Marlene Nahrgang - yukawa.kyoto-u.ac.jp€¦ · QCD phase diagram Understanding the phase diagram of the strong interaction, QCD, is extremely difﬁcult, because there is no global

Fluctuations at the QCD phase transition fromdynamical models

Marlene NahrgangDuke University & Frankfurt Institute for Advanced Studies (FIAS)

New Frontiers in QCD 2013— Insight into QCD matter from heavy-ion collisions —

Kyoto, November 18th, 2013

QCD phase diagram

Understanding the phase diagram of the strong interaction, QCD, isextremely difficult, because

• there is no global analytical approach to solve QCD, and• one cannot produce a long-lived and controlable system of QCD

matter to explore experimentally.

How can we still hope to fill this phase diagram?

T

µB

QCD phase diagram

Understanding the phase diagram of the strong interaction, QCD, isextremely difficult, because

• there is no global analytical approach to solve QCD, and• one cannot produce a long-lived and controlable system of QCD

matter to explore experimentally.

How can we still hope to fill this phase diagram?

T

µB

hadrons

QGP

Approaches to the QCD phase diagram

• QCD calculations in the nonperturbative regime:

LQCD Wuppertal-BudapestJHEP 0203 (2002), JHEP 1104 (2011) DSE C. Fischer, J. Luecker, PLB718 (2013)

• Effective models of QCD:

• Heavy-ion collisions:

Fluid dynamical description of heavy-ion collisions

• The discovery of RHIC: The QGP is an almost ideal stronglycoupled fluid.

• Early hydrodynamic calculations reproduce spectra and ellipticflow P. Kolb, U. Heinz, QGP (2003).

• Long road of improvements during the last decade:(3 + 1d), viscosity, initial conditions, initial state fluctuations, hybrid models

elliptic flow at LHC

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2 2.5 3 3.5 4

v2

pT [GeV]

LHC 2.76 TeV

ALICE h+/-

30-40%

ideal, avg

η/s=0.08, avg

ideal, e-b-e

η/s=0.08, e-b-e

η/s=0.16 , e-b-e

triangular flow at LHC

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3

v3

pT [GeV]

LHC 2.76 TeV, 30-40% central

ideal, e-b-e

η/s=0.08, e-b-e

η/s=0.16, e-b-e

MUSIC by B. Schenke, S. Jeon, C. Gale PLB702 (2011)

What is fluiddynamics?

• Two time scales:• fast processes⇒ local equilibration• slow processes⇒ change of conserved charges (energy,

momentum, charge)

• General dynamics:

∂µT µν = 0 ,

∂µNµ = 0

• Properties of the system enter via the equation of state andtransport coefficients.

What is fluiddynamics?

• Two time scales:• fast processes⇒ local equilibration• slow processes⇒ change of conserved charges (energy,

momentum, charge)

• General dynamics:

∂µT µν = 0 ,

∂µNµ = 0

• Properties of the system enter via the equation of state andtransport coefficients.

Phase transitions are easy to implement in fluiddynamics!

Equation of state - phase transition

• Build an equation of state from the QGP and the hadronic phase.

• Assume an noninteracting gasof hadronic resonances belowTC ⇒ c2

s = ∂p/∂e ∼ 0.15 soft• QGP: gas of noninteracting

quarks and gluons subject toan external bag pressure B⇒eos: p = 1/3e− 4/3B stiff

• Joined together by a Maxwellconstruction.

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

e (GeV/fm3)

p (

GeV

/fm

3)

n=0 fm−3

EOS I

EOS Q

EOS H

P. Kolb, U. Heinz, 2003

The speed of sound vanishes during the phase coexistence.However, no clear experimental signals were found corresponding tothe “softest point” of the eos.C. Hung, E. Shuryak, PRL75 (1995)

Equation of state - lattice QCD

• At µB = 0 the eos can be calculated on the lattice:

Wuppertal-Budapest, JHEP 1011 (2010)

2

4

6

8

100 250 400 550 700

T [MeV]

(ε-3p)/T4

p4, Nτ=6

p4, Nτ=8

asqtad, Nτ=8

P. Huovinen, P. Petreczky, NPA837 (2010)

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á

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á

á

á

á

á

á

á

á

á

á

á

¨¨

¨

àà

ææ

áá ¨

hotQCD results

p4 Nt=8

asqtad Nt=8

Wuppertal-Budapest results

stout Nt=8

stout Nt=10, 12

100 150 200 250 300 350 400

1

2

3

4

5

6

7

T @MeVD

HΕ-3pL�T4

Wuppertal-Budapest, JHEP 1011 (2010)

• Strong increase in the energy density around Tc due to theliberation of color dof.

