chiral and deconfinement aspects of (2+1)-flavor qcd · 1st order cep 0 0 n = 2 n = 3 n = 1 f f f m...

Chiral and Deconfinement Aspects of (2+1)-flavor QCD

Bernd-Jochen Schaefer

Karl-Franzens-Universität Graz, Austria

17th - 23rd January, 2010

XXXVIII Hirschegg Workshop

Strongly Interacting Matter under Extreme Conditions

Hirschegg, Austria

Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)

QCD Phase TransitionsQCD: two phase transitions:

1 restoration of chiral symmetry

SUL+R(Nf )→ SUL(Nf )× SUR(Nf )

order parameter:

〈q̄q〉

> 0⇔ symmetry broken, T < Tc= 0⇔ symmetric phase, T > Tc

associate limit: mq → 0

Tem

pera

ture

µ

early universe

neutron star cores

LHCRHIC

<ψψ> > 0

AGS

SIS

quark−gluon plasma

hadronic fluid

nuclear mattervacuum

SPSFAIR

n = 0 n > 0

<ψψ> ∼ 0

<ψψ> > 0

phases ?

quark matter

crossover

CFLB B

superfluid/superconducting

2SC

crossover

chiral transition: spontaneous restoration of global SUL(Nf )× SUR(Nf ) at high T




2 de/confinement (center symmetry)

order parameter:

Φ

= 0⇔ confined phase, T < Tc> 0⇔ deconfined phase, T > Tc

Φ =1

Nc〈trcPe

iZ β

0dτA0(τ,~x)

〉

associate limit: mq →∞

Tem

pera

ture

µ

early universe

neutron star cores

LHCRHIC

<ψψ> > 0

AGS

SIS


hadronic fluid


SPSFAIR

n = 0 n > 0

<ψψ> ∼ 0

<ψψ> > 0

phases ?

quark matter

crossover

CFLB B


2SC

crossover

Ü related to free energy of a static quark state: Φ = e−βFq




2 de/confinement (center symmetry)

order parameter:

Φ

= 0⇔ confined phase, T < Tc> 0⇔ deconfined phase, T > Tc

Tem

pera

ture

µ

early universe

neutron star cores

LHCRHIC

<ψψ> > 0

AGS

SIS


hadronic fluid


SPSFAIR

n = 0 n > 0

<ψψ> ∼ 0

<ψψ> > 0

phases ?

quark matter

crossover

CFLB B


2SC

crossover

alternative:Ü dressed Polyakov loop (or dual condensate)

it relates chiral and deconfinement transition to spectral properties of Dirac operator


The conjectured QCD Phase DiagramT

empe

ratu

re

µ

early universe

neutron star cores

LHCRHIC

<ψψ> > 0

AGS

SIS


hadronic fluid


SPSFAIR

n = 0 n > 0

<ψψ> ∼ 0

<ψψ> > 0

phases ?

quark matter

crossover

CFLB B


2SC

crossover

At densities/temperatures of interestonly model calculations available

Open issues:related to chiral & deconfinementtransition

B existence of CEP?

B its location?

B additional CEPs?How many?

B coincidence of both transitions atµ = 0?

B quarkyonic phase at µ > 0?

B chiral CEP/deconfinement CEP?

B so far only MFA resultseffect of fluctuations (e.g. size ofcrit. reg.)?

B ...

effective models:1 Quark-meson model (renormalizable) or other models e.g. NJL

2 Polyakov–quark-meson model or PNJL models


Charts of QCD Critical End Pointsmodel studies vs. lattice simulations

Black points: models Lines & green points: lattice Red points: Freezeout points for HIC

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��

��

��

��

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��

��

��

��

��

��

��

��

��

��

CO94

NJL01NJL89b

CJT02

NJL89a

LR04

RM98

LSM01

HB02 LTE03LTE04

3NJL05

INJL98

LR01

PNJL06

130

9

5

2

17

50

0

100

150

200

0 400 800 1000 1200 1400 1600600200

T ,MeV

µB, MeV

Stephanov ’05 & ’07

QM(2)

QM(2) RG

PQM(2)

QM(2+1)

PQM(2+1)

lattice methods:

reweighting

imaginary µB

Taylor expansionaround µB = 0


Phase diagram in (T, µB,mq)-space

Chiral limit: (mq = 0) SU(2)× SU(2) ∼ O(4)-symmetry −→ 4 modes critical σ, ~π

mq 6= 0 : no symmetry remains −→ only one critical mode σ (Ising) (~π massive)

