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Xiaofeng Luo 1 / 31INT Workshop 16-3,Seattle, Sept. 19 -Oct. 14, 2016

Fluctuations of Conserved Quantities in High-EnergyNuclear Collisions at RHIC

Xiaofeng Luo

Central China Normal University

Search for the QCD Critical Point -

Oct. 03, 2016


Outline

Ø Introduction

Ø Data Analysis : Net-P, Net-Q, Net-K1) Centrality determination and Volume Fluctuations.2) Efficiency correction, particle identification.3) Error Estimation.

Ø Results and Discussion: CP Search1) Cumulants and Their Ratios: up to 4th order2) Centrality, rapidity and energy dependence.3) Deuteron Formation, UrQMD, JAM, NJL etc.4) Correlation functions from data and model.

Ø Summary and Outlook


QCD Phase Diagram (Conjectured)

K. Fukushima and T. Hatsuda, Rept. Prog. Phys. 74, 014001(2011);; arXiv: 1005.4814

QCD Phase Structure : Emergent properties of the strong interaction.

Search for the QCD Critical Point in HIC:Challenges:1. finite size/time.2. Contribution of non-CP physics background.3. signal survive after dynamical expansion.


Critical Point and Critical Phenomena

First CP is discovered in 1869 for CO2by Andrews.

Can we discovery the Critical Pointof Quark Matter ? (Put a permanentmark in the QCD phase diagram in text book. )

Critical Phenomena :Ø Density fluctuations and cluster formations.Ø Divergence of Correlation length (ξ).

Susceptibilities (χ), heat capacity (CV)，Compressibility (𝜅) etc.Critical opalescence.

Ø Universality and critical exponents determinedby the symmetry and dimensions of underlyingsystem.

Ø Finite Size and Finite time effects. Tc ~ Trillion (1012 ) °C

2D-Ising model simulation from ISNB4-563-02435-X C33421

Tc = 31°C


Lattice QCD：1): Fodor&Katz, JHEP 0404,050 (2004).(μEB, TE )= (360, 162) MeV (Reweighting)

2): Gavai&Gupta, NPA904, 883c (2013)(μEB, TE )= (279, 155) MeV (Taylor Expansion)

3): F. Karsch (μEB/ TE >2, CPOD2016)

Location of CEP: Theoretical Prediction

DSE:1): Y. X. Liu, et al., PRD90, 076006 (2014).

(μEB, TE ) = (372, 129 ) MeV

2): Hong-shi Zong et al., JHEP 07, 014 (2014).(μEB, TE )= (405, 127) MeV

3): C. S. Fischer et al., PRD90, 034022 (2014).(μEB, TE ) = (504, 115) MeV

Lattice QCD DSE

μEB =266 ~ 504 MeV, TE = 115~162, μE

B/ TE =1.8~4.38


Finite Size Scaling : HBT and Net-P Fluctuations

1) Scaling function validates the location of the CEP and the (static) critical exponents.

2) 2nd order PT (3D Ising Model): ν= 0.66, γ=1.2, T~165 MeV, μB~95 MeV

Roy Lacey et al.,arXiv:1606.08071M. Suzuki, Prog. Theor. Phys. 58, 1142, 1977R. Lacey: Phys.Rev.Lett. 114 (2015) , 142301

HBT C3/C2

Do we understand the non-critical contributions ??


Fluctuations as Signature of Phase TransitionFluctuations are sensitive to the phase transition and critical point.

M. Stephanov, K. Rajagopal, E. Shuryak, Phys. Rev. Lett. 81, 4816 (1998). M. Stephanov, K. Rajagopal, E. Shuryak, Phys. Rev. D 60, 114028 (1999). S. Jeon and V. Koch, Phys. Rev. Lett. 83, 5435 (1999). S. Jeon and V. Koch, Phys. Rev. Lett. 85, 2076(2000). M. Asakawa, U. Heinz and B. Muller, Phys. Rev. Lett. 85, 2072 (2000). Y. Hatta, M. Stephanov，Phys. Rev. Lett. 91, 102003 (2003). V. Koch, A. Majumder, J. Randrup, Phys.Rev.Lett. 95, 182301 (2005).M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009). M. Asakawa, S. Ejiri and M. Kitazawa, Phys. Rev. Lett.103,262301 (2009).M. A. Stephanov, Phys. Rev. Lett. 107, 052301 (2011).