• Compares well to the HRG eos below Tc ⇒ allows forparametrizing the QCD eos for use in hydrodynamic simulations.

• Some differences between two lattice QCD groups.

Equation of state - critical point

• Construct an eos with CP from theuniversality class of the 3d Isingmodel.

• Map the temperature and theexternal magnetic field (r ,h) onto the(T , µ)-plane⇒ critical part of theentropy density Sc.

• Match with nonsingular entropydensity from QGP and the hadronphase:

s = 1/2(1− tanh Sc)sH

+ 1/2(1 + tanh Sc)sQGP

µB

∆TµB∆

hr

T

CEP

critical regioncrit

crit

80

100

120

140

160

180

200

220

T(M

eV)

0 200 400 600 800 1000

B (MeV)

• Focussing of trajectories⇒ Different behavior of p/p yields.

C. Nonaka, M. Asakawa PRC71 (2005); M. Asakawa, S. Bass, B. Mueller, C. Nonaka PRL101 (2008)

Equation of state - effective models

• Equations of state can be obtained from effective modelLagrangians.

• Hadronic SU(3) non-linear sigma model including quark degreesof freedom yields a realistic structure of the phase diagram andphenomenologically acceptable results for saturated nuclearmatter.V. Dexheimer, S. Schramm, PRC81 (2010)

• The influence of the eos on the directed flow or mean transversemomentum spectra is negligible.

0 50 100 150 200 250 300 3500

50

100

150

200

250

0 0

Lines of constant S/A

T [M

eV]

q [MeV]

0 0

SIS 300

SIS 100

0 2 4 6 80

2

4

6

8

10

12

14

16

18

20 Lines of constant S/A

n/n0

/0

0

0.1000

0.2000

0.3000

0.4000

0.5000

cs

0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Proton Data (NA49) Pion Data (NA 49)

ELab = 40 AGeV UrQMD Protons UrQMD hybrid DE Protons hybrid DE hybrid HG Protons hybrid HG

v 1

y/yb

J. Steinheimer, V. Dexheimer, H. Petersen, M. Bleicher, S. Schramm, H. Stoecker, PRC81 (2010)

Phase transitions are easy to implement in fluiddynamics on the levelof the equation of state.

BUT:

Phase transitions are difficult to implement in fluiddynamics on thelevel of fluctuations!

! Conventional fluiddynamics locally propagates thermal averages! Fluctuations really matter at the critical point...

Fluctuations at the critical point• Coupling of the order parameter to

pions gσππ and protons Gσpp ⇒fluctuations in multiplicitydistributions

〈(δN)2〉 ∝ 〈(∆σ)2〉 ∝ ξ2

ξ: correlation length, diverges at theCPM. Stephanov, K. Rajagopal, E. Shuryak, PRL 81 (1998), PRD 60 (1999)

• Higher cumulants are more sensitiveto the CPskewness: 〈(δN)3〉 ∝ ξ4.5

kurtosis: 〈(δN)4〉 − 3〈(δN)2〉2 ∝ ξ7

M. Stephanov, PLB 102 (2009), PRL 107 (2011)

• Experimental difficulties, baryonnumber conservationMN et al. EPJ C72 (2012)A. Bzdak, V. Koch, PRC86 (2012), PRC87 (2013)

(NA49 collaboration J. Phys. G 35 (2008))

(STAR collaboration, QM2012)

Fluctuations at the critical point

• Long relaxation times near a critical point⇒ the system is driven out of equilibrium (critical slowing down)!

