7 01003 (2010)
EPJ Web of Conferences 7,
DOI:10.1051/epjconf/20100701003
© Owned by the authors, published by EDP Sciences, 2010
Effective Field Theories for Hot and Dense Matter
David Blaschke1,2,a
1
2
University of Wrocław, 50-204 Wrocław, Poland
Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, 141980 Dubna, Russia
Abstract. The lecture is divided in two parts. The first one deals with an introduction to the physics of hot,
dense many-particle systems in quantum field theory [1,2]. The basics of the path integral approach to the partition function are explained for the example of chiral quark models. The QCD phase diagram is discussed in the
meanfield approximation while QCD bound states in the medium are treated in the rainbow-ladder approximation (Gaussian fluctuations). Special emphasis is devoted to the discussion of the Mott effect, i.e. the transition
of bound states to unbound, but resonant scattering states in the continnum under the influence of compression
and heating of the system. Three examples are given: (1) the QCD model phase diagram with chiral symmetry
restoration and color superconductivity [3], (2) the Schrödinger equation for heavy-quarkonia [4], and (2) Pions
[5] as well as Kaons and D-mesons in the finite-temperature Bethe-Salpeter equation [6]. We discuss recent applications of this quantum field theoretical approach to hot and dense quark matter for a description of anomalous
J/ψ supression in heavy-ion collisions [7] and for the structure and cooling of compact stars with quark matter
interiors [8].
The second part provides a detailed introduction to the Polyakov-loop Nambu–Jona-Lasinio model [9] for thermodynamics and mesonic correlations [10] in the phase diagram of quark matter. Important relationships of
low-energy QCD like the Gell-Mann–Oakes–Renner relation are generalized to finite temperatures. The effect
of including the coupling to the Polyakov-loop potential on the phase diagram and mesonic correlations is discussed. An outlook is given to effects of nonlocality of the interactions [11] and of mesonic correlations in the
medium [12] which go beyond the meanfield description.
References
1. J.I. Kapusta and C. Gale, Finite Temperature Field
Theory, Cambridge University Press (2006).
2. K. Yagi, T. Hatsuda and Y. Miake, Quark-Gluon
Plasma, Cambridge University Press (2005).
3. D. Blaschke, S. Fredriksson, H. Grigorian, A. M. Oztas and F. Sandin, Phys. Rev. D 72, 065020 (2005).
4. D. Blaschke, O. Kaczmarek, E. Laermann and V. Yudichev, Eur. Phys. J. C 43, 81 (2005).
5. D. Zablocki, D. Blaschke and R. Anglani, AIP Conf.
Proc. 1038, 159 (2008).
6. D. Blaschke, G. Burau, Yu. L. Kalinovsky and
V. L. Yudichev, Prog. Theor. Phys. Suppl. 149, 182
(2003).
7. R. Rapp, D. Blaschke and P. Crochet, “Charmonium
and bottomonium production in heavy-ion collisions,”
arXiv:0807.2470.
8. D. Blaschke, T. Klaehn and F. Weber, in: Astronomy
and Relativistic Astrophysics, World Scientific, Singaprore (2010), pp. 31-47; [arXiv:0808.1279 [astroph]].
9. C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73,
014019 (2006)
10. H. Hansen, W. M. Alberico, A. Beraudo, A. Molinari, M. Nardi and C. Ratti, Phys. Rev. D 75, 065004
(2007).
11. D. Gomez Dumm, D. B. Blaschke, A. G. Grunfeld and
N. N. Scoccola, Phys. Rev. D 78, 114021 (2008).
12. D. Blaschke, M. Buballa, A. E. Radzhabov and
M. K. Volkov, Yad. Fiz. 71, 2012 (2008).
This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which
permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited.
Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100701003
EPJ Web of Conferences
E FFECTIVE
FIELD THEORIES FOR HOT AND DENSE MATTER
(I)
David Blaschke
Institute for Theoretical Physics, University of Wroclaw, Poland
Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia
• Introduction: Many-particle Systems and Quantum Field Theory
• Partition function for QCD: Lattice Simulations vs. Resonance Gas
• Bound states and Mott effect, Color superconductivity
– Heavy Quarkonia - Schrödinger Equation
– Chiral quark model - Color superconductivity
– Pions, Kaons, D-mesons - Chiral Quark Model
• Application 1: J/ψ suppression in Heavy-Ion Collisions
• Application 2: Quark Matter in Compact Stars
• Summary / Outlook to Lecture II
M ANY PARTICLE S YSTEMS & Q UANTUM F IELD T HEORY
Elements
Bound states
System
humans, animals
couples, groups, parties
society
molecules, crystals (bio)polymers
animals, plants
atoms
molecules, clusters, crystals solids, liquids, ...
ions, electrons
atoms
plasmas
nucleons, mesons
nuclei
nuclear matter
quarks, anti-quarks nucleons, mesons
Highly Compressed Matter ⇔ Pauli Principle
Partition function:
01003-p.2
quark matter
Z = Tr e−β(H−μiQi)
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
PARTITION
Q UANTUM C HROMODYNAMICS (QCD)
FUNCTION FOR
• Partition function as a Path Integral (imaginary time τ = i t, 0 ≤ τ ≤ β = 1/T ) ⇒ PS I
β
Z[T, V, μ] = D ψ̄DψDA exp −
dτ
d3x LQCD (ψ, ψ̄, A)
0
V
a
(A) = ∂μ Aa ν − ∂ν Aaμ + g f abc [Abμ , Acν ]
• QCD Lagrangian, non-Abelian gluon eld strength: Fμν
1 a
LQCD (ψ, ψ̄, A) = ψ̄[iγ μ(∂μ − igAμ ) − m − γ 0μ]ψ − Fμν
(A)F a,μν (A)
4
• Numerical evaluation: Lattice gauge theory simulations (Bielefeld group)
4
SB/T
15
• Phase transition at Tc = 170 MeV
10
/T
4
• Equation of state: ε(T ) = −∂lnZ[T, V, μ]/∂β
• Problem: Interpretation ?
Lattice QCD (2+1 flavor)
Tc=170 MeV +/- 10%
5
ε/T 4 =
Karsch et al., PLB 478 (2000) 447
0
0
4
0.5
1
2
1.5
T/Tc
2.5
3
P HASEDIAGRAM
ε/T =
3.5
OF
π2
30 Nπ ∼ 1 (ideal
π2
(NG + 87 NQ ) ∼
30
pion gas)
15.6 (quarks and gluons)
QCD: L ATTICE S IMULATIONS
D
ICA
MP
N
01003-p.3
EPJ Web of Conferences
P HASEDIAGRAM OF QCD: L ATTICE S IMULATIONS
Lattice QCD
Simulations
PD
A
NIC
M
L ATTICE QCD E O S VS . RESONANCE GAS
4
SB/T
Ideal hadron gas mixture ...
p2 + m2i
d3 p
gi
ε(T ) =
3
2
(2π)
exp( p + m2i /T ) + δi
i=π,ρ,...
10
/T
4
15
, m=770 MeV
+
Lattice (2+1 flavor)
missing degrees of freedom below and above Tc
5
0
0
0.5
1
2
1.5
T/Tc
3
2.5
3.5
Resonance gas ...
Karsch, Redlich, Tawk, Eur.Phys.J. C29, 549 (2003)
ε(T ) =
εi(T )
4
SB/T
10
i=π,ρ,...
/T
4
15
Hagedorn gas
Lattice (2+1 flavor)
5
+
r=M,B
β
0
0
0.5
1
2
1.5
T/Tc
2.5
3
3.5
gr
dm ρ(m)
d3 p
p2 + m2
(2π)3 exp( p2 + m2 /T ) + δr
ρ(m) ∼ m exp(m/TH ) ... Hagedorn mass spektrum
too many degrees of freedom above Tc
01003-p.4
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
εR(T, {μj }) =
L ATTICE QCD E O S
AND
i=π,K,...