T

tricritical point, mq = 0

critical line, m q = 0

line, m q = 0

µΒ�

mqO(4) universality

triple

General properties

chiral limittricritical point(Gaussian fixedpoint)

finite mqcritical endpoints(3D-Ising class)


Phase diagram in (T, µB,mq)-space

Chiral limit: (mq = 0) SU(2)× SU(2) ∼ O(4)-symmetry −→ 4 modes critical σ, ~π

mq 6= 0 : no symmetry remains −→ only one critical mode σ (Ising) (~π massive)

T

tricritical point, mq = 0

critical line, m q = 0

triple line, m q = 0

µΒ�

line of end points,mq = 0

surface of 1st ordertransitions

mq

/

3d-Ising universality

O(4) universality General properties

chiral limittricritical point(Gaussian fixedpoint)

finite mqcritical endpoints(3D-Ising class)


Outline

◦ Three-Flavor Quark-Meson Model

◦ ...with Polyakov loop dynamics

◦ Finite density extrapolations


Nf = 3 Quark-Meson (QM) model

Model Lagrangian: Lqm = Lquark + Lmeson

Quark part with Yukawa coupling g:

Lquark = q̄(i∂/− gλa

2(σa + iγ5πa))q

Meson part: scalar σa and pseudoscalar πa nonet

fields: φ =8X

a=0

λa

2(σa + iπa)

Lmeson = tr[∂µφ†∂µφ]− m2tr[φ†φ]− λ1(tr[φ†φ])2 − λ2tr[(φ†φ)2] + c[det(φ) + det(φ†)]

+tr[H(φ+ φ†)]

explicit symmetry breaking matrix: H =X

a

λa

2ha

U(1)A symmetry breaking implemented by ’t Hooft interaction


Phase diagram for Nf = 2 + 1 (µ ≡ µq = µs)Model parameter fitted to (pseudo)scalar meson spectrum:

PDG: f0(600) mass=(400 . . . 1200) MeV→ broad resonance

Ü existence of CEP depends on mσ !

Example: mσ = 600 MeV (lower lines), 800 and 900 MeV (here mean-field approximation)

with U(1)A [BJS, M. Wagner ’09]

0

50

100

150

200

250

0 50 100 150 200 250 300 350 400

T [M

eV]

µ [MeV]

crossover1st order

CEP


In-medium meson masses

Finite temperature axis: µ = 0

masses without U(1)A anomaly

0

200

400

600

800

1000

1200

0 50 100 150 200 250 300 350 400

M [M

eV]

T [MeV]

a0σ

π, η’ 0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300 350 400M

[MeV

]T [MeV]

f0κηK

• At low temperatures: mesons dominate

• At high temperatures: quarks dominate



Finite temperature axis: µ = 0

masses with U(1)A anomaly

0

200

400

600

800

1000

1200

1400

0 50 100 150 200 250 300 350 400

M [M

eV]

T [MeV]

a0ση’π

0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300 350 400M

[MeV

]T [MeV]

f0κηK





slide through CEP: µ = µc

masses with U(1)A anomaly

0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300 350 400

M [M

eV]

T [MeV]

a0σηπ

0

200

400

600

800

1000

1200

1400

1600

0 50 100 150 200 250 300 350 400M

[MeV

]T [MeV]

f0K*η’K




Mass sensitivity

Chiral limit: RG arguments→ for Nf ≥ 3 first-order [Pisarski, Wilczek ’84]

Columbia plot: [Brown et al. ’90]

Tχ ∼ 150 . . . 190 MeV

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

phys. point

1storder

2nd order

2nd orderZ(2)

O(4) ?

crossover

1storder

d

tric

∞

∞Pure TNc=3

d ∼ 270 MeV


Mass sensitivityChiral limit: RG arguments→ for Nf ≥ 3 first-order [Pisarski, Wilczek ’84]

variation of mπ and mK :mπ/m∗π = 0.49 (lower line), 0.6, 0.8 . . . , 1.36 (upper line)

m∗π = 138 MeV , m∗K = 496 MeV , fixed ratio mπ/mK = m∗π/m∗K

with U(1)A, mσ = 800 MeV

0

50

100

150

200

250

0 50 100 150 200 250 300 350 400

T [M

eV]

µ [MeV]

crossover1st order

CEP

00

N = 2

N = 3

N = 1

f

f

f

m s

sm

Gauge

m , mu

phys. point

1storder

2nd order

2nd orderZ(2)

O(4) ?

