On the critical endpoint
in the QCD phase diagram
Bernd-Jochen Schaefer
Austria
th
16 April, 2012
Formal Seminars
Department of Physics and Astronomy
University of Sussex
Brighton, UK
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 1/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 2/30
Heavy-Ion Collision Experiments
aim: create hot and dense QCD matter → elucidate its properties
QCD under extrem conditions: very active field (April 2012)
6 large experiments
√
sNN ∼ 200 GeV)
1
RHIC @ BNL (Au-Au collisions
2
LHC @ CERN (higher energies)
3
LeRHIC @ BNL (low-energy scan to produce
nB >> n0 ∼ 0.17 fm−3 )
4
FAIR @ GSI (Facility for Antiproton and Ion Research –
hopefully SIS-300)
5
NICA @ JINR (Nuclotron-based Ion Collider Facility)
6
J-PARC @ JAERI (Japan Proton Accelerator Research
Complex)
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 3/30
QCD Phase Transitions
QCD → two phase transitions:
early universe
LHC
restoration of chiral symmetry
SUL+R (Nf ) → SUL (Nf ) × SUR (Nf )
order parameter:
> 0 ⇔ symmetry broken, T < Tc
hq̄qi
= 0 ⇔ symmetric phase, T > Tc
2
crossover
order parameter:
Polyakov loop variable
= 0 ⇔ confined phase,
T < Tc
Φ
> 0 ⇔ deconfined phase, T > Tc
trc P exp
<ψψ> ∼ 0
FAIR/JINR
AGS
SIS
Z β
i
dτ A0 (τ,~x)
/Nc
0
quark matter
<ψψ> =/ 0
crossover
hadronic fluid
de/confinement
Φ=
quark−gluon plasma
RHIC
SPS
Temperature
1
nB = 0
vacuum
superfluid/superconducting
phases ?
nB > 0
nuclear matter
µ
2SC
<ψψ> =/ 0 CFL
neutron star cores
At densities/temperatures of interest
only model calculations available
alternative: Ü dressed Polyakov loop (dual condensate)
relates chiral and deconfinement transition → spectral properties of Dirac operator
effective models:
1
Quark-meson model
2
Polyakov–quark-meson model
On the critical endpoint in the QCD phase diagram
or other models e.g. NJL
or PNJL models
B.-J. Schaefer (KFU Graz) 4/30
The conjectured QCD Phase Diagram
Open issues:
early universe
LHC
quark−gluon plasma
RHIC
Temperature
SPS
crossover
<ψψ> ∼ 0
FAIR/JINR
AGS
SIS
quark matter
related to chiral & deconfinement transition
B existence/location of CEP?
How many? Additional CEPs?
B coincidence of both transitions at µ = 0 and
µ > 0 (quarkyonic phase)?
<ψψ> =/ 0
hadronic fluid
nB = 0
vacuum
nB > 0
nuclear matter
µ
crossover
superfluid/superconducting
phases ?
2SC
<ψψ> =/ 0 CFL
neutron star cores
At densities/temperatures of interest
only model calculations available
Ü can one improve the model calculations?
B relation between chiral and deconfinement?
chiral CEP/deconfinement CEP?
B are there finite volume effects?
→ lattice comparison
B mostly only MFA results
effects of fluctuations?
→ size of crit. region
B What are good exp. signatures?
→ higher moments more sensitive
Ü remove model parameter dependency?
non-perturbative functional methods (FunMethods)
→ complementary to lattice
• no sign problem µ > 0
On the critical endpoint in the QCD phase diagram
• chiral symmetry/fermions (small masses/chiral limit) etc...
B.-J. Schaefer (KFU Graz) 5/30
The conjectured QCD Phase Diagram
Open issues:
early universe
LHC
quark−gluon plasma
RHIC
Temperature
SPS
crossover
<ψψ> ∼ 0
FAIR/JINR
AGS
SIS
quark matter
related to chiral & deconfinement transition
B existence/location of CEP?
How many? Additional CEPs?
B coincidence of both transitions at µ = 0 and
µ > 0 (quarkyonic phase)?
<ψψ> =/ 0
hadronic fluid
nB = 0
vacuum
nB > 0
nuclear matter
µ
crossover
superfluid/superconducting
phases ?
2SC
<ψψ> =/ 0 CFL
neutron star cores
At densities/temperatures of interest
only model calculations available
Ü can one improve the model calculations?