1. Derivations of thermodynamic quantities (EoS)are related to E-by-E fluctuations in HIC.EoS

2. It reveals more details: Sign change and diverge.


Higher Order Fluctuations of Conserved Quantities

δN( )3c≈ ξ 4.5 , δN( )4

c≈ ξ 7

1. Higher sensitivity to correlation length (ξ)and probe non-gaussian fluctuations.

M. A. Stephanov,Phys. Rev. Lett. 102, 032301 (2009).M. A. Stephanov,Phys. Rev. Lett. 107, 052301 (2011).M.Asakawa, S. Ejiri andM. Kitazawa,Phys. Rev. Lett. 103, 262301 (2009).Y. Hatta, M. Stephanov，Phys. Rev. Lett. 91, 102003 (2003).

2. Connection to the susceptibility of the system.

χq(n) = 1

VT 3 ×Cn,q =∂ n (p /T ^ 4)

∂ µq( )n,q = B,Q,S

S. Ejiri et al,Phys.Lett. B 633 (2006) 275. Cheng et al, PRD (2009) 074505. B. Friman et al., EPJC 71 (2011) 1694. F. Karsch and K. Redlich , PLB 695, 136 (2011).S. Gupta, et al., Science, 332, 1525(2012). A. Bazavov et al., PRL109, 192302(12) // S. Borsanyi et al., PRL111, 062005(13) // P. Alba et al., arXiv:1403.4903

Lattice

χq4

χq2 =κσ

2 =C4,q

C2,q

χq3

χq2 = Sσ =

C3,q

C2,q

,

C1,x =< x >,C2,x =< (δ x)2 >,

C3,x =< (δ x)3 >,C4,x =< (δ x)

4 > −3< (δ x)2 >2


STAR Detector System TPCMTDMagnet BEMC BBCEEMC TOF

ØLarge, UniformAcceptance at Mid-yØExcellent Particle Identification


√sNN(GeV)

Events (106)

Year *μB (MeV)

*TCH(MeV)

200 350 2010 25 16662.4 67 2010 73 16539 39 2010 112 16427 70 2011 156 16219.6 36 2011 206 16014.5 20 2014 264 15611.5 12 2010 316 1527.7 4 2010 422 140

1) Access broad region of the QCD phase diagram.2) STAR: Large and homogeneous acceptance, excellent PID capabilities.

RHIC Beam Energy Scan- I (2010-2014)

*(μB, TCH) : J. Cleymans et al., PRC73, 034905 (2006)

STAR is a unique detector with huge discovery potential in exploringthe QCD phase structure at high baryon density.

STAR Preliminary


Analysis Details


-50 0 50

-50 0 50

10

210

310

410

510

610

10

210

310

410

510

6100-5%

30-40%70-80%

| < 0.5η < 2 (GeV/c),| Tp0.2 <

= 14.5GeVNNSAu+Au

-50 0 50

-50 0 50

10

210

310

410

510

610

10

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510

610 < 1.6 (GeV/c),|y| < 0.5Tp0.2 <

-50 0 50

-50 0 50

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510

610

10

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510

610 < 2 (GeV/c),|y| < 0.5Tp0.4 <

STAR Preliminary

)KN∆Net-Kaon ( )ChN∆Net-Charge ( )PN∆Net-proton (

Num

ber o

f Eve

nts

Raw Event-By-Event Net-Particle Multiplicity Distribution

Effects needed to be addressed to get final moments/cumulants:1. Auto-correlation effects: Centrality definition.2. Effects of volume fluctuation：Centrality bin width correction3. Finite detector efficiency: Factorial moments

A. Bzdak and V. Koch, PRC86, 044904 (2012);;X.Luo, et al. J. Phys. G40,105104(2013);; X.Luo, Phys. Rev. C 91, 034907 (2015);;A . Bzdak and V. Koch, PRC91, 027901 (2015).V. Skokov et al., Phys. Rev. C 88, 034911 (2013).


Ø We can express the cumulants in terms of the factorialmoments, which can be easily efficiency corrected byassuming binomial response function for efficiency.