εi(T, {μi})+
r=M,B
gr
dm
mr
M OTT-H AGEDORN G AS
ds ρ(m)A(s, m; T )
p2 + s
√
d3 p
(2π)3
p2 +s−μr
T
exp
Hagedorn mass spectrum: ρ(m)
+ δr
Spectral function for heavy resonances:
TH=1.2 Tc
A(s, m; T ) = Ns
resonances
10
/T
4
15
Ansatz with Mott effect at T = TH = 180 MeV:
Γ(T ) = BΘ(T − TH )
quarks + gluons
5
0
mΓ(T )
(s − m2)2 + m2Γ2(T )
m
TH
2.5
6
T
TH
exp
m
TH
No width below TH : Hagedorn resonance gas
Apparent phase transition at Tc ∼ 150 MeV
Blaschke & Bugaev, Fizika B13, 491 (2004)
0.5
1
2
T/Tc
1.5
3
2.5
Prog. Part. Nucl. Phys. 53, 197 (2004)
3.5
Blaschke & Yudichev, in preparation
Bugaev, Petrov, Zinovjev, arXiv:0812.2189
H ADRONIC C ORRELATIONS
Ha d
r on
ABOVE
Tc : L ATTICE QCD
()
2.5
ic
(a)
Co
D
CA
P
M
T = 0.78Tc
2
r
r
e
l
a
T = 1.38Tc
T = 1.62Tc
1.5
t
1
i
o
n
s
0.5
NI
0
(b)
T = 1.87Tc
2
Hadron correlators GH =⇒ spectral densities ρH (ω, T )
∞
cosh(ω(τ − T /2))
GH (τ, T ) =
dωρH (ω, T )
sinh(ω/2T )
0
Maximum entropy method
Karsch et al. PLB 530 (2002) 147
Result:
Correlations persist above Tc !
Karsch et al. NPA 715 (2003)
T = 2.33Tc
1.5
1
0.5
0
0
5
10
15
20
25
30
[GeV]
J/ψ and ηc survive up to T ∼ 1.6Tc
Asakawa, Hatsuda; PRL 92 (2004) 012001
01003-p.5
EPJ Web of Conferences
H ADRONIC C ORRELATIONS
IN THE
P HASEDIAGRAM
OF
QCD
OF
QCD
Ha d
r on
ic
Co
PD
CA
M
rr
e
la
t
i
o
n
s
NI
H ADRONIC C ORRELATIONS
IN THE
A
NIC
MP
D
01003-p.6
P HASEDIAGRAM
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
H EAVY Q UARK P OTENTIAL
FROM
L ATTICE QCD
Color-singlet free energy F1 in quenched QCD
Tr[L(0)L†(r)] = exp[−F1(r)/T ]
1
F1(r)/ , V1(r)/
1/2
Long- and short- range parts
2
F1 (r, T ) = F1,long (r, T ) + V1,short(r)e−(µ(T )r)
1/2
0
F1,long (r, T ) = ′ screened′confinementpot.
4 α(r)
, α(r) = runningcoupl. (1)
V1,short(r) = −
3 r
T=1.013 Tc
T=1.151 Tc
T=1.500 Tc
T=1.684 Tc
T=2.999 Tc
-1
0
T=0
1
2
1/2
r
3
4
Quarkonium (QQ̄) 1S
1P1
2S
Charmonium (cc̄) J/ψ(3097) χc1 (3510) ψ ′ (3686)
Bottomonium (bb̄) Υ (9460) χb1 (9892) Υ′ (10023)
=⇒ Wong (DM 2, 5); Lombardo (DM 3, 7)
Blaschke, Kaczmarek, Laermann, Yudichev,
EPJC 43, 81 (2005); [hep-ph/0505053]
S CHROEDINGER
EQN : BOUND
SCATTERING STATES
Quarkonia bound states at nite T :
300
1S, charmonium
1S, bottomonium
2S, bottomonium
250
EB [MeV]
&
200
[−∇2/mQ + Veff (r, T )]ψ(r, T ) = EB (T )ψ(r, T )
150
100
Binding energy vanishes EB (TMott) = 0: Mott effect
50
0
1
S(k)
3
1.5
Charmonium
1.01
1.10
1.20
1.24
1.50
1.68
2.21
3.00
T / Tc =
2
1
S(k)
6
Bottomonium
1.01
1.05
1.10
1.15
1.30
1.68
2.21
3.00
T / Tc =
4
2
0
0
2
Scattering states:
2
T / Tc
k [GeV]
dδS (k, r, T )
mQVeff
=−
sin(kr + δS (k, r, T ))
dr
k
Levinson theorem:
Phase shift at threshold jumps by π when
bound state → resonance at T = TMott
Blaschke, Kaczmarek, Laermann, Yudichev
EPJC 43, 81 (2005); [hep-ph/0505053]
4
01003-p.7
EPJ Web of Conferences
P HASEDIAGRAM
OF
QCD: C HIRAL M ODEL F IELD T HEORIES
Chira
l Q
ua
rk
Mo
de
l
Fi
el
d
Th
eo
ry
PD
CA
M
NI
C HIRAL M ODEL F IELD T HEORY
FOR
Q UARK M ATTER
• Partition function as a Path Integral (imaginary time τ = i t)
