Exploring the QCD Phase Diagram withFluctuation Observables

Volodymyr Vovchenko

Institut für Theoretische Physik, Goethe University Frankfurt, GermanyFrankfurt Institute for Advanced Studies, Germany

• Lattice QCD context: Going from zero to finite µB• QCD phase structure from experimental measurements

3rd CBM - China Workshop, Yichang, China

April 16, 2018

HGS-HIReHelmholtz Graduate School for Hadron and Ion Research

1/23

Exploring the QCD phase diagram

The QCD phase diagram has many unknowns at finite density

A ton of information is carried by the fluctuation observables

2/23

Fluctuation observables

2/23

Higher-order moments of fluctuations

Let N be a random variable and P(N) its probability distribution.

k-th moment: 〈Nk〉 =∑N

Nk P(N)

k-th cumulant: κk : log〈etN〉 =∞∑n=1

κntnn!

Mean: M = 〈N〉Variance: σ2 = 〈(∆N)2〉 = 〈(N − 〈N〉)2〉

Cumulant ratios:

Scaled variance: σ2

M = κ2κ1

= σ2

〈N〉 width

(Scaled) skewness: Sσ = κ3κ2

= 〈(∆N)3〉σ2 asymmetry

(Scaled) kurtosis: κσ2 = κ4κ2

= 〈(∆N)4〉 − 3 〈(∆N)2〉2

σ2 peakedness

and so on...3/23

Non-Gaussian fluctuations: Skewness

(Normalized) skewness measures the degree of asymmetry of distribution

Sσ = κ3κ2

= 〈(∆N)3〉σ2 .

Baselines:• Gaussian: Sσ = 0• Poisson: Sσ = 1 ← ideal gas in grand canonical ensemble

4/23

Non-Gaussian fluctuations: Kurtosis

(Normalized) kurtosis measures “peakedness” of distribution

κσ2 = κ4κ2

= 〈(∆N)4〉 − 3 〈(∆N)2〉2

σ2 .

Baselines:• Gaussian: κσ2 = 0• Poisson: κσ2 = 1 ← ideal gas in grand canonical ensemble

5/23

Fluctuations in thermodynamic systems

In thermodynamics fluctuations are related to susceptibilities χ(n)

〈Nk〉 =∑

N NkeµNT ZN∑

N eµNT ZN

⇒ κn ∼ χ(n) = ∂n(p/T 4)∂(µ/T )n

σ2

M = χ(2)

χ(1) , Sσ = χ(3)

χ(2) , κσ2 = χ(4)

χ(2) ,

Since they are determined by the derivatives of the thermodynamicpotential, fluctuations can probe regions of the phase diagram which areotherwise inaccessible

6/23

Fluctuation observables and Lattice QCD

6/23

QCD equation of state at µ = 0

Lattice QCD provides the equation of state at µB = 0 from first principles

3p/T4

ε/T4

3s/4T3

0

4

8

12

16

130 170 210 250 290 330 370

T [MeV]

HRG

non-int. limit

TcWB HotQCD

• QCD exhibits crossover-type transition at T ∼ 140− 190 MeV• But not an actual phase transition, at least not at µ = 0• Basic thermodynamic quantities described well by ideal hadronresonance gas model below and in the crossover region

Bazavov et al. [HotQCD Collaboration], PRD 90, 094503 (2014)Borsanyi et al. [Wuppertal-Budapest Collaboration], PLB 730, 99 (2014)

7/23

Lattice QCD susceptibilities at µ = 0

Susceptibilities of the conserved charges at µ = 0 are calculated as well:

χBSQlmn = ∂ l+m+np/T 4

∂(µB/T )l ∂(µS/T )m ∂(µQ/T )n

• Fluctuations probe finer details of the EoS• Ideal HRG breaks down rapidly at T ∼ 150 MeV• This deviation commonly attributed to onset of deconfinement

Bazavov et al. [BNL-Bielefeld-CCNU Collaboration], PRD 95, 054504 (2017)8/23

QCD equation of state at small but finite µLQCD simulations are restricted to µ = 0 region due to the sign problem

Fluctuation observables at µB = 0 provide EoS at finite µB via Taylor expansion:

p(T , µB)T 4 = p(T , 0)

T 4 + χB2 (T , 0)2! (µB/T )2 + χB4 (T , 0)

4! (µB/T )4 + . . .