B relation between chiral and deconfinement?
chiral CEP/deconfinement CEP?
B are there finite volume effects?
→ lattice comparison
B mostly only MFA results
effects of fluctuations?
→ size of crit. region
B What are good exp. signatures?
→ higher moments more sensitive
Ü remove model parameter dependency?
non-perturbative functional methods (FunMethods)
Method of choice: Funtional Renormalization Group Method (FRG)
one needs a truncation: e.g. (Polyakov)-quark-meson model
• good for chiral sector
• what about deconfinement sector
On the critical endpoint in the QCD phase diagram
• baryonic degrees of freedom ?
B.-J. Schaefer (KFU Graz) 5/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 6/30
Chiral Sector:
Three flavor Quark-Meson (QM) Model Lagrangian:
Lqm = Lquark + Lmeson
Quark part with Yukawa coupling h:
Lquark = q̄(i∂
/−h
λa
(σa + iγ5 πa ))q
2
Meson part: scalar σa and pseudoscalar πa nonet
meson fields: M =
8
X
λa
(σa + iπa )
2
a=0
Lmeson = tr[∂µ M † ∂ µ M]−m2 tr[M † M]−λ1 (tr[M † M])2 −λ2 tr[(M † M)2 ]+c[det(M) + det(M † )]
+tr[H(M + M † )]
explicit symmetry breaking matrix: H =
X λa
ha
2
a
U(1)A symmetry breaking implemented by ’t Hooft interaction
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 7/30
Mean-Field Approximation
Model Lagrangian: Lqm = Lquark + Lmeson
Quark part with Yukawa coupling h:
Lquark = q̄(i∂
/−h
λa
(σa + iγ5 πa ))q
2
Meson part: scalar σa and pseudoscalar πa nonet
Lmeson = tr[∂µ M † ∂ µ M]−m2 tr[M † M]−λ1 (tr[M † M])2 −λ2 tr[(M † M)2 ]+c[det(M) + det(M † )]
+tr[H(M + M † )]
Mean-field approximation (MFA):
Integrate over quarks and neglect mesonic fluctuations
Grand potential:
Ω(T, µ) = Ωvac + Ωqq̄ + Uclass
no-sea MFA: neglect Ωvac
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 8/30
Beyond Mean-Field Approximation
Model Lagrangian: Lqm = Lquark + Lmeson
Renormalizable model:
regularization
(Λ regularization scale)
hm i
Nc X 4
f
Ωvac (Λ) = − 2
m log
8π f =u,d,s f
Λ
renormalized grand potential
Ω(T, µ) = Ωrenorm
+ Ωqq̄ + Uclass
vac
all physical observables independent of choice Λ
( remaining model parameters cancel the scale dependence)
later - full FRG treatment
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 9/30
Nf = 2 + 1 Phase Diagram
(µ ≡ µq = µs )
• model parameters fitted to (pseudo)scalar meson spectrum:
• one parameter precarious: f0 (600) ’particle’ (sigma)→ broad resonance
PDG: mass = (400 . . . 1200) MeV
Ü
existence of CEP depends on mσ !
Example: mσ = 600 MeV (lower lines), 800 and 900 MeV (here no-sea MFA)
with U(1)A
250
crossover
1st order
CEP
200
T [MeV]
[BJS, M. Wagner ’09]
150
mσ smaller → CEP towards T axis
including vacuum term Ωvac
(beyond no-sea approximation)
→ CEP moves down
renormalized model has NO CEP
for mσ > 500 MeV
100
here: use smaller mσ to study
influence of fluctuations
50
0
0
50 100 150 200 250 300 350 400
µ [MeV]
On the critical endpoint in the QCD phase diagram
including mesonic fluctuations (full
FRG calculation):
→ CEP moves up again
B.-J. Schaefer (KFU Graz) 10/30
... including ’statistical’ confinement aspects
via Polyakov-loop (not yet dynamical)
Lagrangian LPQM = Lqm + Lpol
with
Lpol = −q̄γ0 A0 q − U (φ, φ̄)
polynomial Polyakov loop potential:
Polyakov 1978, Meisinger 1996, Pisarski 2000
b
2
U (φ, φ̄)
b2 (T, T0 )
b3 3
4
3
=
−
+
φ
φ̄
−
φ
+
φ̄
φφ̄
T4
2
6
16
b2 (T, T0 ) = a0 + a1 (T0 /T) + a2 (T0 /T)2 + a3 (T0 /T)3
beyond MFA:
improve this model by back-coupling of quarks (QCD matter sector) to the Yang-Mills
sector
this yields to a Nf and µ-modifications in presence of dynamical quarks:
T0 = T0 (Nf , µ, mq )
BJS, Pawlowski, Wambach; 2007
Nf
T0 [MeV]
0
270
1
240
2
208
2+1
187
3
178
This becomes clear by considering the FRG
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 11/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 12/30
Functional RG Approach
Γk [φ] scale dependent effective action
;
Rk regulators
t = ln(k/Λ) ;
FRG (average effective action)
∂t Γk [φ] =
1
Tr ∂t Rk
2
k∂k Γk [φ] ∼
On the critical endpoint in the QCD phase diagram
Wetterich 1993
1
(2)
Γk
+ Rk
!
