Equation of state and phase diagram of strongly interacting ...pawlowsk/talks/QM14...Full dynamical QCD Phase structure Braun, Haas, Marhauser, JMP, PRL 106 (2011) 022002 0 50 100

Equation of state and phase diagram of

strongly interacting matter

Jan M. Pawlowski

Universität Heidelberg & ExtreMe Matter Institute

Darmstadt, May 22 2014th

1

C. Fischer ‘Locating the CEP’

A. Tripolt ‘Spectral functions from the functional renormalization group’

N. Strodthoff ‘QCD-like theories at finite density’

M. Mitter ‘Phase Structure of Strongly Interacting Matter: Thermodynamics and Chiral Anomaly’

R. Stiele ‘Thermodynamics and phase structure of strongly-interacting matter’

K. Morita ‘The Chiral Criticality in the Probability Distribution of Conserved Charges’

Related Talks & Posters

L. Fister ‘On the phase structure and dynamics of QCD’

M. Huber ‘Nonperturbative gluonic three-point correlations’

M. Hopfer ‘The role of the quark-gluon vertex function in the QCD phase transition’

M. Strickland ‘Three loop HTL perturbation theory at finite temperature and chemical potential’

2

https://indico.cern.ch/event/219436/session/11/contribution/95https://indico.cern.ch/event/219436/session/11/contribution/95https://indico.cern.ch/event/219436/session/11/contribution/378https://indico.cern.ch/event/219436/session/11/contribution/378https://indico.cern.ch/event/219436/session/2/contribution/438https://indico.cern.ch/event/219436/session/2/contribution/438https://indico.cern.ch/event/219436/session/2/contribution/463https://indico.cern.ch/event/219436/session/2/contribution/463https://indico.cern.ch/event/219436/session/2/contribution/627https://indico.cern.ch/event/219436/session/2/contribution/627https://indico.cern.ch/event/219436/session/2/contribution/590https://indico.cern.ch/event/219436/session/2/contribution/590https://indico.cern.ch/event/219436/session/2/contribution/581https://indico.cern.ch/event/219436/session/2/contribution/581https://indico.cern.ch/event/219436/session/2/contribution/173https://indico.cern.ch/event/219436/session/2/contribution/173https://indico.cern.ch/event/219436/session/2/contribution/398https://indico.cern.ch/event/219436/session/2/contribution/398https://indico.cern.ch/event/219436/session/18/contribution/21https://indico.cern.ch/event/219436/session/18/contribution/21

Tem

per

atu

re

µ

early universe

neutron star cores

LHCRHIC

SIS

AGS

quark−gluon plasma

hadronic fluid

nuclear mattervacuum

FAIR/JINR

SPS

n = 0 n > 0

∼ 0

= 0/

= 0/

phases ?

quark matter

crossover

CFLB B

superfluid/superconducting

2SC

crossover

3

!Phase Structure of QCD and Equation of State

!Spectral Functions & Transport Coefficients

!Outlook

Outline

4

Functional Methods for QCD

hq(x1) · · · q̄(x2n)Aµ(y1) · · ·Aµ(ym)i

quark-gluon correlations -1 -1 -1-1

5


quark-gluon-hadron correlations -1 -1 -1-1hq(x1) · · · q̄(x2n)Aµ(y1) · · ·Aµ(ym)h(z1) · · ·h(zl)i

5


Functional renormalisation group equations

Dyson-Schwinger equations

2PI/nPI hierarchies

...

Xoff-shell

off-shell(

functional relations

' +... ... ... ... ...scattering amplitude/vertex functions

(


5


Xoff-shell

off-shell(



(

properties

access to physics mechanisms

low energy models naturally encorporated

numerically tractable no sign problemsystematic error control via closed form

lattice: see talk of D. Sexty


5


Xoff-shell

off-shell(



(

properties





QCD low energy models


5


Xoff-shell

off-shell(



(

properties





QCD low energy models


e.g. lattice input on rhs

FunMethods complementary to lattice

e.g. volume flucs., finite density, dynamics, ...