• QCD EoS now available up to µB/T ' 2 via Taylor expansion• Deviations from ideal HRG set in earlier for a larger µB/T

Bazavov et al. [BNL-Bielefeld-CCNU Collaboration], PRD 95, 054504 (2017) 9/23

Baryon number susceptibilities and baryonic interactions

Fluctuation observables from lattice constrain phenomenological modelsExample: HRG model with attractive and repulsive van der Waals interactionsbetween baryons [V.V., M. Gorenstein, H. Stoecker, Phys. Rev. Lett. 118, 182301 (2017)]

χBQ11

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 00 . 0 0

0 . 0 1

0 . 0 2

0 . 0 3

0 . 0 4

0 . 0 5

0 . 0 6 L a t t i c e ( W u p p e r t a l - B u d a p e s t p r e l i m i n a r y ) L a t t i c e ( H o t Q C D ) I d - H R G E V - H R G , b = 3 . 4 2 f m 3

V D W - H R G , b = 3 . 4 2 f m 3 , a = 3 2 9 M e V f m 3

( d )

χBQ 11

T ( M e V )

χB4 /χB2

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

L a t t i c e ( W u p p e r t a l - B u d a p e s t ) I d - H R G E V - H R G V D W - H R G

χB 4/χB 2

T ( M e V )Initial deviation from Poisson statistics captured by repulsive interactions

More recent developments:V.V., A. Motornenko, M. Gorenstein, H. Stoecker, Phys. Rev. C 97, 035202 (2018)P. Huovinen, P. Petreczky, Phys. Lett. B 777, 125 (2018)S. Samanta, B. Mohanty, Phys. Rev. C 97, 015201 (2018) 10/23

Cluster Expansion Model (CEM)

Cluster Expansion Model combines repulsive baryonic interactions withdeconfinement at high T [V.V., Steinheimer, Philipsen, Stoecker, 1711.01261]

• Relativistic fugacity expansion for baryon density

ρB(T )T 3 = χB1 (T ) =

∞∑k=1

bk(T ) sinh(k µB/T )

• b1(T ) and b2(T ) are model input imaginary µB lattice data1

• Higher order coefficients given by the excluded volume typeexpression, which is matched to the Stefan-Boltzmann limit ofmassless quarks

bk(T ) = αSBk

[b2(T )]k−1

[b1(T )]k−2 , αSBk = [bSB

1 (T )]k−2

[bSB2 (T )]k−1 b

SBk (T )

• Physical picture: Hadron gas with residual repulsion at moderate T ,weakly interacting quarks and gluons at high T

1V.V., A. Pásztor, Z. Fodor, S. Katz, H. Stoecker, 1708.02852; S. Borsányi, QM201711/23

CEM: Baryon number fluctuations

Baryon number susceptibilities at µB = 0:

χB2n(T ) ≡ ∂2n(p/T 4)∂(µB/T )2n

∣∣∣∣∣µB=0

=∞∑k=1

k2n−1 bk(T ) 'kmax∑k=1

k2n−1 bk(T ).

χB2

LQCD (Wuppertal-Budapest) CEM-LQCD

T [MeV]

B = 0

Implicit evidence for consistency between WB and HotQCD data12/23

CEM: 4th and 6th order ratios

χB2n(T ) ≡ ∂2n(p/T 4)∂(µB/T )2n

∣∣∣∣∣µB=0

=∞∑k=1

k2n−1 bk(T ) 'kmax∑k=1

k2n−1 .

χB4 /χB2

LQCD (Wuppertal-Budapest) CEM-LQCD

T [MeV]

B = 0

χB6 /χB2

100 120 140 160 180 200 220 240-2

-1

0

1

2

3 LQCD (HotQCD prelim., Nt = 8) CEM-LQCD

T [MeV]

B = 0

Lattice QCD data on fluctuation observables validate CEM

LQCD data from 1507.04627 (Wuppertal-Budapest), 1701.04325 & 1708.04897 (HotQCD)13/23

Radius of convergence

Taylor expansion of QCD pressure:p(T , µB)

T 4 = p(T , 0)T 4 + χB2 (T )

2! (µB/T )2 + χB4 (T )4! (µB/T )4 + . . .

