Probing QCD phase diagram with charge fluctuations
Vladimir Skokov
GSI (Darmstadt)

5 September 2011

V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

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Outline

Introduction and motivation
Fluctuations of conserved charges
O(4) scaling and fluctuations
Experimental measurement: ratios of cumulants
Probabilities: comparison to experiment

V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

2 / 42QCD phase diagram
QCD phase diagram canbe studied byfirst-principle QCDcalculations
experimentmodel calculations

Phase transitions
crossover at small µ,T ≈ 165 MeV, O(4)scaling
critical end-point (CEP),
3d Z(2) universality class[expected]
first-order phasetransition [expected]

V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

3 / 42QCD phase diagram: quark mass dependenceµ=0

F. Karsch et. al. LGT QCD:
1.6

Mb/h1/�

1.4
1.2
1
0.8
0.6
0.4
0.2
0

QCD at physical mq ?
ml → 0: O(4) or Z(2)?

V.Skokov@GSI.deN�=4: ml/ms
1/5
1/10
1/20
1/40
1/40
1/80
-2-1O(2)z0123• QCD at physical mπ in O(4)scaling• Nτ = 8 supports these results

Fluctuations and the QCD phase transition

RG workshop, Kyoto

4 / 42QCD phase diagram: additional facts
Experiment + HRG model:
freeze-out curveT(µB ) =µB
T/Tc

a − bµ2b − cµ4B√
= d/(1 + e sNN )≈1 − 0.023(µB /T)2

LGT QCD: curvature ofcrossover line at small µB(κq ≈ 0.06)
T/Tc = 1 − κq
F. Karsch ’10

V.Skokov@GSI.de µ 2qT− O(µ4q )Is there any relation between
freeze-out and crossoverlines?
Fluctuations and the QCD phase transition

RG workshop, Kyoto

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Hadron resonance gas model vs experimentHadron Resonance Gas (HRG) model:
PP
pHRG (T, µB , µS , µQ ) = mesons pi + baryons pi ,mi < 2.5 GeV

A. Andronic et al.

Particle yields are well described by HRG model
V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

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Hadron resonance gas model vs LGT QCD
Low temperature lattice QCD results are in agreement with HRG

C. Ratti et. al. ’09
V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

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Fluctuations of conserved charges: definition
Assume system in equilibrium with external reservoir with respect to both particle and energy
exchange.

Energy density and number ofparticles = random variables
with a certain P
P(N) depends on T, µ and V.
P(N) characterises byP
mean N̄ ≡ hNi = N NP(N)variance σ2 = h(N − N̄)2 i
higher order moments
and/or cumulants χnMGF(y) = heNy i = 1+N̄y+hN 2 iy2 /2+· · ·Heavy-ion collisions:

CGF(y) = ln(heNy i) =Conserved charges do notfluctuaten

Part of phase space (pt -cuts,rapidity cuts, etc)
V.Skokov@GSI.deX cnn!

yn

c1 = N̄, c2 = σ2c4 = h(N − N̄)4 i − 3h(N − N̄)2 i2

Fluctuations and the QCD phase transition

RG workshop, Kyoto

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Fluctuations of conserved charges: definition
Probability theory:

Thermodynamics, GCE:

MGF(y) = heNy i = 1+N̄y+hN 2 i

CGF(y) = ln(heNy i) =X cnnP(N)n!

y2+· · ·2yn

MGF(y) = ZGC (y ≡ µ̄ = µ/T) = Tre− T +µ̄NĤCGF(y) = ln ZGC (µ̄) =ZGC =∞XVp(µ̄)p(µ̄) = (VT 3 )· 4TT

ZC (N) exp(µN)

N=−∞

P(N) = const(T, µ) · ZC (N)eµ̂N

V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

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Fluctuations of conserved charges: how one gets numbers

Experiment:
PP(N) { hN k i = N N k P(N) { cumulants

Theory:p(T, µ) { ∂n /∂µn { χi · (VT 3 ) ≡ cumulants

V.Skokov@GSI.deFluctuations and the QCD phase transition

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10 / 42Cumulants ∝ susceptibilities

Fluctuations of net-quark number χqn and net-baryon charge χBn!n∂n (p/T 4 )1∂n (p/T 4 )B
q| χn ==
χn =
χqn∂(µq /T)n∂(µB /T)n3
Fluctuations of electric charge χQnχQn =∂n (p/T 4 )∂(µQ /T)n

Fluctuations of net-strange number. . .