;
(2)
Γk
=
δ 2 Γk
δφδφ
1
2
B.-J. Schaefer (KFU Graz) 13/30
T0 (Nf , µ) modification
full dynamical QCD FRG flow: fluctuations of
gluon, ghost, quark and meson (via hadronization) fluctuations
Braun, Haas, Marhauser, Pawlowski; 2009
∂t Γk [φ] =
1
2
−
−
+
1
2
in presence of dynamical quarks
gluon propagator modified:
→ pure Yang Mills flow + these modifications
pure Yang Mills flow
replaced by effective Polyakov loop potential:
(fit to YM thermodynamics)
T0 ↔ ΛQCD
:
T0 → T0 (Nf , µ, mq )
BJS, Pawlowski, Wambach; 2007
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 14/30
Nf = 2 + 1 (P)QM phase diagrams
Summary of QM and PQM models in no-sea MFA
chiral transition and ’deconfinement’ coincide
at small densities
for PQM model
250
for QM model
(upper lines)
(lower lines)
T [MeV]
200
150
100
CEP
χ crossover
χ first order
− crossover
Φ,Φ
50
0
0
50 100 150 200 250 300 350 400
µ [MeV]
BJS, M. Wagner; arXiv:1111.6871
Nf = 2: BJS, Pawlowski, Wambach; 2007
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 15/30
Nf = 2 + 1 (P)QM phase diagrams
Summary of QM and PQM models in no-sea MFA
chiral transition and ’deconfinement’ coincide
at small densities
for PQM model
250
with
matter back reaction
in Polyakov loop
potential
i.e. T0 (µ)
T [MeV]
200
150
(upper lines)
→ shrinking of
possible quarkyonic
phase
100
CEP
χ crossover
χ first order
− crossover
Φ,Φ
50
0
0
for QM model
(lower lines)
50 100 150 200 250 300 350 400
µ [MeV]
BJS, M. Wagner; arXiv:1111.6871
Nf = 2: BJS, Pawlowski, Wambach; 2007
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 15/30
Functional Renormalization Group
Polyakov-quark-meson model
no matter back-reaction to YM system (T0 const.)
Herbst, Pawlowski, BJS; 2011
200
T [MeV]
150
100
χ crossover
Φ crossover
—
Φ crossover
χ 1st order
50
0
0
50
100
150
200
250
300
350
µ [MeV]
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 16/30
Functional Renormalization Group
Polyakov-quark-meson model
with matter back-reaction to YM system (T0 (µ))
Herbst, Pawlowski, BJS; 2011
200
T [MeV]
150
100
χ crossover
Φ crossover
—
Φ crossover
χ 1st order
50
0
0
50
100
150
200
250
300
350
µ [MeV]
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 16/30
Functional Renormalization Group
Dyson-Schwinger calculation
matter back-reaction to YM system estimated via HTL
Ch. S. Fischer, J. Luecker, J. A. Mueller; 2011
200
T [MeV]
150
100
50
0
0
dressed Polyakov-loop
dressed conjugate P.-loop
chiral crossover
chiral CEP
chiral first order region
100
On the critical endpoint in the QCD phase diagram
200
µ [MeV]
300
B.-J. Schaefer (KFU Graz) 16/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 17/30
Critical region
contour plot of size of the critical region around CEP
free
defined via fixed ratio of susceptibilities: R = χq /χq
Ü compressed with Polyakov loop
QM model (2+1) no-sea MFA
15
R=2
R=3
R=5
10
0
-10
5
0
-5
-10
-15
-20
-20
-60
R=2
R=3
R=5
10
(T-Tc) [MeV]
20
(T-Tc) [MeV]
PQM model (2+1) no-sea MFA:
T0 dashed
T0 (µ) solid
-40
-20
0
(µ-µc) [MeV]
20
40
-40 -30 -20 -10 0 10
(µ-µc) [MeV]
20
30
40
BJS, M. Wagner; arXiv:1111.6871