A. Bzdak and V. Koch, PRC91, 027901 (2015).X. Luo, PRC91, 034907 (2015);;

Ø Statistical Errors based on Delta Theorem. With same N events: error(net-charge) > error(net-kaon) > error(net-proton)

error(Sσ )∝ σε3/2

error(κσ 2 )∝σ2

ε 2

)εEfficiency (0 0.2 0.4 0.6 0.8 1

Sta

tistic

al E

rro

rs

0

2

4

6

2σ κ

σS

Skellam Distribution (6, 3)Number of Events: 1 Million

Error: Delta Theorem

/M2σ

Efficiency Correlation and Error Estimation

Au+Au 14.5GeV Net-Charge Net-Proton Net-Kaon

Typical Width(σ) 12.2 4.2 3.4

Average efficiency(ε) 65% 75% 38%

σ2/ε2 355 32 82

Those numbers are for illustration purpose and not used in actual analysis


0.4

0.6

0.8

1(1) √sNN = 200 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

(2) √sNN = 62.4 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

(3) √sNN = 39 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

(4) √sNN = 27 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

0.4

0.6

0.8

1

0 20 40 60 80

(5) √sNN = 19.6 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

0 20 40 60 80

(6) √sNN = 11.5 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

0 20 40 60 80

(7) √sNN = 7.7 GeV

0.4<pT<0.8 (GeV/c)

0.8<pT<2 (GeV/c)

Efficiencies(|y| < 0.5)

Protonlow pT (TPC)high pT (TPC+ToF)

Anti-protonlow pT (TPC)high pT (TPC+ToF)

Fraction of Collision Centralities (%)

Au + Au Collisions at RHIC

Mid

-rapi

dity

p a

nd p

bar E

ffici

enci

es

Efficiencies for Protons and Anti-protons

TPC

TPC + TOF

STAR Preliminary

Ø Due to TOF matching eff., high pT efficiency (~50%) are smaller than low pT (~80%).Ø Efficiency decrease with increasing energies and centralities.Ø Proton Efficiency > Anti-proton Efficiency


Efficiencies for K+ and K-

Ø 0.2< pT <0.4 (GeV/c), TPC onlyØ 0.4< pT <1.6 (GeV/c), TPC+TOF Efficiency=Efficiency(Tracking)*Efficiency(TOF match)

STAR Preliminary

TPC

Ji Xu, SQM 2016.

0.2

0.4

0.6 0-5%

0.5 1 1.5 20

0.2

0.4

0.6 40-50%

5-10%

0.5 1 1.5 2

50-60%

10-20%

0.5 1 1.5 2

60-70%

20-30%

0.5 1 1.5 2

70-80%

30-40%

0.5 1 1.5 2

+κ -κ

39GeV

AuAu collision

Combined Efficiency

(GeV/c)T

p

Effic

ienc

y


Net-Proton Cumulants (C1~C4 ) Vs. Centrality

1. In general, cumulants are linearly increasing with <Npart>.2. Efficiency corrections are important.3. At low energies, the proton cumulants are close to net-proton.

STAR Preliminary


Cumulants for Net-Kaon

0

5

107.7GeV

0

20

40

Au+Au Collisions

<1.6(GeV/c)T

0.2<p|y|<0.5

Net-kaon

0

5

10

0 100 200 300

50−

0

50

11.5GeV

0 100 200 300

14.5GeV

0 100 200 300

19.6GeV

0 100 200 300

27GeV

0 100 200 300

39GeV

0 100 200 300

62.4GeV

0 100 200 300

200GeV

Eff. Corrected

Poisson

0 100 200 300

⟩partN⟨Average Number of Participant Nucleons

1C2C

3C4C

C3 and C4 generally consistent with Poisson expectation.

STAR Preliminary

Ji Xu, SQM 2016.


5 10 20 100 2000

10

20

30

40 )1

( CMost Central (0-5%)Au+Au Collisions

5 10 20 100 2000

20

40

60

80 )2

( C KaonAnti-kaonNet-kaon

5 10 20 100 2000

20

40 )

3( C Poisson

KaonAnti-kaonNet-kaon

5 10 20 100 200

50−

0

50

)4

( C

<1.6(GeV/c)T

0.2<p|y|<0.5

(GeV)NNsCu

mula

nts

Cumulants vs. Poisson (Protons) Cumulants vs. Poisson (Kaons)

Cumulants vs. Baselines

Ø The higher the order of cumulants, the larger deviations from Poissonexpectations for net-proton and proton.

Ø In general, the cumulants for net-kaon, kaon and antikaon are consistent with Poisson baseline within uncertainties.