5

Functional Methods for QCDFunctional RG

@t�k[�] =

free energy/grand potential

gluequantum fluctuations

quark quantum fluctuations

hadronic quantum fluctuations

! "#$ S"#$ !k

k=%

k

"#$

k 0

IR UV

k- k&

RG-scale k: t = ln k

free energy at momentum scale k

JMP, AIP Conf.Proc. 1343 (2011)

FRG QCD surveyJMP, Aussois ’12

Phase diagram surveyJMP, Schladming ’13

6

http://www.thphys.uni-heidelberg.de/~pawlowsk/talks/schladming13_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/schladming13_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/erg12_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/erg12_pawlowski.pdf

Functional Methods for QCDFunctional RG

@t�k[�] =

free energy/grand potential

gluequantum fluctuations

quark quantum fluctuations

hadronic quantum fluctuations

! "#$ S"#$ !k

k=%

k

"#$

k 0

IR UV

k- k&

RG-scale k: t = ln k

free energy at momentum scale k


FRG QCD surveyJMP, Aussois ’12

Phase diagram surveyJMP, Schladming ’13

dynamical Gies, Wetterich ’01 JMP ’05 Flörchinger, Wetterich ’09

Dynamical hadronisation

· · ·

6

http://www.thphys.uni-heidelberg.de/~pawlowsk/talks/schladming13_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/schladming13_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/erg12_pawlowski.pdfhttp://www.thphys.uni-heidelberg.de/~pawlowsk/talks/erg12_pawlowski.pdf


∂t = 2 + + +2

@t = � 3 +6 +3 � 6

� 12

+

∂t−1

= +

∂t−1

= − −1/2

+

2PI-resummation

DSE-flowYang-Mills propagators

2

3

4

1

0 5 64321

FRG: Fischer, Maas, JMP, Annals Phys. 324 (2009) 2408

lattice: Sternbeck et al, PoS LAT2006 (2006) 076

p2�A A⇥(p2)

p [GeV]

Yang-Mills

... +

see poster of M. Huber

7

QCD

@t�k[�] =

0

5

10

15

20

25

30

35

0.1 1 10

Yuka

wa

inte

ract

ion

h(k

)

k [GeV]

FRG-QCD: Fister, Herbst, Mitter, Rennecke, Strodthoff, JMP

preliminary

quark-meson coupling

hk

90

8

QCD

@t�k[�] =

0

5

10

15

20

25

30

35

0.1 1 10

Yuka

wa

inte

ract

ion

h(k

)

k [GeV]


preliminary

quark-meson coupling

hk

Low energy models

FRG:

(com

plet

ely)

fixe

d fro

m Q

CD

Model results on the phase structure of QCD

∂tΓk[φ] =1

2− − + 1

2+ ∂tΓk[φ] = 12 − − + 12+

PQM-model PNJL-model QM-model NJL-model

90

8


- + 12

present best approximation

∂t−1

= +

∂t−1

= − −1/2

+

momentum dependence

∂t = 2 + + +2

@t = � 3 +6 +3 � 6

� 12

+

momentum dependence

2PI-resummed momentum dependence

�

momentum dependence

momentum dependence

+matter-contributions

all tensor structures

h[⇥,⇤�]

full mesonic field-dependence

momentum dependence

Aconst0

full field-dependence

+ ...

... +

Ve� [⇥,⇤�;A0]all tensor structures


DSE: see poster of M. Hopfer

9

Phase Structure and Equation of State

10

Confinement

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

0.0

0.2

0.4

0.6

0.8

1.0

T/Tc

Polyakov loop

SU(3)

�gA02⇥

-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

0 0.2 0.4 0.6 0.8 1

!4 V

(! <"

0>)

! /(2#)

0.3 0.5 0.7

276 MeV

295 MeV

286 MeV

280 MeV

276 MeV

271 MeV

�4 VYM[A0]