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 18/30
Critical region
contour plot of size of the critical region around CEP
free
defined via fixed ratio of susceptibilities: R = χq /χq
Ü compressed with Polyakov loop
40
R=2
R=3
R=5
30
(T-Tc) [MeV]
PQM model (2+1) renormalized:
T0 dashed
T0 (µ) solid
20
10
0
-10
-20
-35 -30 -25 -20 -15 -10 -5
(µ-µc) [MeV]
0
5
10 15
(T-Tc) [MeV]
QM model (2+1) renormalized
60
50
40
30
20
10
0
-10
-20
-30
-40
-50
-50
R=2
R=3
R=5
-40
-30
-20 -10
(µ-µc) [MeV]
0
10
20
BJS, M. Wagner; arXiv:1111.6871
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 18/30
Anatomy of the critical region
γ=1
T
A
B
ε=2/3
C
crossover
ε=2/3
crit. exponents depend on path (cf.
different slopes)
→ reason for the elongation
CEP
C'
first order
B'
log10(χ / [MeV2])
QM model
11
10.5
10
9.5
9
8.5
8
7.5
7
6.5
6
µ
A
B
C
-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2
log10(r)
On the critical endpoint in the QCD phase diagram
BJS, M. Wagner; 2012
PQM model (T0 const)
log10(χ / [MeV2])
A'
11
10.5
10
9.5
9
8.5
8
7.5
7
6.5
A
B
C
-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2
log10(r)
B.-J. Schaefer (KFU Graz) 19/30
T
Anatomy of the critical region
γ=1
A
B
ε=2/3
crit. exponents depend on path (cf.
different slopes)
C
crossover
→ reason for the elongation
CEP
ε=2/3
another influence: fluctuations
C'
first order
B'
A'
1
(P)QM models with/without vacuum
(zero modes) quark contribution
Mean-field approximations vs.
RG calculation
µ
BJS, J. Wambach; 2006
Mean Field
RG
0.8
(T-Tc)/Tc
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8 -0.6 -0.4 -0.2 0
(µ-µc)/µc
On the critical endpoint in the QCD phase diagram
0.2 0.4 0.6
B.-J. Schaefer (KFU Graz) 19/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 20/30
Higher moments
limited theoretical guidance for experimental search of CEP
(e.g. STAR @ BNL/RHIC)
unfortunately model predictions vary a lot
possible signatures are based on singular behavior of thermodynamic functions
BUT: realistic heavy-ion collision correlation length ξ is always finite!
(finite volume and critical slowing down)
example estimates: ξ ∼ 2 − 3 fm near a critical point (only factor 3 larger)
hope: non-monotonic behavior in the fluctuations of particle numbers (near CEP)
might serve as a probe
Volker Koch: The Details are in the tails
more sensible quantities (on ξ) are needed
→ higher (cumulants) moments ≡ generalized susceptibilities
higher order moments depend on higher powers of ξ; example: χ4 ∼ ξ 7 near CEP
generalized susceptibities are related to Taylor expansion coefficients
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 21/30
12
D at nonzero densityTaylor
(I)
expansion
coefficients
sure
Taylor expansion:
µd , µs ) =
�
i,j,k
� µ �i � µ ∞
�j � �k
u µ)
dX µs µ n
p(T,
cu,d,s
=
i,j,k
TT 4
T cn (T)
T T
n=0
t fixed temperature
�
�
�
�
k
) �
T
with cn (T) =
convergence-radius has to
be determined nonperturbatively
method for
small µ/T
quark-gluon
plasma
∼ 190M eV
µ=0
convergence radii:
limited by first-order line?
c2
ρ2n = c
deconfined,
χ- symmetric
µu,d,s =0
1 ∂ n p(T, µ)/T 4
n!