crossover

1storder

d

tric

∞

∞Pure


Mass sensitivity (lattice, Nf = 3, µB 6= 0)

Standard scenario: mc(µ) increasing

* QCD critical point

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

T

µ

confined

QGP

Color superconductor

Tc

•

Nonstandard scenario: mc(µ) decreasing

QCD critical point DISAPPEARED

crossover 1rst0

∞

Real world

X

Heavy quarks

mu,dms

µ

Tµ

confined

QGP

Color superconductor

Tc

[de Forcrand, Philipsen: hep-lat/0611027]


Chiral critical surface (mσ = 800 MeV)

Ü standard scenario for mσ = 800 MeV (as expected)

with U(1)A

0 25

50 75

100 125

150 175

0 100

200 300

400 500

600

100

200

300

400

µ [MeV]

mπ [MeV]mK [MeV]

µ [MeV]

physical point

without U(1)A

0 25

50 75

100 125

150 175

0 100

200 300

400 500

600

100

200

300

400

µ [MeV]

mπ [MeV]mK [MeV]

µ [MeV]

physical point

[BJS, M. Wagner, ’09]

Note: ’t Hooft coupling µ-independent

PNJL with (unrealistic) large vector int. → bending of surface


Chiral critical surface for different mσ

B CEP vanishes for mσ > 800 MeV→ non-standard scenario possible?

No Ü three cuts of critical surface along fixed mπ/mK ratio through physical point

critical µc

0 50

100 150 200 250 300 350 400 450

0 0.5 1 1.5 2

µ [M

eV]

mπ/mπ*

mσ= 700MeVmσ= 800MeVmσ= 900MeV

critical Tc

0 20 40 60 80

100 120 140 160 180 200

0 0.5 1 1.5 2

T [M

eV]

mπ/mπ*

mσ= 700MeVmσ= 800MeVmσ= 900MeV

[BJS, M. Wagner, ’09]


Outline





Polyakov-quark-meson (PQM) modelLagrangian LPQM = Lqm + Lpol with Lpol = −q̄γ0A0q− U(φ, φ̄)

polynomial Polyakov loop potential: Polyakov 1978, Meisinger 1996, Pisarski 2000

U(φ, φ̄)

T4= −

b2(T, T0)

2φφ̄−

b3

6

“φ3 + φ̄3

”+

b4

16

`φφ̄´2

withb2(T, T0) = a0 + a1(T0/T) + a2(T0/T)2 + a3(T0/T)3




U(φ, φ̄)

T4= −

b2(T, T0)

2φφ̄−

b3

6

“φ3 + φ̄3

”+

b4

16

`φφ̄´2


logarithmic potential: Rössner et al. 2007

Ulog

T4= −

12

a(T)φ̄φ+ b(T) lnh

1− 6φ̄φ+ 4“φ3 + φ̄3

”− 3

`φ̄φ´2i

witha(T) = a0 + a1(T0/T) + a2(T0/T)2 and b(T) = b3(T0/T)3




U(φ, φ̄)

T4= −

b2(T, T0)

2φφ̄−

b3

6

“φ3 + φ̄3

”+

b4

16

`φφ̄´2


Fukushima Fukushima 2008

UFuku = −bTn

54e−a/Tφφ̄+ lnh

1− 6φ̄φ+ 4“φ3 + φ̄3

”− 3

`φ̄φ´2io

witha controls deconfinement b strength of mixing chiral & deconfinement




U(φ, φ̄)

T4= −

b2(T, T0)

2φφ̄−

b3

6

“φ3 + φ̄3

”+

b4

16

`φφ̄´2


in presence of dynamical quarks: T0 = T0(Nf , µ) BJS, Pawlowski, Wambach, 2007

Nf 0 1 2 2 + 1 3T0 [MeV] 270 240 208 187 178

µ 6= 0 : φ̄ > φ

since φ̄ is related to free energy gain of antiquarks

in medium with more quarks→ antiquarks are more easily screened.