Radius of convergence rµ/T of the expansion is the distance to the nearestsingularity of p/T 4 in the complex µB/T plane at a given temperature T

If the nearest singularity is at a real µB/T value, this could point to theQCD critical point

Lattice QCD strategy: Estimate rµ/T from few leading termsM. D’Elia et al., 1611.08285; S. Datta et al., 1612.06673; A. Bazavov et al., 1701.04325

Baryon number fluctuations at µB = 0 determine rµ/T

rµ/T = limn→∞

inf∣∣∣∣∣(2n)!χB2n

∣∣∣∣∣1/(2n)

14/23

Radius of convergence in CEM

0 2 4 6 8 1080

100

120

140

160

180

200

220

240

Fodor, Katz (LQCD, reweighting) Lacey (HIC, finite-size scaling) Fischer et al. (Dyson-Schwinger) Critelli et al. (holographic) Li et al. (holographic)

r /T (CEM-LQCD)

r /T (CEM-HRG, b = 1 fm3)T

[MeV

]

B/T

Roberge-Weiss CP estimates:

• LQCD susceptibilities at µ = 0 are consistent with the Roberge-Weisstype transition in the complex µB/T plane [Roberge, Weiss, NPB ’86]

• No evidence for a phase transition at µB/T . π 15/23

Where to look for a critical point

CBM may be in the right spot!

16/23

Where to look for a critical point

CBM may be in the right spot!

16/23

Probing phase diagram with fluctuationmeasurements in heavy-ion collisions

16/23

Fluctuations and critical point

Lattice QCD data on susceptibilities

χ(n) = ∂n(p/T 4)∂(µ/T )n

at µ = 0 may not locate the critical point and/or a phase transition.

In the vicinity of a critical point, however, cumulants are proportional toincreasing powers of the correlation length [M. Stephanov, PRL (2009)]

χ(2) ∼ ξ2

χ(3) ∼ ξ4.5

χ(4) ∼ ξ7

Infinite system: ξ →∞ at CPHeavy-ion collisions: ξ . 2− 3 fm [B. Berdnikov, K. Rajagopal, PRD (2000)]

Fluctuation observables at finite µB probe critical behavior and can bemeasured in heavy-ion collisions see also next talk by X. Luo

17/23

Example: Critical point of nuclear matter

The QCD phase diagram

is known to contain the critical point of nuclear matter at Tc ∼ 15 MeVand real (µB/T )c ∼ 40

18/23

Quantum van der Waals model of nuclear matter

8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

l i q u i d

T (Me

V)

µ ( M e V )

g a s

n ( f m - 3 )

0 . 0 0

0 . 0 2

0 . 0 4

0 . 0 7

0 . 0 9

0 . 1 1

0 . 1 3

0 . 1 6

0 . 1 8

( a )8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

χ 2 / χ 1

1 0

2

1 0 . 5

l i q u i d

T (Me

V)

µ ( M e V )

g a s0 . 1

ω[ N ]

0 . 0 10 . 1

1

1 0

8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

χ 3 / χ 2- 1

1 0

1

0 - 1

l i q u i d

T (Me

V)

µ ( M e V )

g a s

- 1 0

S σ

- 4 0- 1 0

- 101

1 04 0

8 8 0 8 9 0 9 0 0 9 1 0 9 2 0 9 3 005

1 01 52 02 53 03 54 0

χ 4 / χ 2

- 1 0 01 0 01 0 01 0

1

1 0

10- 1

l i q u i d

T (Me

V)

µ ( M e V )

g a s

- 1 0κσ2

- 4 0- 1 0

- 101

1 04 0

V.V., D. Anchishkin, M. Gorenstein, R. Poberezhnyuk, PRC 91, 064314 (2015)19/23

QvdW-HRG model: fluctuations in T -µB plane

χB3 /χB2

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

140

160

180

(a)

QvdW-HRG, Q/B = 0.4, S = 0

1

-12

0

T [M

eV]

B [MeV]

S

-10

0

1

2

10

χB4 /χB2

0 100 200 300 400 500 600 700 800 900 10000

20

40

60

80

100

120

140

160

180

1

(b)

QvdW-HRG, Q/B = 0.4, S = 0

22

-10

0

T [M

eV]