STAR Preliminary

STAR Preliminary


Forth Order Fluctuations of Net-P, Net-Q, Net-K

-15

-10

-5

0

5

10

[

[

[

[

[

[

[

[

[

[

[

[

[

[ [ [

Net

-Q κσ

2

(a) STAR Net-chargeδφ = 2π

|η|<0.5; 0.2<pT<2(GeV/c)70 - 80% 0 - 5%

NBD

STAR Prilimnary

-1

0

1

2 [

[[

[

[

[

[

[

[

[

[

[

Net

-Q κσ

2

(b) PHENIX Net-chargeδφ = π/2+π/2

|η|<0.35; 0.3<pT<2(GeV/c)

0 - 5%

NBD

-4

-2

0

2 [

[

[

[

[

[

[

[

[

[ [

[

[

[

[

[

10 20 505 100 200

√sNN (GeV)

Net

-K κσ

2

(c) STAR Net-Kaonδφ = 2π

|yK|<0.5; 0.2<pT<1.6(GeV/c)

70 - 80% 0 - 5%

STAR Prilimnary

0

1

2

3

4

[

[

[

[[

[

[

[

[

[

[

[

[

[

[

[

10 20 505 100 200

√sNN (GeV)

Net

-p κσ

2

(d) STAR Net-protonδφ = 2π

|yp|<0.5; 0.4<pT<2(GeV/c)

70 - 80% 0 - 5%

STAR Prilimnary

1) Within errors, the results of net-Q and net-Kaon show flat energy dependence.2) More statistics are needed at low energies.

In STAR:

σ(Q) > σ(K) > σ(p)


Theoretical and Model Calculations

Motivation:1):Signals from Criticality.Ø NJL, VDW liquid-gas EoS, PQM, σ model etc.Ø DSE, Lattice QCD.

Theoretical predictions are important for us !

2) Study non-CP background in HIC.Transport model: UrQMD, AMPT, JAM etc.


M.A. Stephanov, PRL107, 052301 (2011).

JW Chen et al., PRD93, 034037 (2016)arXiv: 1603.05198.

Vovchenko et al., PRC92, 054901 (2015)Schaefer&Wanger,PRD 85, 034027 (2012) Non-monotonic energy dependence is

observed for 4th order net-protonfluctuations in most central Au+Au collisions.

Model STAR BES Data

Forth Order Fluctuations: Net-protonκσ2 =C4/C2

STAR,PoS(CPOD14)019;; QM (15).


NJL Model CalculationsBaryon (B) Charge (Q) Strangeness (S)

C3/C2

C4/C2

W. Fan, X. Luo, H. Zong, arXiv: 1608. 07903 JW Chen et al., PRD93, 034037 (2016)；arXiv: 1603.05198.

1) CP Signals from baryon fluctuations are much stronger than Q and S.2): Forth and third order fluctuations have very different behavior.


B

C3/C2 C4/C2

C3/C2

C4/C2

m1=C3/C2m2=C4/C2

Comparison Between NJL Model and STAR Data

W. Fan, X. Luo, H. Zong, arXiv: 1608. 07903

Along the assumed freeze-out curve,the NJLModel can qualitativelydescribe the non-monotonic behaviorobserved in data.


Model Calculations

At √sNN ≤ 10 GeV: Data: κσ2 > 1 Model: κσ2 < 1Ø Model simulation indicates: Baryon conservations, Mean-field potential,Deuteron formation, Softening of EOS. All suppress the net-proton fluctuations.

1) Z. Feckova, J. Steonheimer, B. Tomasik, M. Bleicher, PRC92, 064908(2015). J. Xu, S. Yu, F. Liu, X. Luo, PRC94, 024901(2016).X. Luo et al, NPA931, 808(14), P.K. Netrakanti et al. 1405.4617, NPA947, 248(2016), P. Garg et al. Phys. Lett. B726, 691(2013).

2) S. He, X. Luo, Y. Nara, S. Esuimi, N. Xu, PLB 2016, arXiv: 1607.06376.

Effects of Deuteron Formation JAM

0 1 2 3

σS

-4

-2

0

2 Net Proton

0-5%Au+Au: 5 GeV

0 1 2 3

2 σκ

-20

-10

0

CascadeMean FieldAttractive

0 1 2 3

-4

-2

0

2Net Baryon

0 1 2 3-20

-10

0

y∆


Acceptance Dependence : Test Power Law Behavior

volcano eruption

B. Ling, M. Stephanov, Phys. Rev. C 93, 034915 (2016).Adam&Volker, arXiv:1607.07375

Δycorr : The correlation range in rapidity

Acceptance dependence of the critical contributionC1 =< N >C2 =< N > +κ 2

C3 =< N > +3κ 2 + κ 3

C4 =< N > +7κ 2 + 6κ 3 + κ 4

κ 2,κ 3,κ 4 : 2,3,4-particlecorrelation function

Generating function for thefactorial cumulants: (corr. fun.):

Δycorr

If Δy<<Δycorr :

If Δy>>Δycorr :

Cn ∝κ n ∝< N >n~ (Δy)n

Cn ∝κ n ∝< N >~ (Δy)

κ n


0

2

4

6

8

10

12

14

0 0.5 1 1.5 2Proton Rapidity Width 6yp

Net

-pro

ton

k*m

2TPC iTPC

BES-II Error

f = 1 - p0*x + p1*x3

STAR Data: 0-5% Au+Au Collisions at 7.7 GeV

0.4 < pT < 0.8 GeV/c0.4 < pT < 2 GeV/c

AMPT-SM

Net-Proton Fluctuations vs. Rapidity

1) BES-I results: Poisson + Baryon conservation + v3, criticality?