Finite temperature and finite µ (PQM Nf = 2)

without T0(µ)-modifications in Polyakov loop potential:

order parameters [BJS, Pawlowski, Wambach ’07]

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250 300 350

T [MeV]

µ=0 MeVµ=168 MeVµ=270 MeV


Phase diagrams Nf = 2

[BJS, Pawlowski, Wambach ’07]

in mean field approximation

chiral transition and ’deconfinement’ coincide

0

50

100

150

200

0 50 100 150 200 250 300 350

T [M

eV]

µ [MeV]

1st ordercrossover

CEP

for PQM modelNf = 2






0

50

100

150

200

0 50 100 150 200 250 300 350

T [M

eV]

µ [MeV]

1st ordercrossover

CEP

for PQM modelNf = 2

for QM model Nf = 2(lower lines)






0

50

100

150

200

0 50 100 150 200 250 300 350

T [M

eV]

µ [MeV]

1st ordercrossover

CEP

for PQM modelNf = 2

for PQM modelNf = 2

withT0(µ)-modificationin Polyakov looppotential(lower lines)


Phase diagram Nf = 2 + 1[BJS, M. Wagner; in preparation ’10]

influence of Polyakov loop

Logarithmic Polyakov loop potential

Mean-field approximation

T0 = 270 MeV (constant) T0(µ) (i.e. with µ corrections)

0

50

100

150

200

250

0 50 100 150 200 250 300 350 400

T [M

eV]

µ [MeV]

chiral crossoverchiral first orderdeconfinementCEP

0

50

100

150

200

250

0 50 100 150 200 250 300 350 400

T [M

eV]

µ [MeV]

chiral crossoverchiral first orderdeconfinementCEP

shrinking of possible quarkyonic phase


Critical region

contour plot of size of the critical region around CEP

defined via fixed ratio of susceptibilities: R = χq/χfreeq

Ü compressed with Polyakov loop

QM model (2+1) PQM model (2+1)

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.15 -0.1 -0.05 0 0.05

(T-T

c)/T

c

(µ-µc)/µc

R=2R=3R=5

-0.01

-0.005

0

0.005

0.01

-0.04 -0.02 0 0.02 0.04

(T-T

c)/T

c

(µ-µc)/µc

R=2R=3R=5

[BJS, M. Wagner; in preparation ’10]


Critical regionsimilar conclusion if fluctuations are included

fluctuations via Renormalization Group

comparison: Nf = 2 QM model

Mean Field ↔ RG analysis

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

(T-T

c)/T

c

(µ-µc)/µc

Mean FieldRG1st order

[BJS, J. Wambach ’06]


Isentropes s/n = const and Focussing

[E. Nakano, BJS, B.Stokic, B.Friman, K.Redlich ’10]

here: Nf = 2 QM model: kink in MFA are washed out in FRG

→ no focussing if fluctuations taken into account

a) influence of Dirac term b) smallnest of critical region

MFA (no Dirac term) MFA (with Dirac term)

ææ

CEP

15.9

3.6

3.85.58.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

Μ @MeVD

T@M

eVD

ææ

10 3.3 5 2.5

1.4

0 50 100 150 200 250 300 3500

50

100

150

200

250

Μ @MeVD

T@M

eVD


Isentropes s/n = const and Focussing

[E. Nakano, BJS, B.Stokic, B.Friman, K.Redlich ’10]



a) influence of Dirac term b) smallnest of critical region

MFA (no Dirac term) FRG (full grid and Taylor expansion)

ææ

CEP

15.9

3.6

3.85.58.1

0 50 100 150 200 250 300 3500

50

100

150

200

250

Μ @MeVD

T@M

eVD

ææ

: CEP HTaylorL

1.2

2.0

2.53.24.36.613.9

0 50 100 150 200 250 300 3500

50

100

150

200

250

Μ @MeVD

T@M

eVD


Isentropes s/n = const and Focussing[E. Nakano, BJS, B.Stokic, B.Friman, K.Redlich ’10]



a) influence of Dirac term b) smallnest of crit region

kink structure at boundary in mean field approximation

⇒ remnant of first-order transition in chiral limit

if Dirac term neglected


QCD Thermodynamics Nf = 2 + 1[BJS, M. Wagner, J. Wambach; arXiv:0910.5628]

SB limit:pSB

T4= 2(N2

c − 1)π2

90+ Nf Nc

7π2

180

(P)QM models (three different Polyakov loop potentials) versus QCD lattice simulations

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3

p/p S

B

T/Tχ

QMPQM logPQM polPQM Fukup4asqtad

B solid lines:PQM with lattice masses(HotQCD)

mπ ∼ 220,mK ∼ 503 MeV

B dashed lines:(P)QM with realistic masses

lattice data: [Bazavov et al. ’09]


QCD Thermodynamics Nf = 2 + 1[BJS, M. Wagner, J. Wambach; arXiv:0910.5628]

SB limit:pSB

T4= 2(N2

c − 1)π2

90+ Nf Nc

7π2

180

(P)QM models (three different Polyakov loop potentials) versus QCD lattice simulations

0

2

4

6

8

10

12

14

16

0.5 1 1.5 2 2.5

ε/T

4

T/Tχ


0

1

2

3

4

5

6

7

8

0.5 1 1.5 2 2.5 3(ε

-3p)

/T4

T/Tχ


solid lines: mπ ∼ 220,mK ∼ 503 MeV (HotQCD)[Bazavov et al. ’09]


Outline





Finite density extrapolations Nf = 2 + 1

Taylor expansion:

p(T, µ)

T4=∞X

n=0

cn(T)

„µ

T

«n

with cn(T) =1n!