β
dτ
d3x[ψ̄(iγ μ∂μ − m − γ 0μ)ψ − Lint]
Z[T, V, μ] = D ψ̄Dψ exp −
V
• Current-current interaction (4-Fermion coupling)
Lint = M=π,σ,... GM (ψ̄ΓM ψ)2 +
D
GD (ψ̄ C ΓD ψ)2
• Bosonization (Hubbard-Stratonovich Transformation)
Z[T, V, μ] =
DMM DΔ†D DΔD exp −
|ΔD |2 1
M2
M
−
+ Tr lnS −1 [{MM }, {ΔD }]
4GM
4GD
2
M
D
• Collective (stochastic) elds: Mesons (MM ) and Diquarks (ΔD )
• Systematic evaluation: Mean elds + Fluctuations
– Mean-eld approximation: order parameters for phase transitions (gap equations)
– Lowest order uctuations: hadronic correlations (bound & scattering states)
– Higher order uctuations: hadron-hadron interactions
01003-p.8
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
NJL
MODEL FOR NEUTRAL
3- FLAVOR
Thermodynamic Potential Ω(T, μ) = −T ln Z[T, μ]
Ω(T, μ) =
φ2u + φ2d + φ2s |Δud|2 + |Δus|2 + |Δds|2
+
−T
8GS
4GD
n
InverseNambu − GorkovPropagator
−1
S (iωn , p) =
QUARK MATTER
d3p 1
1 −1
Tr ln
S (iωn , p) + Ωe − Ω0.
(2π)3 2
T
p)
Δ(
γμ pμ − M (p) + μγ 0
†
μ
Δ (p)
γμp − M (p) − μγ 0
p) = iγ5ǫαβγ ǫijk Δkγ g(p) ; Δkγ = 2GD q̄iα iγ5ǫαβγ ǫijk g(q)q C .
Δ(
jβ
Fermion Determinant (Tr ln D = ln det D): lndet[β S −1 (iωn , p)] = 2
18
2 2
a=1 ln{β [ωn
,
+ λa (p)2]} .
Result for the thermodynamic Potential (Meaneld approximation)
18
d3p
φ2 + φ2d + φ2s |Δud|2 + |Δus|2 + |Δds|2
λa + 2T ln 1 + e−λa /T + Ωe − Ω0.
+
−
Ω(T, μ) = u
8GS
4GD
(2π)3 a=1
Color and electric charge neutrality constraints: nQ = n8 = n3 = 0, ni = −∂Ω/∂μi = 0,
Equations of state: P = −Ω, etc.
O RDER PARAMETERS : M ASSES
D IQUARK G APS
Left: Gap in excitation spectrum (T = 0)
Right: ’Gapless’ excitations (T = 60 MeV)
Masses (M ) and Diquark gaps (Δ) as a
function of the chemical potential at T = 0
600
500
500
400
Ms
Mu
Md
Mu, Md
ud
us, ds
400
300
E [MeV]
, M [MeV]
AND
ug-dr
ub-sr, db-sg
ur-dg-sb
ub-sr
db-sg
300
200
200
100
100
300
350
400
450
[MeV]
500
550
0
0
01003-p.9
200
400
p [MeV]
0
200
400
p [MeV]
600
EPJ Web of Conferences
M OTT
EFFECT:
NJL
MODEL PRIMER
RPA-type resummation of quark-antiquark scattering
in the mesonic channel M,
Meson masses [MeV]
600
TMott=186 MeV
M
M
400
+... =
=
1 J M (P ,P; T)
denes Meson propagator
200
M
DM (P0, P ; T ) ∼ [1 − JM (P0 , P ; T )]−1,
0
0
by the complex polarization function JM
→ Breit-Wigner type spectral function
50
200
100
150
T [MeV]
250
1
Im DM (P0, P ; T )
π
ΓM (T )MM (T )
1
∼
2 (T ))2 + Γ2 (T )M 2 (T )
π (s − MM
M
M
AM (P0 , P ; T ) =
2.5
Meson Masses, Widths [GeV]
M
M
+
NJL with IR cutoff
2
1.5
bound state mass
resonance mass
resonance width
K
1
D
For T < TMott: Γ → 0, i.e. bound state
2
(T ))
AM (P0, P ; T ) = δ(s − MM
D*
Light meson sector:
Blaschke, Burau, Volkov, Yudichev: EPJA 11 (2001) 319
0.5
Mott
T
0
0
= 186 MeV
100
200
Charm meson sector:
Blaschke, Burau, Kalinovsky, Yudichev,
Prog. Theor. Phys. Suppl. 149 (2003) 182
300
T [MeV]
P HASEDIAGRAM
QCD: H EAVY-I ON C OLLISIONS
OF
Lattice QCD
Simulations
D
ICA
MP
N
01003-p.10
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
P HASEDIAGRAM
OF
QCD: L ATTICE
VS .