Polyakov loop Potential

�[A0] =1

3

(1 + 2 cos

1

2

�gA0)

lattice : Tc/p

� = 0.646

Tc/p

� = 0.658± 0.023

�

Braun, Gies, JMP, PLB 684 (2010) 262

Fister, JMP, PRD 88 (2013) 045010

FRG:

FRG, DSE, 2PI:

see also talk of C. Sasaki

11

Confinement

0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25

0.0

0.2

0.4

0.6

0.8

1.0

T/Tc

Polyakov loop

SU(3)

�gA02⇥

-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

0 0.2 0.4 0.6 0.8 1

!4 V

(! <"

0>)

! /(2#)

0.3 0.5 0.7

276 MeV

295 MeV

286 MeV

280 MeV

276 MeV

271 MeV

�4 VYM[A0]

Polyakov loop Potential

�[A0] =1

3

(1 + 2 cos

1

2

�gA0)

lattice : Tc/p

� = 0.646

Tc/p

� = 0.658± 0.023

Fister, JMP, arXiv:1112.5440

0.0 0.5 1.0 1.5 2.00

1

2

3

4

5

p [GeV]

Transversal Propagator GT

FRG: T = 0

FRG: T = 0.361Tc

FRG: T = 0.903Tc

FRG: T = 1.81Tc

Lattice: T = 0

Lattice: T = 0.361Tc

Lattice: T = 0.903Tc

Lattice: T = 1.81Tc

transversal gluon propagator

gauge independence

from the full propagators

confinement criteria

�

Braun, Gies, JMP, PLB 684 (2010) 262

Fister, JMP, PRD 88 (2013) 045010

FRG:

FRG, DSE, 2PI:

see also talk of C. Sasaki

11

Full dynamical QCDPhase structure

Braun, Haas, Marhauser, JMP, PRL 106 (2011) 022002

0

50

100

150

200

250

300

0 π/3 2π/3 π 4π/3

T [M

eV

]

2πθ

TconfTχ

imaginary chemical potential

=2Nf

T� ' Tconf

chiral limit

0

0.2

0.4

0.6

0.8

1

150 160 170 180 190 200 210 220 230

T [MeV]

fπ(T)/fπ(0)

Dual density

Polyakov Loop

160 180 200

χL,d

ual

12

Full dynamical QCDPhase structure

Braun, Haas, Marhauser, JMP, PRL 106 (2011) 022002

0

50

100

150

200

250

300

0 π/3 2π/3 π 4π/3

T [M

eV

]

2πθ

TconfTχ

imaginary chemical potential

=2Nf

T� ' Tconf

chiral limit

0

0.2

0.4

0.6

0.8

1

150 160 170 180 190 200 210 220 230

T [MeV]

fπ(T)/fπ(0)

Dual density

Polyakov Loop

160 180 200

χL,d

ual

see talk of C. Fischer

Luecker, Fischer, Fister, JMP, PoS CPOD2013 (2013) 057 Fischer, Luecker, Welzbacher, arXiv:1405.4762

50 100 150 200 250T [MeV]

0

0.2

0.4

0.6

0.8

1

∆l,

s(T

)/∆

l,s(

0)

Lattice QCDQuark Condensate

=2+1Nf

0

0.2

0.4

0.6

0.8

1

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

∆l,s

t

Wuppertal-Budapest, 2010

PQM FRG

PQM eMF

PQM MF

Herbst, Mitter et al, PLB 731 (2014) 248-256

QCD-improved PQM model DSE

12

Glue Potential

Yang-Mills Potential

Polyakov loop potential in full QCD

0 0.2 0.4 0.6 0.8 1βgA0 / 2π

-0.8

-0.6

-0.4

-0.2

0

β4V

[A0]

glueYang-Mills

t = -0.05

t = 0

t = 0.05

�4V [A0]


Haas, Stiele et al, PRD 87 (2013) 076004

Full dynamical QCDImproving models towards QCD

U [�, �̄]

�[A0]