• Phenomenological equation:

ddt

mσ(t) = −Γ[mσ(t)](mσ(t)−1

ξeq(t))

with Γ(mσ) =Aξ0(mσξ0)

z

z = 3(dynamic) critical exponentfrom model H in Hohenberg-Halperin

⇒ ξ ∼ 1.5− 2.5 fm

(B. Berdnikov and K. Rajagopal, PRD 61 (2000)); D.T.Son, M.Stephanov, PRD 70 (2004); M.Asakawa, C.Nonaka, Nucl. Phys. A774 (2006))

Fluctuations at the phase transition in heavy-ioncollisions

• Large nonstatistical fluctuations in nonequilibrium situations ofsingle events.

• Instability of slow modes inthe spinodal region(spinodal decomposition)I. Mishustin, PRL 82 (1999)C. Sasaki, B. Friman, K. Redlich, PRD 77 (2008)

• Significant amplification of initial density irregularities

0 2 4 6 8 10 12

Compression ρ/ρs

-500

0

500

1000

1500

2000

2500

Me

V/f

m3

nucleons

quarks

fT=0

(ρ)

Maxwell

T=0

free energ

y densit

y

pressure

0

5

10

15std EoS

Maxw EoS

0 1 2 3 4 5 6Beam energy E

lab (A GeV)

0

1

2

3

M

axim

um

en

ha

nce

me

nt

of

<ρN>

/<ρ>N

std EoS

Maxw EoS

N = 5

N = 7

J. Steinheimer, J. Randrup, PRL 109 (2012), arXiv:1302.2956

Heavy-ion collisions are• inhomogeneous• finite in space and time• and highly dynamic.

? Can nonequilibrium effects become strong enough to developsignals of the first order phase transition?

? Do enhanced equilibrium fluctuations at the critical point survivethe dynamics?

Goal:Combine the fluid dynamical description of heavy-ion collisions withfluctuation phenomena at the phase transition!

• Explicit propagation of the order parameter(s) – NχFD• Fluid dynamical fluctuations

Nonequilibrium chiral fluid dynamics - NχFD• Langevin equation for the sigma field: damping and noise from

the interaction with the quarks (QM model)

∂µ∂µσ +δUδσ

+ gρs + η∂t σ = ξ

• For PQM: phenomenological dynamics for the Polyakov-loop

η`∂t `T 2 +∂Veff

∂`= ξ`

• Fluid dynamic expansion of the quark fluid = heat bath, includingenergy-momentum exchange

∂µT µνq = Sν = −∂µT µν

σ

⇒ includes a stochastic source term!• Nonequilibrium equation of state p = p(e, σ)

Selfconsistent approach within the 2PI effective action!

MN, S. Leupold, C. Herold, M. Bleicher, PRC 84 (2011); MN, S. Leupold, M. Bleicher, PLB 711 (2012);MN, C. Herold, S. Leupold, I. Mishustin, M. Bleicher, arXiv:1105.1962; C. Herold, MN, I. Mishustin, M. Bleicher PRC 87 (2013)

Dynamics versus equilibration

• Quantify the fluctuations of the order parameter:

dNσ

d3k=

(ω2k |σk |2 + |∂t σk |2)(2π)32ωk

0

2

4

6

8

10

12

0 1 2 3 4 5

dNσ

d|k|

( GeV

−1)

|k| (GeV)

first order transitionrs

rs

rs

rs

rs

rs

rs rs rsrs rs rs

rsrs rs rs

rsrsrsrsrs

rsrsrs rs rs

rs

critical point

ut

ut ut

ututut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut

ut

00.51

1.52

2.53

3.5

0 1 2 3 4 5

dNσ

d|k|

( GeV

−1)

|k| (GeV)

first order transition

rs

rsrsrs rs rs rs

rs rs rs rs rs rs rs rs rsrs rs

rs rs rsrs rs rs rs rs

rs

critical point

ut

ut ut

ut

ut

ututut ut ut ut

ut utut ut

utut ut ut ut ut ut ut ut ut ut

ut

• Strong enhancement of the intensities for a first order phasetransition during the evolution.

• Strong enhancement of the intensities for a critical point scenarioafter equilibration.

C. Herold, MN, I. Mishustin, M. Bleicher PRC 87 (2013)

Trajectories and isentropes at finite µB

Isentropes in the PQM model

0

50

100

150

200

50 100 150 200 250 300 350

T(M

eV)

µ (MeV)

rs

Fluid dynamic trajectories

0

50

100

150

200

50 100 150 200 250 300 350

〈T〉(

MeV

)

〈µ〉 (MeV)

• Fluid dynamic trajectories differ from the isentropes due tointeraction with the fields.