H EAVY-I ON C OLLISIONS
QGP Signal: Anomalous J/ suppression
Lattice QCD
A
NIC
MP
J/ Suppression factor
Simulations
D
CERN - SPS
1.4
1.2
1
0.8
pp(d) NA51
pA NA38
SU NA38
Pb Pb 1996
Pb Pb 1998
0.6
0.4
0.2
Ratios
0
0
p /p
/
/
-
+
-/+ K /K
-
K /-
*0
-
*0
p /- K /h K /h
-
1
T
f.o.
10
-1
= 174 MeV
f.o.= 46 MeV
STAR
PHENIX
PHOBOS
BRAHMS
10
0.5
1
1.5
2
2.5
3
[GeV/fm ]
3
3.5
Statistical model describes composition of hadron yields in
Heavy-Ion Collisions with few freeze-out parameters.
gi ∞
dp p2 ln[1 ± λi exp(−βεi(p))]
ln Z[T, V, {μ}] = ±V
2π 2 0
i
λi (T, {μ}) = exp[β(μB Bi + μS Si + μQ Qi )]
Braun-Munzinger, Redlich, Stachel, in QGP III (2003)
RHIC
Brookhaven
-2
A SNAPSHOP
OF THE S QGP
⇐⇒
The Picture: String-ip (Rearrangement)
Pair correlation
g(r)
1
1
r NN
2
Horowitz et al. PRD (1985), D.B. et al. PLB (1985),
Röpke, Blaschke, Schulz, PRD (1986)
Thoma,[hep-ph/0509154]
Gelman et al., PRC 74 (2006)
• Strong correlations present: hadronic spectral functions above Tc (lattice QCD)
• Finite width due to rearrangement collisions (higher order correlations)
• Liquid-like pair correlation function (nearest neighbor peak)
01003-p.11
r
EPJ Web of Conferences
Q UANTUM
KINETIC APPROACH TO
Tqq
Uex
τ −1(p) = Γ(p) = Σ>(p) ∓ Σ<(p)
<
>
>
>
′
<
<
(2π)4δp,p′ ;p1 ,p2 |M|2 G>
Σ<(p, ω) =
π (p ) GD1 (p1 ) GD2 (p2 )
Uex
TqQ
p′
TQQ
G>
h (p)
10
1/2
Mott effect ( ~(TTc) )
Mott Effect ( ~(TTc))
no Mott effect
<v> [mb]
1
0.1
p2
= [1 ± fh (p)]Ah(p) and G<
h (p) = fh (p)Ah (p)
3 ′
d
p
τ −1 (p) =
ds′ fπ (p′, s′ ) Aπ (s′)vrel σ ∗ (s)
(2π)3
Medium effects in spectral functions Ah and σ(s; s1, s2)
Ah (s) =
0.001
0.05
0.1
0.15
0.2
1
Γh (T ) Mh (T )
−→ δ(s − Mh2 )
π (s − Mh2 (T ))2 + Γ2h (T )Mh2(T )
resonance ⇐ Mott-effect ⇐ bound state
0.25
T [GeV]
Blaschke et al., Heavy Ion Phys. 18 (2003) 49
“A NOMALOUS ” J/ψ
Suppression factor
p1
In-medium breakup cross section
σ ∗(s) = ds1 ds2 AD1 (s1) AD2 (s2) σ(s; s1, s2)
0.01
J/
BREAKUP
Inverse lifetime for Charmonium states
TqQ
TQQ
J/ψ
SUPPRESSION IN
M OTT-H AGEDORN
1.4
Survival probability for J/ψ
1.2
S(ET )/SN (ET ) = exp −
1
0.4
0.2
0
0
tf
dt τ
−1
(n(t))
t0
0.8
0.6
GAS
Threshold: Mott effect for hadrons
pp(d) NA51
pA NA38
SU NA38
Pb Pb 1996
Pb Pb 1998
this work
0.5
Blaschke and Bugaev, Prog.