∂ (µ/T)n
2n
1/(2n−2)
hadron gas
confined,
χ- broken
nuclear matter
density
color superconductor
µB
C. Schmidt 2009
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 22/30
Taylor coefficients for Nf = 2 + 1 PQM model
New method: based on algorithmic differentiation
M. Wagner, A. Walther, BJS; 2010
Taylor coefficients cn numerically known to high order, e.g. n = 22
3000
4
2
0
2000
0
-50
1000
-2
-100
-4
c6
0
-1000
-150
c8
c10
-2000
-3000
5e+07
-6
50000
B investigation of convergence
properties of Taylor series
1e+06
0
0
c12
-50000
3e+09
2e+09
1e+09
0
-1e+09
-2e+09
-3e+09
-4e+09
B technique applied to PQM model
c14
c16
-1e+06
5e+10
0
c18
0,98
c20
1
1,02
0,98
c22
-5e+10
1
T/Tχ
1,02
0,98
1
1,02
0
properties of cn
-5e+07
oscillating
2e+12
increasing amplitude
0
no numerical noise
-2e+12
small outside transition region
-4e+12
number of roots increasing
26th order
Can we use these coefficients to locate the CEP experimentally ?
F. Karsch, BJS, M. Wagner, J. Wambach; 2010
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 23/30
Generalized Susceptibilities ≡ higher moments
Higher moments are increasingly sensitive to critical behavior even at µ = 0
Example: Kurtosis RB4,2 =
1 c4
9 c2
→
probe of deconfinement?
It measures quark content of particles carrying baryon number B
in Hadron Resonance Gas (HRG) model R4,2 = 1 (always positive)
PQM three flavor calculation (no-sea MFA)
60
200
40
100
c4 / c2
20
R4,2
0
0.0
0.1
0.2
0.3
0.4
-20
-40
-60
-80
196
198
100
0
-100
0
-100
200
200
202
204
206
203
206
T [MeV]
209
20
0 10
50 60
30 40
µ [MeV]
208
T [MeV]
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 24/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
→ turn negative
Karsch, Redlich, Friman et al.; 2011
higher moments: Rqn,m = cn /cm
regions where Rn,2 are negative along crossover line in the phase diagram
PQM with T0 (µ) no-sea MFA
QM model no-sea MFA
210
160
205
150
140
195
R4,2
R6,2
R8,2
R10,2
R12,2
crossover
CEP
190
185
180
175
170
0
20
40 60
80 100 120 140 160 180
µ [MeV]
T [MeV]
T [MeV]
200
130
R4,2
R6,2
R8,2
R10,2
R12,2
crossover
CEP
120
110
100
90
80
0
50
100
150
200
µ [MeV]
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 25/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
→ turn negative
Karsch, Redlich, Friman et al.; 2011
higher moments: Rqn,m = cn /cm
regions where Rn,2 are negative along crossover line in the phase diagram
renormalized PQM with T0 (µ)
renormalized QM model
220
160
200
140
120
160
T [MeV]
T [MeV]
180
R4,2
R6,2
R8,2
R10,2
R12,2
crossover
CEP
140
120
100
80
60
0
50
100
100
R4,2
R6,2
R8,2
R10,2
R12,2
crossover
80
60
40
20
0
150
µ [MeV]
200
250
300
0
50
100
150
200
250
300
µ [MeV]
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 25/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
can we exclude the existence of the CEP ?
→ linear fit
PQM yes - QM no
distance ∆T ≡ Tn − Tχ of the first root in Rn,2 to the chiral temperature Tχ
PQM with T0 (µ)
QM model
0.2
0
∆T [MeV]
∆T [MeV]
-0.2
-0.4
-0.6
-0.8
n=3
n=5
n=7
n=9
-1
-1.2
-1.4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
µ/T
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
n=3
n=5
n=7
n=9
0
0.5
1
1.5
2
2.5
µ/T
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 26/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
can we exclude the existence of the CEP ?
→ linear fit
PQM yes - QM no
distance ∆T ≡ Tn − Tχ of the first root in Rn,2 to the chiral temperature Tχ
renormalized PQM with T0 (µ)
renormalized QM model
5
0
0
-2
-5
-4
-6
n=3
n=5
n=7
n=9
-8
-10
-12
0
0.5
1
1.5
µ/T
2
2.5
3
∆T [MeV]
∆T [MeV]
2
-10
-15
n=3
n=5
n=7
n=9
-20
-25
-30
0
1
2
3
4
5
6
7
8
9
µ/T
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 26/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
can we exclude the existence of the CEP ?