• No significant features in the trajectories left of the critical point.• Right to the critical point: system spends significant time in the

spinodal region! ⇒ possibility of spinodal decomposition!

Irregularities in net-baryon densities

first-order phase transition

-10 -5 0 5 10

x (fm)

-10

-5

0

5

10

y (

fm)

0

0.5

1

1.5

2

2.5

3

3.5

4

n/<n>

-10 -5 0 5 10

x (fm)

-10

-5

0

5

10

y (

fm)

0

0.5

1

1.5

2

2.5

3

3.5

4n/<n>

critical point

-10 -5 0 5 10

x (fm)

-10

-5

0

5

10

y (

fm)

0

0.5

1

1.5

2

2.5

3

3.5

4

n/<n>

-10 -5 0 5 10

x (fm)

-10

-5

0

5

10

y (

fm)

0

0.5

1

1.5

2

2.5

3

3.5

4

n/<n>

C. Herold, MN, I. Mishustin, M. Bleicher, arxiv:1304.5372

Dynamic enhancement of event-by-event fluctuations

temperature averaged sigma field

event-by-event fluctuations

• Initial fluctuations of the sigmafield from initial fluctuations in T .

• Enhanced event-by-eventfluctuations of the orderparameter at Tc .

Relativistic theory of fluid dynamical fluctuations

Conventional fluiddynamics propagates thermal averages of theenergy density, pressure, velocities, charge densities, etc.

However, ...• ... already in equilibrium there are thermal fluctuations• ... the fast processes, which lead to local equilibration also lead

to noise!Conventional ideal fluid dynamics:

T µν = T µνeq

Nµ = Nµeq




to noise!Conventional viscous fluid dynamics:

T µν = T µνeq + ∆T µν

visc

Nµ = Nµeq + ∆Nµ

visc




to noise!Stochastic viscous fluid dynamics:

T µν = T µνeq + ∆T µν

visc + Ξµν

Nµ = Nµeq + ∆Nµ

visc + Iµ


The noise terms are such that averaged quantities exactly equal theconventional quantities:

〈T µν〉 = T µνeq + ∆T µν

visc with 〈Ξµν〉 = 0

〈Nµ〉 = Nµeq + ∆Nµ

visc with 〈Iµ〉 = 0

The two formulations will, however, differ when one calculatescorrelation functions:

〈T µν(x)T µν(x ′)〉〈Nµ(x)Nµ(x ′)〉


In linear response theory the retarded correlator 〈T µν(x)T µν(x ′)〉gives the viscosities and 〈Nµ(x)Nµ(x ′)〉 the charge conductivities viathe dissipation-fluctuation theorem (Kubo-formula)!

It means that when dissipation is included also fluctuations need tobe included!


recent interest in fluid dynamical fluctuations in relativistic (e.gheavy-ion collisions) and non-relativistic fluids (ultra-cold gases).

• J. Kapusta, B. Mueller, M. Stephanov PRC85 (2012)• P. Kovtun, J.Phys. A45 (2012)• J. Kapusta, J. Torres-Rincon PRC86 (2012)• C. Young, arXiv:1306.0472• K. Murase and T. Hirano, arXiv:1304.3243• C. Chafin and T. Schafer, PRA87 (2013)• P. Romatschke and R. E. Young, PRA87 (2013)


As an example: consider only the conservation equation ∂µT µν = 0(no baryon current).

Procedure:• Write down the linearized fluid dynamical equations.• Solve the coupled equations in linear response theory⇒

retarded Green’s functions.• Construct 〈T µν(x)T µν(x ′)〉 from the conservation equation and

via the dissipation-fluctuation theorem.


first-order (second-order) relativistic viscous fluid dynamics

T µν = euµuν − (p + Π)∆µν + πµν

Nµ = nBuµ + νµB

∆µν = gµν − uµuν

πµν = ∆µα∆νβ

(∂αuβ + ∂βuα −

23

gαβ∂µuµ

)−τηπµν

Π = ζ∂µuµ−τζ Π

In principle, one needs to solve the equations for (second-order)relativistic viscous fluid dynamics to preserve causality.

Here, as an example: first-order!