Nucl. Phys. 53 (2004) 197
1
1.5
2
2.5
3
[GeV/fm ]
3
3.5
In progress: full kinetics with gain processes (D-fusion), HIC simulation
01003-p.12
Part.
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
P HASEDIAGRAM
OF DEGENERATE
Q UARK M ATTER
Chira
l Q
ua
rk
Mo
de
l
Fi
el
d
Th
eo
ry
PD
CA
M
NI
P HASEDIAGRAM
Chiral
OF DEGENERATE
Q UARK M ATTER
Qu
ark
Mo
del
Fi
eld
PD
CA
M
NI
01003-p.13
Th
eo
ry
EPJ Web of Conferences
Q UARK MATTER IN COMPACT STARS
The phases are characterized by 3 gaps:
80
70
NQ
g2SC
60
• NQ-2SC: Δud = 0, Δus = Δds = 0, 0≤ χ2SC ≤1;
• 2SC: Δud = 0, Δus = Δds = 0;
guSC
50
• uSC: Δud = 0, Δus = 0, Δds = 0;
2SC
• CFL: Δud = 0, Δds = 0, Δus = 0;
40
Result:
1.0
0.9
20
0.8
0.7
0
• CFL only at high chemical potential,
s
10
• Gapless phases only at high T,
175
2SC =
0 MeV
30
M =20
T [MeV]
• NQ: Δud = Δus = Δds = 0;
gCFL
NQ-2SC
400
350
450
[MeV]
500
CFL
• At T ≤25-30 MeV: mixed NQ-2SC phase,
550
• Critical point (Tc ,μc)=(48 MeV, 353 MeV),
• Strong coupling, η = 1, changes?.
Rüster et al: PRD 72 (2005) 034004
Blaschke et al: PRD 72 (2005) 065020
Abuki, Kunihiro: NPA 768 (2006) 118
=⇒ Zhuang (DM 12, 17)
Q UARK MATTER IN COMPACT STARS : M ASS -R ADIUS CONSTRAINT
Solve TOV Eqn. → Hybrid stars fulll constraint!
• Isolated Neutron star RX J1856:
M-R constraint from thermal
emission
2.5
RX J1856
M [Msun]
2.0
c
1.5
au
i
sal
ty
lim
it
4U 1636 -536
4U 0614 +09
1.0
• Low-mass X-ray binary 4U 1636:
Mass constraint from ISCO obs.
NJL330 D = 0.75, V = 0.00
NJL330 D = 1.03, V = 0.25
0.5
DBHF
0
6
8
10
R [km]
12
14
16
Klähn et al: Constraints on the high-density EoS ...
PRC 74 (2006); [nucl-th/0602038], [astro-ph/0606524]
01003-p.14
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
Q UARK
MATTER IN COMPACT STARS :
C OOLING
CONSTRAINT
Quark matter in compact stars: color superconducting
• Neutrinos carry energy off the star,
Cooling evolution (schematic) by
2
6,4
Crab
Model IV
ǫγ + j=Urca,... ǫjν
dT (t)
=−
i
dt
i=q,e,γ,... cV
Vela
5,8
5,6
1.05
1.13
1.22 (critical)
1.28
1.35
1.45
1.55
1.65
1.75
5,2
5
1.5
2
3
4
5
log10(t[yr])
• Most efcient process: Urca
-
e-
1
hybrid stars
5,4
Nstar
hybrid
M [Msol]
3C58
typical stars
6
RX J1856
log10(Ts [K])
6,2
d
6
10
12
R [km]
0.5
Popov et al: Neutron star cooling constraints ...
