ★ If If λλ ‡ 0 ‡ 0 ★ ★ If If λλ ‡ 0 ‡ 0

0( , ) aQ s L L( , ) aQ s L L

is a parameteris a parameter is a parameteris a parametera


Taking logarithm in both sides of FSS,Taking logarithm in both sides of FSS,

0

ln ( , ) ln ln ( )QQ s L L F L

At critical pointAt critical point , ,

ln ( , ) lncQ s L L C

★ ★ Straight line behavior:Straight line behavior: ★ ★ Straight line behavior:Straight line behavior:

is linear function of at given !is linear function of at given !ln ( , )cQ s L ln L cs

1

FSS: ( , ) ( )QQ s L L F L

☞ ☞ Fixed-point and best straight-line behaviorFixed-point and best straight-line behavior☞ ☞ Fixed-point and best straight-line behaviorFixed-point and best straight-line behavior

1313

,cs

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STARSTAR

3. 3. Critical behaviour of pCritical behaviour of ptt corr. at RHIC corr. at RHIC

, ,1 , 1

1

1( , )

( 1)

k k

event

k

N N

t i t t j tNi j i j

Nevent k k

p p p p

P s LN N N

Au + Au collisions Au + Au collisions

at at 9 L9 L (centralities) (centralities)

for each offor each of 4 4√s√s . .

Au + Au collisions Au + Au collisions

at at 9 L9 L (centralities) (centralities)

for each offor each of 4 4√s√s . .

★ ★ pptt corr. as one of critical related observable corr. as one of critical related observable ★ ★ pptt corr. as one of critical related observable corr. as one of critical related observable

H. Heiselberg, Phys. Rept. 351, 161(2001);M. Stephanov, J. of Phys. 27, 144(2005).

H. Heiselberg, Phys. Rept. 351, 161(2001);M. Stephanov, J. of Phys. 27, 144(2005).

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Number of Participants

Impact Parameter

★ ★ System size:System size:

★ ★ System size:System size:

Initial mean size: Initial mean size: Initial mean size: Initial mean size: partN

2

part

A

NL

NScaled initial mean Scaled initial mean

size : size :

Scaled initial mean Scaled initial mean size : size :

System size at transition:System size at transition:System size at transition:System size at transition:' 1 , 0.L cL

☞☞Whether Whether L’L’, or , or LL is taken, the critical exponents will be is taken, the critical exponents will be different, but the critical point is the same!different, but the critical point is the same!

☞☞Whether Whether L’L’, or , or LL is taken, the critical exponents will be is taken, the critical exponents will be different, but the critical point is the same!different, but the critical point is the same!

Initial Initial Near transition Near transition Transition Transition

expansionexpansion critical slowing downcritical slowing down

zt

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★ ★ √√s dependence of ps dependence of ptt corr. at different system sizes corr. at different system sizes ★ ★ √√s dependence of ps dependence of ptt corr. at different system sizes corr. at different system sizes

FSS is valid forFSS is valid for L > 10 fm.L > 10 fm.

L: 10 to 2 fm for most central and peripheral coll.L: 10 to 2 fm for most central and peripheral coll.

L: 10 to 5 fm for 6 more central coll.L: 10 to 5 fm for 6 more central coll.

So 6 more central collisions are chosen !So 6 more central collisions are chosen !

FSS is valid forFSS is valid for L > 10 fm.L > 10 fm.

L: 10 to 2 fm for most central and peripheral coll.L: 10 to 2 fm for most central and peripheral coll.

L: 10 to 5 fm for 6 more central coll.L: 10 to 5 fm for 6 more central coll.

So 6 more central collisions are chosen !So 6 more central collisions are chosen !

B. Klein & J. Braun, arXiv:0710.1161.

B. Klein & J. Braun, arXiv:0710.1161.

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★ ★ Fixed-point behavior of pFixed-point behavior of ptt correlation. correlation. ★ ★ Fixed-point behavior of pFixed-point behavior of ptt correlation. correlation.

Ratios of critical Ratios of critical exponents : exponents :

Ratios of critical Ratios of critical exponents : exponents :

62, 200 GeVs Two fixed-point around:Two fixed-point around: Two fixed-point around:Two fixed-point around:

222

1

1, , ,

LNa a a

j jjL

P s L L P s L L P s L LN

At given √s, At given √s, the width of the width of : : At given √s, At given √s, the width of the width of : : , aP s L L

☞ ☞ Precise position of the minimum can be obtained by Precise position of the minimum can be obtained by additional collisions around 62 and 200 GeV.additional collisions around 62 and 200 GeV.

☞ ☞ Precise position of the minimum can be obtained by Precise position of the minimum can be obtained by additional collisions around 62 and 200 GeV.additional collisions around 62 and 200 GeV.

0,1 0,2, 2.09, 2.08a a

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★ ★ Straight-line behavior of pStraight-line behavior of ptt correlation. correlation. ★ ★ Straight-line behavior of pStraight-line behavior of ptt correlation. correlation.