• Linearized hydro equations: small fluctuations e + δe, p + δp andδv i

with: δT 00 = δe and δT ij = mi = (e + p)v i = wv i

∂tm⊥ + η/wk2m⊥ = 0∂t δe + ik ·m|| = 0

∂tm|| + iv2s kδe + γv k2m|| = 0

• Speed of sound vs from the equation of state,γv = (4/3η + ζ)/w .

• Transverse momentum densities decouple, remaining equationscan be written in compact form: ∂t φa + Mab = 0


⇒ then the retarded Green’s function can be calculated via:

Gretab (ω,k) = −(1+ iω(−iωδab + M)−1)χ

• The static susceptibilities χab - response to an external source.• sources contribute time-dependent parts to the Hamiltonian,

here:

δH = −∫

d3x(

δTT

δe + v||m||

)⇒ λ = (δT /T , v||)

• then χ = ∂φa∂λb

=

(cv T 0

0 w

)


• retarded Green’s function for δe and m||:

Gretab (ω,k) =

wω2 − v2

s k2 + iωγsk2

(k2 ω|k|

ω|k| v2s k2 − iωγsk2

)

• including the transverse momentum density:

Gretmi ,mj

(ω,k) =(

δij −kikj

k2

)ηk2

iω− γηk2 +kikj

k2w(v2

s k2 − iωγsk2)

ω2 − v2s k2 + iωγsk2

• Kubo-formulas for viscosities:

η = − ω

2k2

(δij −

kikj

k2

)=Gret

mi mj(ω,k→ 0)

ζ +43

η = −ω3

k4 =Gretee (ω,k→ 0)


∂

∂µx

∂

∂µx ′〈Ξµ0(x)Ξµ0(x ′)〉S = − ∂

∂µx

∂

∂µx ′〈T µ0(x)T µ0(x ′)〉S

=∫ dω

2π

∫ d3k(2π)3 eik(x−x′)e−iω(t−t ′)×

×

ω2GS

ee(ω,k)︸︷︷︸FDT

− 2ω|k|GSem|| (ω,k)︸︷︷︸

FDT

+ k2GSm||m|| (ω,k)︸︷︷︸

FDT

GSab(ω,k) = −2T

ω=Gret

ab (ω,k)

= 0

⇒ no noise term in the first fluid dynamical equation!


many more terms in the second fluid dynamical equation, but sameprocedure:

∂

∂µx

∂

∂µx ′〈Ξµi (x)Ξµj (x ′)〉S = − ∂

∂µx

∂

∂µx ′〈T µi (x)T µj (x ′)〉S

= 2T[(

ζ +43

η

)∂i ∂j + η(δij∇2 − ∂i ∂j )

]δ4(x − x ′)

⇒

〈Ξij (x)Ξkl (x ′)〉S = 2T[(

ζ − 23

η

)δij δkl + η(δik δjl + δil δjk ))

]δ4(x − x ′)

boost back to the Landau-Lifschitz frame

〈Ξµν(x)Ξαβ(x ′)〉S = 2T[(

ζ − 23

η

)∆µα∆νβ + η(∆µβδνα + ∆µν∆αβ))

]δ4(x − x ′)

To do list

• Derive fluid dynamical fluctuations for the coupled systemincluding the net-baryon number current in second-order viscousfluid dynamics and implement it numerically.

• Make a realistic choice for the equation of state and the transportparameters.

• For a comparison to experiment, acknowledge that there othersources of fluctuations and correlations in heavy-ion collisions,e.g. initial fluctuations, flow...

• Final state interactions. Would any signal survive?

Summary

• Heavy-ion collisions can successfully be described byfluiddynamics (possibly viscous and as a part of hybrid models)

• Equations of state from the universality class and effective modelapproaches.

• Nonequilibrium effects can become strong enough to developsignals of the first order phase transition.

• There are indications from NχFD for a dynamical enhancementof event-by-event-fluctuations of the order parameter (σ) at thecritical point

• Fluid dynamical fluctuations play an important role at the criticalpoint!

Documents

Marlene Nahrgang - yukawa.kyoto-u.ac.jp€¦ · QCD phase diagram Understanding the phase diagram of the strong interaction, QCD, is extremely difﬁcult, because there is no global