∂n `p(T, µ)/T4´∂ (µ/T)n

˛̨̨̨˛µ=0

high temperature limits:

c0(T →∞) =7NcNfπ

2

180,

c2(T →∞) =NcNf

6,

c4(T →∞) =NcNf

12π2

cn(T →∞) = 0 for n > 4.



Taylor expansion:

p(T, µ)

T4=∞X

n=0

cn(T)

„µ

T

«n

with cn(T) =1n!

∂n `p(T, µ)/T4´∂ (µ/T)n

˛̨̨̨˛µ=0

•Taylor-expansion of the pressure p

T 4=

1V T 3

lnZ(V, T, µu, µd, µs) =�i,j,k

cu,d,si,j,k

�µu

T

�i �µd

T

�j �µs

T

�k

nuclear matter density

T

∼ 190MeV

µB

quark-gluon plasma

deconfined, symmetric

hadron gasconfined,

broken color super-conductor

χ-

χ-

method for small µ/T

convergence-radius has to be determined non-

perturbatively

•no sign-problem: all simulations at µ = 0

cu,d,si,j,k ≡ 1

i!j!k!

1

V T 3

· ∂i∂j∂k ln Z

∂(µu

T)i∂(µd

T)j∂(µs

T)k

��µu,d,s=0

•method is conceptually easy

Allton et al., PRD66:074507,2002;Allton et al., PRD68:014507,2003;Allton et al., PRD71:054508,2005.

•calculate Taylor coefficients at fixed temperature

12Lattice QCD at nonzero density (I)

[C. Schmidt ’09]

convergence radii:

limited by first-order line?

ρ2n =

˛̨̨̨c2

c2n

˛̨̨̨1/(2n−2)



Taylor expansion:

p(T, µ)

T4=∞X

n=0

cn(T)

„µ

T

«n

with cn(T) =1n!

∂n `p(T, µ)/T4´∂ (µ/T)n

˛̨̨̨˛µ=0

first three coefficients:

c0: pressure at µ = 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2

c 2

T/T0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.5 1 1.5 2

c 4

T/T0



Taylor expansion:

p(T, µ)

T4=∞X

n=0

cn(T)

„µ

T

«n

with cn(T) =1n!

∂n `p(T, µ)/T4´∂ (µ/T)n

˛̨̨̨˛µ=0

Non-zero density QCD by the Taylor expansion method Chuan Miao and Christian Schmidt

0

0.1

0.2

0.3

0.4

0.5

0.6

150 200 250 300 350 400 450

T[MeV]

c2u

SB

filled: N! = 4

open: N! = 6

nf=2+1, m"=220 MeV

nf=2, m"=770 MeV

0

0.02

0.04

0.06

0.08

0.1

150 200 250 300 350 400 450

T[MeV]

c4u

SB

filled: N! = 4

open: N! = 6

nf=2+1, m"=220 MeV

nf=2, m"=770 MeV

-0.02

-0.01

0

0.01

0.02

0.03

150 200 250 300 350 400 450

T[MeV]

c6u

SB

nf=2+1, m"=220 MeV

nf=2, m"=770 MeV

Figure 1: Taylor coefficients of the pressure in term of the up-quark chemical potential. Results are obtained

with the p4fat3 action onN! = 4 (full) andN! = 6 (open symbols) lattices. We compare preliminary results of(2+1)-flavor a pion mass of m" ! 220 MeV to previous results of 2-flavor simulations with a correspondingpion mass of mp ! 770 [2].

1. Introduction

A detailed and comprehensive understanding of the thermodynamics of quarks and gluons,

e.g. of the equation of state is most desirable and of particular importance for the phenomenology

of relativistic heavy ion collisions. Lattice regularized QCD simulations at non-zero temperatures

have been shown to be a very successful tool in analyzing the non-perturbative features of the

quark-gluon plasma. Driven by both, the exponential growth of the computational power of re-

cent super-computer as well as by drastic algorithmic improvements one is now able to simulate

dynamical quarks and gluons on fine lattices with almost physical masses.