→ linear fit
PQM yes - QM no
relative temperature distance ∆τ ≡ Tn+2 − Tn of the first root
PQM with T0 (µ)
QM model
0
0
-0.2
-0.4
-0.2
∆τ [MeV]
∆τ [MeV]
-0.1
-0.3
-0.4
n=3
n=5
n=7
n=9
-0.5
-0.6
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
µ/T
-0.6
-0.8
-1
-1.2
n=3
n=5
n=7
n=9
-1.4
-1.6
-1.8
1
0
0.5
1
1.5
2
2.5
µ/T
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 26/30
Generalized Susceptibilities ≡ higher moments
Fluctuations of higher moments exhibit strong variation from HRG model
can we exclude the existence of the CEP ?
→ linear fit
PQM yes - QM no
relative temperature distance ∆τ ≡ Tn+2 − Tn of the first root
renormalized QM model
2
0
-2
n=3
n=5
n=7
n=9
0
0.5
1
1.5
µ/T
2
2.5
3
∆τ [MeV]
∆τ [MeV]
renormalized PQM with T0 (µ)
0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-4
-6
-8
n=3
n=5
n=7
n=9
-10
-12
-14
0
1
2
3
4
5
6
7
8
9
µ/T
BJS, M.Wagner; 2012
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 26/30
Outline
◦ QCD Phase Diagram
◦ Chiral QCD-like models with ’statistical’ confinement
◦ Importance of fluctuations:
B Matter back-coupling to Yang-Mills sector
◦ Anatomy of the critical region around the CEP
◦ Higher moments
◦ QCD for Nc = 2
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 27/30
Role of baryonic degrees of freedom
baryonic dof important for larger densities!
Nc = 2: scalar diquarks ←→ bosonic baryons
Quark-Meson-Diquark (QMD) model
hqqi& hq̄qi condensates
MFA with vacuum term
lattice data: Hands 2000
LO χPT
hqqi depends on mσ
variation in diquark due to finite mσ
finite mσ : beyond LO χPT
LO χPT corresponds mσ → ∞
(non-linear σ model)
diquark condensation: µc = mB /Nc
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 28/30
Role of baryonic degrees of freedom
baryonic dof important for larger densities!
Nc = 2: scalar diquarks ←→ bosonic baryons
Quark-Meson-Diquark (QMD) model
hqqi& hq̄qi condensates
RG and MFA
lattice data: Hands 2000
LO χPT
hqqi depends on mσ
variation in diquark due to finite mσ
finite mσ : beyond LO χPT
LO χPT corresponds mσ → ∞
(non-linear σ model)
diquark condensation: µc = mB /Nc
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 28/30
Role of baryonic degrees of freedom
Nc = 2: scalar diquarks ←→ bosonic baryons
Quark-Meson-Diquark (QMD) model
NJL and QMD phase diagrams
MFA with & without vacuum term
NJL: continuous hqqi condensation
QMD: MFA 2nd order & TCP
Ü NLO χ PT predicts also TCP
BUT this is a MFA artifact!
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 29/30
Role of baryonic degrees of freedom
Nc = 2: scalar diquarks ←→ bosonic baryons
Quark-Meson-Diquark (QMD) model
QMD phase diagrams
RG and MFA
QMD: MFA 2nd order & TCP
Ü NLO χ PT predicts also TCP
BUT this is a MFA artifact!
QMD: RG no endpoint
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 29/30
Role of baryonic degrees of freedom
Nc = 2: scalar diquarks ←→ bosonic baryons
Quark-Meson-Diquark (QMD) model
QMD phase diagrams
RG with/without baryonic fluctuations (∆ = 0)
QMD: RG no endpoint
No 1st order T = 0!
Ü more tests are needed
first step towards covariant
quark-diquark description with a
quark-meson-baryon model for QCD
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 29/30
Summary and Outlook
chiral (Polyakov)-quark-meson model study (three flavor)
Ü role of fluctuations: important
(location, existence of the CEP, critical region)
Ü experimental signatures: higher moments
Ü role of baryonic degrees of freedom
PRD 85 (2012) 034027 arXiv:1111.6871
and
PRD 85 (2012) 074007 arXiv:1112.5401
functional approaches (such as the FRG) are suitable and controllable tools
to investigate the QCD phase diagram and its phase boundaries
Ü FunMethods guide the way towards full QCD
On the critical endpoint in the QCD phase diagram
B.-J. Schaefer (KFU Graz) 30/30