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. - PowerPoint PPT Presentation
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Locating critical point of QCD phase Locating critical point of QCD phase transition transition by finite-size scaling by finite-size scaling Chen Lizhu Chen Lizhu 1 , X. S. Chen , X. S. Chen 2 , , Wu Yuanfang Wu Yuanfang 1 1 1 IOPP, Huazhong Normal University, Wuhan, China IOPP, Huazhong Normal University, Wuhan, China 2 2 ITP, Chinese Academy of Sciences, Beijing 100190, China ITP, Chinese Academy of Sciences, Beijing 100190, China 1. Motivation 1. Motivation 2. Finite-size scaling form and 2. Finite-size scaling form and how to locate critical how to locate critical point by it point by it 3. Critical behaviour of p 3. Critical behaviour of p t corr. at RHIC corr. at RHIC 4. Discussions and suggestions 4. Discussions and suggestions nks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou De nks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou De
Transcript
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. Finite-size scaling form and 2. Finite-size scaling form and how to locate critical point by ithow to locate critical point by it3. Critical behaviour of p3. Critical behaviour of ptt corr. at RHIC corr. at RHIC 4. Discussions and suggestions4. Discussions and suggestions5. Summary5. Summary
Thanks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou DefuThanks to: Prof. Liu Lianshou, Dr. Li Liangshen and Prof. Hou Defu
1. Motivation (II)1. Motivation (II) ★ ★ Current status of relativistic heavy ion experiments:Current status of relativistic heavy ion experiments:RHICRHIC at BNL, the at BNL, the SPSSPS at CERN, and future at CERN, and future FAIRFAIR at GSI at GSIare aimed to find critical point.are aimed to find critical point.
Question:Question: How to locate the critical point from observable?How to locate the critical point from observable?
★ ★ Limited size of formed matter Limited size of formed matter
☞ ☞ The effect of finite size is not negligible!The effect of finite size is not negligible!
1. Motivation 1. Motivation (III)(III)★ ★ Non-monotonous behavior, and why it is not enoughNon-monotonous behavior, and why it is not enough
At critical point,At critical point, ● ● in infinite system: in infinite system: correlation lengthcorrelation lengthξξ → ∞. → ∞. ●●in finite system: in finite system: finite and have a maximum, finite and have a maximum, i.e., non-monotonous behaviori.e., non-monotonous behavior
☞☞However, the position of the maximum of non-However, the position of the maximum of non- monotonous behavior of observable changes withmonotonous behavior of observable changes with system size and deviates from the true critical point.system size and deviates from the true critical point.
1. Motivation (IV)1. Motivation (IV)
☞☞The absence of non-The absence of non- monotonous behavior monotonous behavior does not mean no CPOD.does not mean no CPOD.
Order parameter in 2D-IsingOrder parameter in 2D-Ising
2.27cT
☞☞Non-monotonous behavior Non-monotonous behavior is not always associated is not always associated with CPOD.with CPOD.
Specific heat in 1D- IsingSpecific heat in 1D- Ising
☞☞The reliable criterion of critical behavior isThe reliable criterion of critical behavior is finite-size scaling of the observable.finite-size scaling of the observable.
2. Finite-size scaling form (I)2. Finite-size scaling form (I)
( , ) ( )QQ s L L F L
: reduced variable, : reduced variable, likelike T T, or , or hh in thermal-dynamic in thermal-dynamic system. system.
: critical exponents: critical exponents( )F L
: scaling function with scaled variable, : scaling function with scaled variable,
c
c
s ss
: critical exponent of correlation length, : critical exponent of correlation length, 0
A observable in relativistic heavy ion collision is a function A observable in relativistic heavy ion collision is a function of incident energy of incident energy √s √s and system size and system size L, √sL, √s like T, or h. like T, or h.Finite-size scaling Finite-size scaling form:form:
L
★ ★ Fixed point:Fixed point:
(0) ( , ) ,F Q s L L
0. At critical At critical point ,point ,
Scaling Scaling function:function:
becomesbecomes a a constant.constant.
It behaves as a It behaves as a fixed pointfixed point,,where all curves converge to.where all curves converge to.
( , ) .Q s L L vs L
0,L Scaled variable:Scaled variable: is independent of size L.is independent of size L.
In the plot:In the plot:
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.
☞ ☞ Reversely, Reversely, if if √s√scc is unknown, the observable at diff. is unknown, the observable at diff. L L can help us to find the position of critical point .can help us to find the position of critical point .
( , ) aQ s L L 0( , ) aQ s L L 0a
Fixed point
★ ★ If If λλ‡ 0 ‡ 0
is a justable parameteris a justable parametera
is linear function of !is linear function of !
Taking logarithm in both sides of FSS,Taking logarithm in both sides of FSS,
0
ln ( , ) ln ln ( )Q s L L F L
At critical pointAt critical point , ,
ln ( , ) lnQ s L L C
ln ( , )Q s L ln L
★ ★ Straight line behavior:Straight line behavior:
☞☞The critical point can also be found from the system sizeThe critical point can also be found from the system size dependence of the observable.dependence of the observable.