At non-zero chemical potential, lattice QCD is harmed by the “sign-problem”, which makes

direct lattice calculations with standard Monte Carlo techniques at non-zero density practically

impossible. However, for small values of the chemical potential, some methods have been success-

fully used to extract information on the dependence of thermodynamic quantities on the chemical

potential. For a recent overview see, e.g. [1].

2. The Taylor expansion method

We closely follow here the approach and notation used in Ref. [2]. We start with a Taylor

expansion for the pressure in terms of the quark chemical potentials

p

T 4= #

i, j,k

cu,d,si, j,k (T )

!µuT

"i!µdT

" j !µsT

"k. (2.1)

The expansion coefficients cu,d,si, j,k (T ) are computed on the lattice at zero chemical potential, using

stochastic estimators. Some details on the computation are given in [3, 4]. Details on our cur-

rent data set and the number of random vectors used for the stochastic random noise method are

summarized in Table 1.

In Fig. 1 we show results on the diagonal expansion coefficients with respect to the up-quark

2

[Miao et al. ’08]


Finite density extrapolations Nf = 2 + 1New method: based on algorithmic differentiation [M. Wagner, A. Walther, BJS, CPC ’10]

Taylor coefficients cn numerically known to high order, e.g. n = 22

-6

-4

-2

0

2

4

c6-150

-100

-50

0

c8

-3000-2000-10000100020003000

c10

-50000

0

50000

c12 -1e+06

0

1e+06

c14 -5e+07

0

5e+07

c16

0,98 1 1,02-4e+09-3e+09-2e+09-1e+09

01e+092e+093e+09

c18

0,98 1 1,02T/Tχ

-5e+10

0

5e+10

c20

0,98 1 1,02

-4e+12

-2e+12

0

2e+12

c22



-6

-4

-2

0

2

4

c6-150

-100

-50

0

c8

-3000-2000-10000100020003000

c10

-50000

0

50000

c12 -1e+06

0

1e+06

c14 -5e+07

0

5e+07

c16

0,98 1 1,02-4e+09-3e+09-2e+09-1e+09

01e+092e+093e+09

c18

0,98 1 1,02T/Tχ

-5e+10

0

5e+10

c20

0,98 1 1,02

-4e+12

-2e+12

0

2e+12

c22

B this technique applied to PQM model

B investigation of convergenceproperties of Taylor series

B properties of cn

oscillating

increasing amplitude

no numerical noise

small outside transition region

number of roots increasing

26th order[F. Karsch, BJS, M. Wagner, J. Wambach; in preparation ’10]

Can we locate the QCD critical endpoint with the Taylor expansion ?


Susceptibility Nf = 2 + 1 PQM model

µ/T = 0.8

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

24th16th8thPQM

µ/T = 3.

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

26th16th8thPQM

µ/T = µc/Tc

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

26th16th8thPQM



µ/T = 0.8

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

24th16th8thPQM

µ/T = 3.

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

26th16th8thPQM

µ/T = µc/Tc

0

2

4

6

8

10

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

χ/T

2

T/Tχ

26th16th8thPQM

convergence radius

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

T/T

χ

µ/Tχ

n=16n=20n=24



Findings:

simply Taylor expansion: slow convergence

high orders needed

disadvantage for lattice simulations

Taylor applicable within convergence radius

also for µ/T > 1

but 1st order transition not resolvable

expansion around µ = 0


SummaryNf = 2 and Nf = 2 + 1 chiral (Polyakov)-quark-meson model study

Ü Mean-field approximation and FRG

with and without axial anomaly

novel AD technique: high order Taylor coefficients, here: n = 26

Findings:

B Parameter in Polyakov loop potential:T0 ⇒ T0(Nf , µ)

B Chiral & deconfinement transition possiblycoincide for Nf = 2 with T0(µ)-correctionsbut possibly not for Nf = 2 + 1

B Mean-field approximation encouraging

but effects of Dirac term point to interesting physicsif fluctuations are considered

→ FRG with PQM truncation

B Taylorcoefficient cn(T)→ high order

⇒ convergence properties of Taylorexpansion

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3

p/p S

B

T/Tχ


Outlook:include glue dynamics with FRG→ full QCD


chiral and deconfinement aspects of (2+1)-flavor qcd · 1st order cep 0 0 n = 2 n = 3 n = 1 f f f m...

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