2) BES-II: iTPC extend the rapidity coverage to Δy = 1.6, allowing to studying kinematic dependence and precision measurement of higher moments

STAR BES-II Whitepaper: https://drupal.star.bnl.gov/STAR/starnotes/public/sn0598


Contributions from Four- Particle Correlation

Without the four particle correlation, the non-monotonic behaviorobserved in forth order net-proton fluctuations disappears.

X. Luo (for the STAR Collaboration), PoS(CPOD2014)019 [arXiv:1503.02558].

6 10 20 100 200

0

2

4

6

<N>4C

(a)

STAR Preliminary

(GeV)NNs6 10 20 30 100 200

0

2

4

6

<N>4κ(b)

Au+Au: 0-5%

<2 (GeV/c),|y|<0.5T

0.4<pProton

6 10 20 30 100 200

0

2

4

6

<N>4κ-4C

(c)

Cum

ulan

ts a

nd C

orr.

Func

.


Proton Cumulants and Correlation Functions from JAM Model

Ø The cumulants can be strongly suppressed due to baryon conservationsA. Bzdak, V. Koch, V. Skokov, Phys. Rev. C 87 (2013) 014901.

Ø The two and three- proton correlation function are negative.Ø How about the experimental data ? Stay tuned for QM 2017.

50 100 1500

50

100

150 1CProton

50 100 1500

50

100

150 1κ50 100 150

0

50

100 2C

50 100 150

-40

-20

0 2κ

<2.0 (GeV/c)T

0.4<p|y|<0.1, ..., 1.2

Mean Field: 0-5%JAM: Au+Au 5 GeV

50 100 150

0

10

20

30

403C

50 100 150-15

-10

-5

0

3κ50 100 150-300

-200

-100

04C

50 100 150-100

-50

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504κ

>p<N

Cum

ulan

tsC

orr.

Func

.


More DataRHIC Luminosity Upgrade for Low Energies

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2Proton Rapidity Width 6yp

Net

-pro

ton

k*m

2

TPC iTPC

BES-II Error

f = 1 - p0*x + p1*x3

STAR Data: 0-5% Au+Au Collisions at 7.7 GeV

0.4 < pT < 0.8 GeV/c0.4 < pT < 2 GeV/c

AMPT-SM

BES II at RHIC (2019-2020)

1) Event statistics driven by QCD CP search and di-electron measurements.2) The STAR Fix-target mode is also planed in BESII. (√sNN: 4.5, 3.9, 3.6, 3.0 GeV)

iTPC upgrade extends the rapidity coverage to Δy = 1.6

STAR Preliminary


Future Experiments for High Baryon Density

0

1

2

3

4

10 20 505 100 200

net-protonanti-protonproton

I. P.

(a) 0-5% centrality

Colliding Energy 3sNN (GeV)

Au + Au Collisions at RHIC|y| < 0.5, 0.4 < pT < 2 (GeV/c)

g*m

2

STAR Preliminary

net-protonanti-protonproton

I. P.

10 20 505 100 200

(b) 70-80% centrality

STAR Preliminary

√sNN (GeV)

Longer future: search for the “peak structure” at lower energies350 < μB < 750 MeV ( 2 < √sNN < 8 GeV ). FXT experiment is more effective.


Summary

Ø We show cumulants of net-P, net-K and net-Q for Au+Aucollisions at 7.7,11.5,14.5, 19.6, 27, 39, 62.4 and 200 GeV.

ØNon-monotonic energy dependence is observed at centralAu+Au collisions for net-proton kurtosis, which is consistentwith the presence of critical point. Observation of the criticality ?

ØAcceptance Studies : Looking for the power law behavior.Understand the background contribution to corr. func.

ØStudy the QCD phase structure at high baryon density with highprecision:(1) BES-II at RHIC (2019-2020, both collider and fix target mode).(2) Fix-target at low energies: : FAIR/CBM(starting at 2022), JPARC.


Thank you !

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