3. 3. Critical behaviour of pCritical behaviour of ptt corr. at RHIC corr. at RHIC
STAR Coll.
, ,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 pP s L
N N N
Au + Au collisions atAu + Au collisions at 4 incident energies:4 incident energies: 20, 62, 130, 200 GeV20, 62, 130, 200 GeV and 9 centralities (sizes).and 9 centralities (sizes).
( , ) ( )PP s L L F L
If √sc is in the RHIC energy, If √sc is in the RHIC energy, its scaling form should be:its scaling form should be:
★ ★ Pt corr. as one of critical related observablePt corr. as one of critical related observableH. Heiselberg, Phys. Rept. 351, 161(2001);M. Stephanov, J. of Phys. 27, 144(2005).
Number of Participants
Impact Parameter
★ ★ System size:System size:
Initial mean size: Initial mean size: partN
2part
A
NL
NScaled mean size of initial system: Scaled mean size of initial system:
System size at transition should be a monotonically System size at transition should be a monotonically increasing function of :increasing function of :L ' 1 , 0.L cL
It will modifies the scaling exponents, but not the position It will modifies the scaling exponents, but not the position of critical point. So we take of critical point. So we take LL instead of instead of L’L’ in the following. in the following.
★ ★ System size dependence of pSystem size dependence of ptt correlation. correlation.
1.1. Change the centrality Change the centrality dependence of pdependence of pt t corr. corr. at diff. incident energies at diff. incident energies to the collision energy to the collision energy dependence at diff. sizes.dependence at diff. sizes.
3. The influence of finite size is obvious.3. The influence of finite size is obvious.
2. Choose 6 centralities at 2. Choose 6 centralities at mid-central and central mid-central and central collisions to do the analysis.collisions to do the analysis.
★ ★ 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,, 2
2 1 0(ln ) lnc L c L c
√s(GeV) 20 62 130 200
2c
1c1.86 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
☞ ☞ the better straight-line the better straight-line behavior happen to be behavior happen to be atat√√s =62 and 200 GeVs =62 and 200 GeV
☞ ☞ the slopes of lines are the slopes of lines are
obtained by the fixed points.obtained by the fixed points.0,1 2.09, and 2.08 respectively,a
★ ★ Same analysis for normalized pSame analysis for normalized ptt correlation. correlation.
,1 1
1
( , )( , ) ,
event k
event
N N
t ik i
t t Nt
kk
pP s LR s L pp N
Two fixed-point behavior around: Two fixed-point behavior around: 62, 200 GeVs With the ratios of critical exponents : With the ratios of critical exponents : 0,1 0,2, 0.55a a
4. 4. Discussions and suggestionsDiscussions and suggestions..
1.1. √√sscc =62, and 200 GeV, are =62, and 200 GeV, are both in the range estimated both in the range estimated by lattice-QCD. They mayby lattice-QCD. They may imply that deconfinement imply that deconfinement and chiral symmetry and chiral symmetry restoration occur at diff. restoration occur at diff. TcTc. .
☻ ☻ DiscussionsDiscussions
2. The similar ratios of critical exponents at two critical points is2. The similar ratios of critical exponents at two critical points is consistent with current theoretical estimation, which shows that consistent with current theoretical estimation, which shows that all critical exponents in 3D-Ising are very close to that of 3D-O(4).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).
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.
4. 4. Discussions and suggestions Discussions and suggestions (II)(II).. ☻ ☻ SuggestionsSuggestions
2. To determine precisely the critical incident energy and2. To determine precisely the critical incident energy and critical exponents, additional collisions around critical exponents, additional collisions around √√s =62 and 200 GeV are required.s =62 and 200 GeV are required.
1.1. More data onMore data on : :
greatly helpful in confirming the results. So, thegreatly helpful in confirming the results. So, the√s√s and and centrality dependence of those observable are called centrality dependence of those observable are called for.for.
5. Summary. 5. Summary. 1.1. It is pointed out that in relativistic heavy ion collisions, critical It is pointed out that in relativistic heavy ion collisions, critical
related observable in the vicinity of critical point should follow related observable in the vicinity of critical point should follow the finite-size scaling. the finite-size scaling. 2. The method of finding and locating critical point is established by 2. The method of finding and locating critical point is established by
finite-size scaling and its critical characteristics, in particular, finite-size scaling and its critical characteristics, in particular, fixed point and straight line behavior. fixed point and straight line behavior.
3. As an application, the data of p3. As an application, the data of ptt correlation from RHIC/STAR are correlation from RHIC/STAR are analyzed. Two fixed-point and straight-line behavior are both analyzed. Two fixed-point and straight-line behavior are both observed aroundobserved around√√s =62 and 200 GeV. This demonstrates two s =62 and 200 GeV. This demonstrates two critical points of QCD phase transition at RHIC.critical points of QCD phase transition at RHIC.
4. To precisely determine the critical endpoints and critical 4. To precisely determine the critical endpoints and critical exponents, more and better data on other critical related exponents, more and better data on other critical related observable at current collision energies, and a few additional observable at current collision energies, and a few additional collisions around √s = 62 and 200 GeV are called for.collisions around √s = 62 and 200 GeV are called for.