V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

11 / 42Fluctuations of conserved charges: HRG
HRG (µS = µQ = 0):• p/T 4 =i f (mi /T) cosh(µB /T) + g(T)P

• χ2n ∝ cosh(µB /T)• χ2n /χ2 = 1

χ2n+1 ∝ sinh(µB /T)

χ2n+1 /χ1 = 1• χ2n > 0
V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

12 / 42Fluctuations of conserved charges: why?

Properties:
At CEP: for n ≥ 2, χn ∝ ξnβδ/ν−3 ≈ ξ5n/2−3 , e.g. χ4 ∼ ξ7(M. Stephanov ’09)
Diverging χ2 can signal spinodal decomposition of anon-equilibrium 1st order transition (C. Sasaki et. al.
’07)

V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

13 / 42Fluctuations of conserved charges: why?
mπ = 0 (critical line):at µB = 0, χn ∝ ξ(n+2α−4)/(2ν) for even n ≥ 6,e.g. χ6 ∼ ξ1.1
at µB , 0, χn ∝ ξ(n+α−2)/ν for n ≥ 3,e.g. χB3 ∼ ξ1.1mπ , 0: rapid change in crossover region(B. Friman et. al. ’11)
Negative values of high order cumulants close tocrossover(B. Friman et. al. ’11)
V.Skokov@GSI.deFluctuations and the QCD phase transition

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Toy mean-field model: net-quark number fluctuations
Trestored phase���a=0

Landau theory for 2d-orderphase transition (mπ = 0):λaΩ = σ2 + σ424σ - order parameter2 1Ta = t0 Tc − 1 + κq µq /T

broken phase�����

Minimization of Ω: ∂Ω/∂σ=0 leads toσ2min = − λa for a < 0 and σ2min = 0 for a > 0.
2aPressure: p = −Ω(σ = σmin ) = 4λSecond-order cumulant µ = 0: χ2 ∼ (T − Tc )θ(T − Tc )Higher order cumulants n > 4 χn = 0
V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

15 / 42Beyond mean-field
Fluctuations of order parameter { non-trivial exponents
Pressure: p ∼ a2a = t10{ p ∼ a2−α2 T−1+κµ/Tqq
Tc

α is non-integer number.
3-dimensional O(4) universality class: α ≈ −0.21
1µ = 0: higher cumulants are non-trivial: χn ∼ (T − Tc )− 2 (n−4+2α)χ6 ∼ 1/(T − Tc )1+α

χ8 ∼ 1/(T − Tc )2+αdivergent nµ , 0: higher cumulants are non-trivial: χn ∼ Tµ a−(n−2+α) 4χ4 ∼ Tµ /a2+α · · · divergent

V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

16 / 42Beyond mean-field: O(4) scaling functions on lattice
Based on: J. Engels, F. Karsch, arXiv:1105.0584 andB. Friman et. al., arXiv:1103.3511

Lattice simulations of O(4) models { singular part ofp/T 4 ∝ −f (a, h)/T 4 ,h ∝ mq0

0.2-(h0h)−(1+α)/βδ f(3)f (z)-(h0h)−α/βδ f(2)
f (z)

-0.2
-0.4
-0.6
-0.8
-1-1.2-1.4h0h=1
0.5
0.25
0.05
0.0

0-0.2
-0.4
-0.6

h0h=1
0.5
0.25
0.15
0.0

-0.8

-1.6
-5 -4 -3 -2 -1 0

1 2

3

4 5z0(T-Tc)/Tc-5 -4 -3 -2 -1 0

1 2

3

4 5z0(T-Tc)/Tc

χ4 (µ = 0)
χ6 (µ = 0)
Does singular part dominates in QCD?
V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

17 / 42Beyond mean-field: O(4) scaling functions on latticeχ6 (µ = 0)
O(4) scaling:

Lattice QCD:
8B-(h0h)−(1+α)/βδ f(3)f (z)0.2
604-0.2-0.4-0.6B

χ4 /χ2B Bχ6 /χ2HRG

2h0h=1
0.5
0.25
0.15
0.0

0-2-0.8-5 -4 -3 -2 -1 0

1 2z0(T-Tc)/Tc

V.Skokov@GSI.de

3

4 5

0.85

T/Tc
0.9

0.95

11.05

1.1

1.15

C. Schmidt, 2010
Fluctuations and the QCD phase transition

RG workshop, Kyoto

18 / 42Modeling QCDLattice QCD restrictions
continuum limit for cumulants
non-zero chemical potential
QCD inspired modelO(4) symmetry in limit of vanishing mass for light quarks
simulation of confinement properties (ratios of cumulants are
sensitive to degrees of freedom)
Polyakov loop-extended NJL or QM modelO(4) symmetry
quark interaction with Polyakov loops { statistical confinementin O(4) scaling regime for physical pion mass, as QCD
V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

19 / 42Modeling QCD
Mean-field approximation (α = 0)?

NO!

Importance of non-trivial critical exponents.

Mean-field, mπ = 0, µ = 0 T = Tc :χsing=06
Beyond mean-field:
V.Skokov@GSI.de

χsing
=∞
6Fluctuations and the QCD phase transition

RG workshop, Kyoto

20 / 42Modeling QCD: beyond mean-field approximation
Functional Renormalization Groupp(T, µ, k), k defines IR cut off {p(T, µ, k) includes modes with momentum > k.
Functional renormalization group equation (exact and general):p(T, µ, k − dk) = p(T, µ, k) + Exact FRG flow
Iterating towards k → 0: p(T, µ, k = 0) includes all momentummodes
Exact FRG is useless, approximations (leading order in gradient
expansion):p(T, µ, k − dk) = p(T, µ, k) + Approximate FRG flow
FRG review: J. Berges, N. Tetradis & C. Wetterich, Phys.Rept.363:223-386, ’02
FRG formulation of PQM model: V. S., B. Stokic, B. Friman & K. Redlich, PRC, ’10
V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

21 / 42Functional renormalization groupThe general flow equation for the effective action(−1 )
−1 
1(2,0)∂k Γk [Φ, ψ] =Tr ∂k RkB Γk [Φ, ψ] + RkB− Tr ∂k RkF Γ(0,2)[Φ, ψ] + RkF
k2The flow equation for the PQM model( "#31 2k4[σ + π2 ]) =1+2n(E;T)+B π
212π2 Eπ"#"#)
4Nf Nc1
1 + 2nB (Eσ ; T) −1 − N(`, `∗ ; T, µq ) − N̄(`, `∗ ; T, µq )EσEq∂k Ω(k, ρ ≡nB (E; T) is the boson distribution functions
N(`, `∗ ; T, µq ) are fermion distribution function modified owing to coupling to gluons
Eσ andpEπ are the functions of k, ∂Ω/∂ρ and ρ∂2 Ω/∂ρ2Eq = k2 + 2gρ
FRG defines Ω(k, ρ; T, µQ , µB ).
Physically relevant quantity is the thermodynamical potentialΩ̄(T, µQ , µB ) ≡ Ω(k → 0, ρ → ρ0 ; T, µQ , µB ), where ρ0 is the minimum of Ω.
V.Skokov@GSI.deFluctuations and the QCD phase transition

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Functional renormalization group: after approximations
accounts for universal critical behaviour near chiral transition

reproduces scaling properties and critical exponents(Berges ’00, B. Stokic et. al. ’10)

respects symmetries(Goldstone theorem fulfilled, second-order phase transition in O(4)model)

V.Skokov@GSI.de

Fluctuations and the QCD phase transition

RG workshop, Kyoto

23 / 42Phase diagram in FRG PQM
1.2
1.1T/Tpc

1µq/T=0.5µq/T=1crossover

0.9µq/T=1.5

0.8
0.7
0.6
CEP

00.2 0.4 0.6 0.8µq/Tpc

11.2

Crossover: |∂σ/∂T| > 0.95 · max(|∂σ/∂T|)