�gA02⇥

see poster of R. Stiele

@t�k[�] =

@t�k[�] =Yang-Mills

glue

13

Glue Potential

Yang-Mills Potential


0 0.2 0.4 0.6 0.8 1βgA0 / 2π

-0.8

-0.6

-0.4

-0.2

0

β4V

[A0]

glueYang-Mills

t = -0.05

t = 0

t = 0.05

�4V [A0]

Full dynamical QCDImproving models towards QCD

U [�, �̄]

�[A0]

�gA02⇥

Mitter, Schaefer, Phys.Rev. D89 (2014) 054027

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300

mass

es

[MeV

]

T [MeV]

π, η’σ

a0η

without anomalous breaking of axial U(1)

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300

mass

es

[MeV

]

T [MeV]

π σ a0 η’ η

with anomalous breaking of axial U(1)

see poster of M. Mitter

QM-model

13

Full dynamical QCD

Herbst, Mitter, JMP, Schaefer, Stiele, Phys.Lett. B731 (2014) 248-256

Pressure

Shaded area: systematic error estimate due to low initial scale 1 GeV

Interaction measure

0

1

2

3

4

5

6

7

8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6(ε

- 3

P)/

T4

t


HotQCD Nt=8, 2012

HotQCD Nt=12, 2012

PQM FRG

PQM eMF+π

PQM MF+π

PQM eMF

0

0.5

1

1.5

2

2.5

3

3.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

P/T

4

t


PQM FRG

PQM eMF+π

PQM MF+π

PQM eMF

Equation of state

see poster of M. Mitter

lattice: see talk of A. Bazavov

high T: see talk of M. Strickland

14

0 0.2 0.4 0.6 0.8 1βgA0 / 2π

-0.8

-0.6

-0.4

-0.2

0

β4V

[A0]

glueYang-Mills

t = -0.05

t = 0

t = 0.05

Full dynamical QCDPhase structure at finite density

0 50

100

150

200

0 50 100 150 200 250 300 350

T [

Me

V]

µ [MeV]

mπ=138 MeV

χ crossoverσ(T=0)/2Φ crossover—Φ crossoverχ 1st orderCEP


FRG QCD results at finite densityHaas, Braun, JMP ’09, unpublished

Herbst, JMP, Schaefer, PLB 696 (2011) 58-67 PRD 88 (2013) 1, 014007

�4V [A0]

Phase diagram of quantised PQM-model

15

0 0.2 0.4 0.6 0.8 1βgA0 / 2π

-0.8

-0.6

-0.4

-0.2

0

β4V

[A0]

glueYang-Mills

t = -0.05

t = 0

t = 0.05


0 50

100

150

200

0 50 100 150 200 250 300 350

T [

Me

V]

µ [MeV]

mπ=138 MeV



�4V [A0]

Phase diagram of quantised PQM-model µBT

= 2

see poster of N. Strodthoff

diquarks, baryons,

see poster of K. Morita

higher moments

inhomogeneous phases

0

50

100

150

200

0 100 200 300 400 500

T[M

eV]

µ [MeV]

Chiral density wavehom. spinodalshom. 1st orderhom. 2nd order

Müller, Buballa, Wambach, PLB 727 (2013) 240 Carignano, Buballa, Schaefer, arXiv:1404.0057

0

30

60

90

120

150

180

0 50 100 150 200 250 300 350 400 450

T (M

eV)

µ (MeV)

15

µBT

= 3



0

50

100

150

200

0 50 100 150 200 250 300 350

T [

Me

V]

µ [MeV]

mπ=138 MeV


Phase diagram of 2+1 flavor QCD

DSE

Polyakov loop at finite density

0 50 100 150 200µq [MeV]

0

50

100

150

200

T [M

eV]

Lattice: curvature range κ=0.0066-0.0180DSE: chiral crossoverDSE: critical end pointDSE: deconfinement crossover

µB/T=2µB/T=3

lattice curvature

µBT

< 2

Critical point unlikely for

µBT

= 2

see talk of C. Fischer

Fischer, Luecker, PLB 718 (2013) 1036

Fischer, Fister, Luecker, JMP, PLB732 (2014) 248

Kaczmarek at al. ’11Endrodi, Fodor, Katz, Szabo ’11Cea, Cosmai, Papa ’14

16

µBT

= 3



0

50

100

150

200

0 50 100 150 200 250 300 350

T [

Me

V]