B [MeV]

-10

0

1

2

10

Chemical freeze-out curve from Cleymans et al., Phys. Rev. C 73, 034905 (2006)

Critical point of nuclear matter shines brightly in fluctuation observables,across the whole region of phase diagram probed by heavy-ion collisions

20/23

QvdW-HRG model: collision energy dependence

Calculating fluctuations along the “freeze-out” curveAcceptance effects (protons instead of baryons, momentum cut) modeledschematically, by applying the binomial filter [M. Kitazawa, M. Asakawa, PRC ’12;A. Bzdak, V. Koch, PRC ’12]

χB3 /χB2

(a)

net proton, 0-5% Au+Au, |y| < 0.5 0.4 < pT < 0.8 GeV/c, STAR publ. 0.4 < pT < 2.0 GeV/c, STAR prelim.

QvdW-HRG, net baryons (q = 1) QvdW-HRG, net protons in acc. (q = 0.2) Skellam (IHRG)

S

s1/2NN [GeV]

χB4 /χB2

(b)

net proton, 0-5% Au+Au, |y| < 0.5 0.4 < pT < 0.8 GeV/c, STAR publ. 0.4 < pT < 2.0 GeV/c, STAR prelim.

QvdW-HRG, net baryons (q = 1) QvdW-HRG, net protons in acc. (q = 0.2) Skellam (IHRG)

s1/2NN [GeV]

Effects of nuclear liquid-gas criticality:• Non-monotonic collision energy dependence• Net proton quite different from net baryon 21/23

Scenarios for collision energy dependence

baseline

scenario I: QCD critical point dominates

sNN

baseline

scenario II: nuclear matter critical point dominates

sNN

Can the scenarios be distinguished? Need data at lower energies...Opportunities for HADES, CBM, NA61/SHINE, STAR!

M. Stephanov, JPG ”11

V.V., L. Jiang, M. Gorenstein,H. Stoecker, 1711.07260

22/23

Summary

• Fluctuation observables from lattice QCD provide the QCD equationof state at small µB/T and constrain phenomenological models

• Onset of deviations from Poisson statistics is explained by repulsivebaryonic interactions

• Susceptibilities at µ = 0 are consistent with a Roberge-Weiss liketransition in the complex µB/T plane, no hints for critical point atµB/T . π

• Signals from nuclear liquid-gas criticality are surprisingly strong

• Net proton fluctuations 6= Net baryon fluctuations, for more reasonsthan just baryon number conservation

Thanks for your attention!

23/23

Summary

• Fluctuation observables from lattice QCD provide the QCD equationof state at small µB/T and constrain phenomenological models

• Onset of deviations from Poisson statistics is explained by repulsivebaryonic interactions

• Susceptibilities at µ = 0 are consistent with a Roberge-Weiss liketransition in the complex µB/T plane, no hints for critical point atµB/T . π

• Signals from nuclear liquid-gas criticality are surprisingly strong

• Net proton fluctuations 6= Net baryon fluctuations, for more reasonsthan just baryon number conservation

Thanks for your attention!23/23

Backup slides

23/23

Search for CP in heavy-ion collision experiments

Experimental search for QCD CP using non-Gaussian fluctuations isunderway

Measurements at STAR and NA61/SHINE experimentsX. Luo, CPOD2014, QM2015

Interpretation challenging, many “background” things contributeNo definitive conclusions regarding the location or existence of QCD CP

yet

23/23

Nucleon-nucleon interaction

Nuclear liquid-gas transition appears due to the vdW type structure of thenucleon-nucleon interaction

Nucleon-nucleon potential:• Repulsive core at small distances• Attraction at intermediate distances• Suggestive similarity to vdWinteractions

Could nuclear matter be described by the van der Waals equation?

23/23

van der Waals equation

P(T ,V ,N) = NTV − bN − a N2

V 2

Formulated in1873.

Simplest model which containsattractive and repulsive interactions

Contains 1st order phase transitionand critical point Nobel Prize in

1910.