V.Skokov@GSI.de

Fluctuations and the QCD phase transitionRG workshop, Kyoto

24 / 42Net-quark number density fluctuations δNq = Nq − hNq i
χq2 = VT1 3 h(δNq )2 i →4
2 
1 + π32 µq /T2N N
χq4 = VT1 3 h(δNq )4 i − 3h(δNq )2 i2 → πc2 f :20qq

χ210201-10
0.4

0.40.6

χq3 = VT1 3 h(δNq )3 i →0.8

1T/Tpc

1.4

0.6

0.8

1T/Tpc

1.2

1.4

V.S., B. Friman and K. Redlich PRC’11µq/T=0.5µq/T=1µq/T=1.54

χq2 : non-monotonic structure(diverges at CEP)2

χq4 : negative for nonzero µq6q1.2

2Nc Nf µq /T :π28

χ3µq/T=0µq/T=0.5µq/T=1µq/T=1.530µq/T=0µq/T=0.5µq/T=1µq/T=1.5

χ43Nc Nf
3

0.4

0.6

V.Skokov@GSI.de

0.8

1T/Tpc1.2

1.4Fluctuations and the QCD phase transition

RG workshop, Kyoto

25 / 42High-order cumulants of the baryon number density
32-0.1Bµq/T=0µq/T=0.25µq/T=0.5

1χ8B0χ6µq/T=0µq/T=0.25µq/T=0.5

0-1-0.2
0.4

0.6

0.8

1T/Tpc

1.2

1.4

0.4

0.6

0.8

1T/Tpc1.2

1.4

Negative also at µq = 0
Temperature range of negative cumulants correlates with crossover
temperature
Many other constraints from O(4) scaling: B. Friman et. al. ’11

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Kurtosis of net-quark number densityqχh(δN )4 i

Kurtosis Rq4,2 = χq4 = h(δNqq )2 i − 3h(δNq )2 i2(S. Ejiri, F. Karsch and K. Redlich ’05):
quark content of effective degrees of freedom that carry baryon numberi F(mi /T) cosh(3µq /T){ Rq4,2 = 9High-temperaturephase:  Pqq̄ /T 4 ≈ Nf Nc

112π2µq 4+ 16
T2µq 2+ 7π
T180qPbaryons /T ≈41098R4,2Low temperature phase: dominance ofeffective three-quarkstates:
PPQM
QM

642

0.4

{ Rq4,2 = (6/π2 ) ≈ 120PQM: statistical confinementmπ = 0, µq , 0: kurtosis diverges µ 4Rq4,2 ∼ Tq /t2+α (t ∝ distance to chiralcritical line)

15V.Skokov@GSI.deµq=0

Fluctuations and the QCD phase transition

0.6

0.8

1T/Tpc1.4µq/T=0µq/T=0.5µq/T=1µq/T=1.5

10q

R4,2

1.2

50-5
0.4

0.6

0.8

1T/Tpc1.2

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1.427 / 42

Sign of kurtosis
M. Stephanov ’11: 3d Ising universality class { kurtosis is negative
close to CEPT, GeVQGP
criticalpointtH
0.1

hadron gas

freezeout
curve
nuclearmatter0V.Skokov@GSI.deFluctuations and the QCD phase transition

1µB, GeV

RG workshop, Kyoto

28 / 42Does negative kurtosis signal CEP?
Negative kurtosis is necessary, but not sufficient condition of CEP.
CEP → RB4,2 < 0, but RB4,2 < 0 9 CEP
80B
χB
4 /χ2BχB
6 /χ26HRGχ6B

4µq/T=0µq/T=0.25µq/T=0.5-0.1

20-0.2
0.4 0.6 0.8-2

1T/Tpc1.2 1.4

sign(RB4,2 ) = signχB4 ,0.85

T/Tc
0.90.95

11.05

1.11.15χB4 (µ/T) ≈ χB4 (0) + 21 χB6 (0) · (µ/T)2 + O((µ/T)4 )RB4,2 < 0 → non-trivial phase diagram
V.Skokov@GSI.deFluctuations and the QCD phase transition