µ [MeV]

mπ=138 MeV


Phase diagram of 2+1 flavor QCD

DSE

Polyakov loop at finite density

0 50 100 150 200µq [MeV]

0

50

100

150

200

T [M

eV]


µB/T=2µB/T=3

lattice curvature

µBT

< 2

Critical point unlikely for

µBT

= 2

16

Spectral functions & Transport Coefficients

17

Viscosity in pure glue

Complex DSEs

Tripolt, Strodthoff, von Smekal, Wamach, PRD 89 (2014) 034010 Kamikado, Strodthoff, von Smekal, Wambach, EPJ C74 (2014) 2806

spectral functions

T=100

FRG

gluon spectral function

T=100 MeV - 1 GeVT=0

analytic complex FRG

M. Haas, Fister, JMP, arXiv:1308.4960

FRG+MEM

Strauss, Fischer, Kellermann, PRL 109 (2012) 252001

pion and sigma spectral functions

see talk of A. Tripolt

see poster of L. Fister

18

Viscosity in pure glue

pure glue

PHSD spectral functions

T=1.44 Tc

=2+1Nf

spectral functions

T=1.44 Tc

transversalM. Haas, Fister, JMP, arXiv:1308.4960

see talk of E. Bratkovskaya

see poster of L. Fister

19

Viscosity in pure glueshear viscosity

T . 2Tc

Shaded area: MEM error estimates

: MEM+optimised RG-scheme systematic error estimates


H. Meyer ’09

entropy lattice

Kubo relation

Diagrammatic representation

=⇢⇡⇡ + +

+ ... 3-loop closed form

⌘ =1

20

d

d!

��!=0

⇢⇡⇡(!, 0)

H. Meyer ’09

20


T . 2Tc

Shaded area: MEM error estimates

: MEM+optimised RG-scheme systematic error estimates

minimum at T = 1.25 : Tc⌘

s= 1.45

1

4⇡

scale matching with QCD: ⌘

s= 2.27

1

4⇡


20



s= 1.45

1

4⇡


s= 2.27

1

4⇡

Christiansen, M. Haas, JMP, Strodthoff, in prep.

2-loop terms

21



s= 1.45

1

4⇡


s= 2.27

1

4⇡


Chen, Deng, Dong, Wang ’11

Yang-Mills

2-loop terms

21



s= 1.45

1

4⇡


s= 2.27

1

4⇡



Yang-Mills


scale matching

2-loop terms

21

Summary & Outlook

22

!Phase structure and Equation of State

Summary & outlook

0

0.5

1

1.5

2

2.5

3

3.5

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

P/T

4

t


PQM FRG

PQM eMF

PQM MF

0 50 100 150 200µq [MeV]

0

50

100

150

200

T [M

eV]


µB/T=2µB/T=3

0

50

100

150

200

250

300

0 π/3 2π/3 π 4π/3

T [

Me

V]

2πθ

TconfTχ

µBT

= 3

0

50

100

150

200

0 50 100 150 200 250 300 350

T [M

eV

]

µ [MeV]

mπ=138 MeV


µBT

= 2

0

1

2

3

4

5

6

7

8

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(ε -

3P

)/T

4

t


HotQCD Nt=8, 2012

HotQCD Nt=12, 2012

PQM FRG

PQM eMF+π

PQM MF+π

PQM eMF

23


!Spectral functions and Transport Coefficients

Summary & outlook

24


!Spectral functions and Transport Coefficients

!Towards quantitative precision

!Baryons, high density regime, dynamics

!Hadronic properties

!hadron spectrum & in medium modifications

! low energy constants

Summary & outlook

25

Documents

Equation of state and phase diagram of strongly interacting ...pawlowsk/talks/QM14...Full dynamical QCD Phase structure Braun, Haas, Marhauser, JMP, PRL 106 (2011) 022002 0 50 100