1. Short-range repulsion: excluded volume (EV) procedure

V → V − bN, b = 44πr3c

32. Intermediate range attraction in mean-field approx.

P → P − a n2, a = π

∫ ∞

2rc|U12(r)|r2dr

23/23

Scaled variance for classical VDW equation

Particle number fluctuations in a classical vdW gas within the GCE

ω[N] = 〈N2〉 − 〈N〉2〈N〉 = χ2

χ1=[ 1

(1− bn)2 −2anT

]−1

0 1 2 30

1

2

3

1 02

1 0 . 5

m i x e d p h a s e

l i q u i d

T/Tc

n / n c

g a s

0 . 1 ω[ N ]

0 . 0 10 . 1

1

1 0

• Repulsive interactions suppress N-fluctuations• Attractive interactions enhance N-fluctuations

V.V., Anchishkin, Gorenstein, J. Phys. A 48, 305001 (2015) 23/23

QvdW gas of nucleons: pressure isothermsa and b fixed to reproduce saturation density and binding energy:

n0 = 0.16 fm−3, E/A = −16 MeV ⇒ a ∼= 329 MeV fm3 and b ∼= 3.42 fm3

5 10 15 20 25 30 35 40 45 50 55 60 65 70-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a)

T=15 MeV

T=18 MeV

T=19.7 MeV

P (M

eV/fm

3 )

v (fm3)

T=21 MeV

0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.280.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b)

T=15 MeV

T=18 MeV

T=19.7 MeV

P (M

eV/fm

3 )

n (fm-3)

T=21 MeV

Behavior qualitatively same as for Boltzmann caseMixed phase results from Maxwell construction

Critical point at Tc ∼= 19.7 MeV and nc ∼= 0.07 fm−3

Experimental estimate1: Tc = 17.9± 0.4 MeV, nc = 0.06± 0.01 fm−3

1J.B. Elliot, P.T. Lake, L.G. Moretto, L. Phair, Phys. Rev. C 87, 054622 (2013)23/23

QvdW gas of nucleons: skewness and kurtosisQvdW Skewness

880 890 900 910 920 9300

5

10

15

20

25

30

35-1

10

1

0 -1

liquid

T (M

eV)

(MeV)

gas

-10

S

-40.00

-10.00

-1.000.001.00

10.00

40.00

QvdW Kurtosis

880 890 900 910 920 9300

5

10

15

20

25

30

35

-100100

10010

1

10

10-1

liquid

T (M

eV)

(MeV)

gas

-10

-40.00

-10.00

-1.000.001.00

10.00

40.00

NJL, J.W. Chen et al., PRD 93, 034037 (2016) PQM, V. Skokov, QM2012

Fluctuation patterns in vdW very similar to effective QCD modelsFluctuation signals from nuclear matter critical point and from QCD critical pointmay very well look alike 23/23

QvdW-HRG model

Further applications require treatment of full hadron spectrumQvdW-HRG model [V.V., M.I. Gorenstein, H. Stoecker, PRL 118, 182301 (2017) ]

• Hadron Resonance Gas (HRG) with attractive and repulsive vdWinteractions between baryons

• vdW parameters a = 329 MeV fm3 and b = 3.42 fm3 tuned to nuclearground state properties

• Critical point of nuclear matter at Tc ' 19.7 MeV, µc ' 908 MeV

Three independent subsystems: mesons + baryons + antibaryons

p(T ,µ) = PM(T ,µ) + PB(T ,µ) + PB̄(T ,µ),

PM(T ,µ) =∑j∈M

pidj (T , µj) and PB(T ,µ) =

∑j∈B

pidj (T , µB∗j )− a n2

B

23/23

QvdW-HRG model at µB = 0

At µ = 0 the model can be confronted with the lattice data

1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 00 . 0 00 . 0 50 . 1 00 . 1 50 . 2 00 . 2 50 . 3 00 . 3 50 . 4 0

L a t t i c e ( W u p p e r t a l - B u d a p e s t ) L a t t i c e ( H o t Q C D ) I d - H R G E V - H R G V D W - H R G

c2 S

T ( M e V )1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 00 . 0

0 . 20 . 40 . 60 . 81 . 01 . 21 . 41 . 6

L a t t i c e ( W u p p e r t a l - B u d a p e s t ) L a t t i c e ( B i e l e f e l d - B N L - C C N U ) I d - H R G E V - H R G V D W - H R G

χB 4/χB 2

T ( M e V )

Strong effects even at µB = 0!What about intermediate collision energies?

23/23