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29 / 42High-order baryon cumulant
Temperature interval of negative cumulants closest to hadronic phase:
1.2

1.2
ch. cross.
Bχ 6<01.1
0.9
0.8

1
0.9
0.8

0.7

0.7

0.6

0.6

0.50

0.2 0.4 0.6 0.8µq/Tpc11.2

ch. cross.
Bχ 8<01.1T/Tpc

T/Tpc

10.50

0.2 0.4 0.6 0.8µq/Tpc11.2

Negative values (in broken phase!) of high-order cumulants:
indicates proximity of freeze-out to crossover
Accessible experimentally

V.Skokov@GSI.deFluctuations and the QCD phase transition

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30 / 42Experiment
106

Au+Au 39 GeV
0.4<pT<0.8 (GeV/c), |y|<0.5

5100-5%

Events420-30%1050-60%

31070-80%

102
10STAR Preliminary

1-10

01020

Net-proton

3040

Xiaofeng Luo, ’11V.Skokov@GSI.deFluctuations and the QCD phase transition

RG workshop, Kyoto

31 / 42Comparison of the HRG model with experiment10.0(1)
χ(2)
B /χB

(2)
χ(4)
B /χB

1.0
HRG: 3/2
4/22/1
STAR: 3/2
4/2
2/10.1510(2)
χ(3)B /χB
1/2sNN [GeV]
2050

100200F. Karsch and K. Redlich, ’10
V.Skokov@GSI.deFluctuations and the QCD phase transition

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32 / 42Comparison of HRG model with experiment720210420

� (MeV)
B

1S�

54

2010Au+Au : 0-5%
Net-proton

0.4<p <0.8 (GeV/c)
T|y|<0.50.5
Lattice QCD

κσ2 = 1N�=6, T c=175 MeV� �202HRG

10-1

STAR Preliminary

3 4 56

Xiaofeng Luo, ’11
V.Skokov@GSI.deSσ = tanh(µB /T)

1020 30100

Deviation from HRG:
• Critical properties?
• Conservation laws?
M. Nahrgang, QM’11
arXiv:0903.2911• Statistics?

200sNN (GeV)

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HRG probabilities
Probability disctribution for HRG (Skellam distribution):!N/2 pNb
IN 2 Nb Nb̄ exp −(Nb + Nb̄ )P(N) =Nb̄
Nb is mean number of baryons (protons)
Nb̄ is mean number of anti-baryons (anti-protons)

For double charge particles and in case of quantum
statistics P(N) is to be corrected
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Events

HRG probabilities

Net-proton
Experimental data: STAR Preliminary, Xiaofeng Luo, ’11
HRG line: Np and Np̄ are corrected for efficiency
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Events

HRG probabilities

Net-proton
Experimental data: STAR Preliminary, Xiaofeng Luo, ’11
HRG line: Np and Np̄ are corrected for efficiency
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Events

HRG probabilities

Net-proton
Experimental data: STAR Preliminary, Xiaofeng Luo, ’11
HRG line: Np and Np̄ are corrected for efficiency
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Events

HRG probabilities

Net-proton
arXiv:1107.4267
HRG line: Np and Np̄ are corrected for efficiency
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Events

HRG probabilities

Net-proton
arXiv:1107.4267
HRG line: Np and Np̄ are corrected for efficiency
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Events

HRG probabilities

Net-proton
Conclusions are premature { efficiency
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40 / 42Conclusions

Fluctuations of conserved charges are sensitive probes of the phase structure.
They carry information about deconfinement and chiral phase transitions.
Fluctuations can be used to identify order and universality class of a phasetransition.
The negative values of χB6 may indicate the proximity of the chemical freeze-out
to the crossover line.
Experimentally, deviations from HRG values of the cumulants or thecorresponding probabilities may signal remnants of the transition.
At the moment, the published data is not conclusive.

V.Skokov@GSI.deFluctuations and the QCD phase transition

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41 / 42Thank you for attention

Collaborators: B. Friman, F. Karsch, K. Morita andK. Redlich

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