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
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
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
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
Ü related to free energy of a static quark state: Φ = e−βFq
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
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
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
alternative:Ü dressed Polyakov loop (or dual condensate)
it relates chiral and deconfinement transition to spectral properties of Dirac operator
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
The conjectured QCD Phase DiagramT
empe
ratu
re
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
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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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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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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)
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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)
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
Outline
◦ Three-Flavor Quark-Meson Model
◦ ...with Polyakov loop dynamics
◦ Finite density extrapolations
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
In-medium meson masses
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
• At low temperatures: mesons dominate
• At high temperatures: quarks dominate
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
In-medium meson masses
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
• At low temperatures: mesons dominate
• At high temperatures: quarks dominate
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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 and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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 and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
Outline
◦ Three-Flavor Quark-Meson Model
◦ ...with Polyakov loop dynamics
◦ Finite density extrapolations
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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.
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
for QM model Nf = 2(lower lines)
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
for PQM modelNf = 2
withT0(µ)-modificationin Polyakov looppotential(lower lines)
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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) 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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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 crit region
kink structure at boundary in mean field approximation
⇒ remnant of first-order transition in chiral limit
if Dirac term neglected
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
QMPQM logPQM polPQM Fukup4asqtad
0
1
2
3
4
5
6
7
8
0.5 1 1.5 2 2.5 3(ε
-3p)
/T4
T/Tχ
QMPQM logPQM polPQM Fukup4asqtad
solid lines: mπ ∼ 220,mK ∼ 503 MeV (HotQCD)[Bazavov et al. ’09]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
Outline
◦ Three-Flavor Quark-Meson Model
◦ ...with Polyakov loop dynamics
◦ Finite density extrapolations
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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.
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
•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)
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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]
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
Finite density extrapolations Nf = 2 + 1
-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 ?
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
Susceptibility Nf = 2 + 1 PQM model
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
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)
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
QMPQM logPQM polPQM Fukup4asqtad
Outlook:include glue dynamics with FRG→ full QCD
Chiral and Deconfinement Aspects of (2+1)-flavor QCD B.-J. Schaefer (KFU Graz)