A parabola fit for data at give A parabola fit for data at give √s√s,,A parabola fit for data at give A parabola fit for data at give √s√s,, 22 1 0(ln ) lnc L c L c

√s(GeV) 20 62 130 200

2c

1c

1.86 0.93 0.6 0.09 1.56 0.41 0.77 0.1

3.9 0.89 2.59 0.09 3.43 0.41 2.74 0.1

Parameters of parabola fitsParameters of parabola fits Parameters of parabola fitsParameters of parabola fits

☞ ☞ better straight-line fit better straight-line fit atat√√s =62 and 200 GeVs =62 and 200 GeV

☞ ☞ better straight-line fit better straight-line fit atat√√s =62 and 200 GeVs =62 and 200 GeV

☞ ☞ the slopes of lines are the slopes of lines are ：：

☞ ☞ the slopes of lines are the slopes of lines are ：： 0,1 2.09, and 2.08 respectively,a

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★ ★ Critical behavior of normalized pCritical behavior of normalized ptt correlation. correlation. ★ ★ Critical behavior of normalized pCritical behavior of normalized ptt correlation. correlation.

,1 1

1

1

( , )( , ) ,

event k

event

N N

t ik ik

t t Nt

kk

pNP s L

R s L pp

N

☞☞Same fixed-point and best straight-line behavior !Same fixed-point and best straight-line behavior !☞☞Same fixed-point and best straight-line behavior !Same fixed-point and best straight-line behavior !

2009.8.112009.8.11 Weihai’09Weihai’09 20202020

4. 4. Discussions and suggestionsDiscussions and suggestions..

1.1.√√sscc ~ 62, and 200 ~ 62, and 200 GeV, GeV,

are both within theare both within the range estimated byrange estimated by lattice-QCD.lattice-QCD.

☻ ☻ SupportsSupports☻ ☻ SupportsSupports

2. The similar ratios of critical exponents at two 2. The similar ratios of critical exponents at two critical critical points is consistent with current theoretical points is consistent with current theoretical estimation.estimation.Jorge Garca, Julio A. Gonzalo, Physica A 326,464(2003).

Jens Braun1 and Bertram Klein, Phys. Rev. D77, 096008(2008).

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, andK.K. Szabo, Phys. Lett. B643, 46(2006); F. Karsch, PoS CFRNC2007.

* Roy A. Lacey, et al, PRL , 98, 092301(2007).

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4. 4. Discussions and suggestions Discussions and suggestions

(II)(II).. ☻ ☻ UncertaintiesUncertainties☻ ☻ UncertaintiesUncertainties

It is absent at moment.It is absent at moment.

1.1. Quality of data Quality of data

Poor statistics for data at 20GeV and 130GeV.Poor statistics for data at 20GeV and 130GeV.

2. Error of the system size 2. Error of the system size

2009.8.112009.8.11 Weihai’09Weihai’09 22222222


(II)(II).. ☻ ☻ Confirmation of CEPConfirmation of CEP☻ ☻ Confirmation of CEPConfirmation of CEP

1. Boundary of phase diagram1. Boundary of phase diagram

Y.Aoki, G. Endrodi, Z. Fodor, S. D. katz, & K.K. Szabo, Nature 443,05120(2006)

11stst-order: the finite size scaling of susceptibility is determined by -order: the finite size scaling of susceptibility is determined by the geometric dimension, the height and width are the geometric dimension, the height and width are proportional to volume V and 1/V, respectively. proportional to volume V and 1/V, respectively.

22ndnd-order : the singular behavior is given by some power of V, -order : the singular behavior is given by some power of V, defined by the critical exponents. defined by the critical exponents.

Crossover: There would be no singular behavior and the susceptibilityCrossover: There would be no singular behavior and the susceptibility peak would not get sharper when increasing the volume V;peak would not get sharper when increasing the volume V; instead, its height and width will be V independent for large instead, its height and width will be V independent for large volumes.volumes.

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(II)(II)..

RHIC energy scanRHIC energy scan

2. Other measurements2. Other measurements

and the third order moments of conserved charges.and the third order moments of conserved charges.

more data onmore data on,,

3. More collisions at RHIC 3. More collisions at RHIC

( ), ( ) , / ,

inp C n Kt i in

Additional collisions around √s = 62 and 200 GeV.Additional collisions around √s = 62 and 200 GeV.It is advantage of RHIC.It is advantage of RHIC.

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5. Summary. 5. Summary.

1.1. It is pointed out that critical related It is pointed out that critical related observable should observable should

follow the finite-size scaling. follow the finite-size scaling.

2. The method of finding and locating critical 2. The method of finding and locating critical point is point is

established by critical characteristics of finite-established by critical characteristics of finite-size scaling. size scaling.

3. As an application, critical behavior in current 3. As an application, critical behavior in current available data from STAR are demonstrated. available data from STAR are demonstrated.

4. Confirmation of CEP from QCD phase diagram, 4. Confirmation of CEP from QCD phase diagram, more and better data on other critical related more and better data on other critical related observable at current collision energies, and a observable at current collision energies, and a few additional collisions are suggested.few additional collisions are suggested.

Documents

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