PRC 74, 025803 (2006); [nucl-th/0512098]
u
• Exponential suppression by pairing
gaps! Δ ∼ 10...100 keV
S UMMARY
• Mott-Hagedorn model as alternative interpretation of Lattice data
• Microscopic formulation of the hadronic Mott effect within a chiral quark model
• Mesonic (hadronic) correlations important for T > Tc
• Step-like enhancement of threshold processes due to Mott effect
• Reaction kinetics for strong correlations in plasmas applicable @ SPS and RHIC
• Prospects for LHC: Plasma diagnostics with bottomonium
L ECTURE II: NJL
MODEL AND ITS RELATIVES
• Polyakov-loop Nambu–Jona-Lasinio (NJL) model
• Nonlocal NJL models
• Schwinger-Dyson Equation approach at nite T, μ
• Walecka model - towards a unied model of quark-hadron matter
01003-p.15
EPJ Web of Conferences
E FFECTIVE
FIELD THEORIES FOR HOT AND DENSE MATTER
NJL
(II)
MODEL AND ITS RELATIVES
David Blaschke
Institute for Theoretical Physics, University of Wroclaw, Poland
Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia
• NJL Model and its Polyakov-Loop Extension:
– Mesonic correlations - Mott Effect
– Polyakov-Loop NJL Model
• Nonlocal, separable NJL Model
– 3D Formfactors, 4D Formfactors and Phase Diagram
– Rank-2 Extension - Schwinger-Dyson type Approach
• Summary / Outlook to a Unied Quark-Hadron Approach
Literature: Hansen et al., Phys. Rev. D75, 065004 (2007); Gomez Dumm et al., Phys. Rev.
D73, 114019 (2006); arXiv:0807.1660; Blaschke et al., arXiv:0705.0384; Schmidt et al., Phys.
Rev. C50, 435 (1994); Zablocki at al., arXiv:0805.2687
H ADRONIC C ORRELATIONS
IN THE
D
ICA
MP
N
01003-p.16
P HASEDIAGRAM
OF
QCD
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
H ADRONIC C ORRELATIONS
P HASEDIAGRAM
IN THE
OF
QCD
Ch
iral
Qu
ark
Mod
el
Field
Theo
ry
D
ICA
MP
N
C HIRAL M ODEL F IELD T HEORY
FOR
Q UARK M ATTER
• Partition function as a Path Integral (imaginary time τ = i t)
β
Z[T, V, μ] = D ψ̄Dψ exp −
dτ
d3x[ψ̄[iγ μ ∂μ − m − γ 0(μ + λ8μ8 + iλ3φ3]ψ − Lint + U (Φ)]
V
Polyakov loop: Φ =
Nc−1 Trc[exp(iβλ3φ3)]
Order parameter for deconnement
• Current-current interaction (4-Fermion coupling)
Lint = M=π,σ,... GM (ψ̄ΓM ψ)2 +
D
GD (ψ̄ C ΓD ψ)2
• Bosonization (Hubbard-Stratonovich Transformation)
⎧
⎫
⎨ M2
⎬
2
|Δ
|
1
D
M
−
+ Tr lnS −1 [{MM }, {ΔD }, Φ] + U (Φ)
Z[T, V, μ] = DMM DΔ†D DΔD exp −
⎩
⎭
4GM
4GD
2
M,D
• Collective quark elds: Mesons (MM ) and Diquarks (ΔD ); Gluon mean eld: Φ
• Systematic evaluation: Mean elds + Fluctuations
– Mean-eld approximation: order parameters for phase transitions (gap equations)
– Lowest order uctuations: hadronic correlations (bound & scattering states)
– Higher order uctuations: hadron-hadron interactions
01003-p.17
EPJ Web of Conferences
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (I)
SU (Nc ) pure gauge sector: Polyakov line
β
L (x) ≡ P exp i
dτ A4 (x, τ ) ; A4 = iA0 = λ3φ3 + λ8φ8
0
Polyakov loop
1
TrL(x) , l(x) = e−βΔFQ (x).
Nc
ZNc symmetric phase: l(x) = 0 =⇒ ΔFQ → ∞: Connement !
Polyakov loop eld:
1
Φ(x) ≡ l(x) =
Trc L(x)
Nc
Potential for the PL-meaneld Φ(x) =const., which ts quenched QCD lattice thermodynamics
b4 2
U Φ, Φ̄; T
b3 3
b2 (T )
Φ̄Φ −
=−
Φ̄Φ ,
Φ + Φ̄3 +
4
T
2
6
4
l(x) =
T0
T
b2 (T ) = a0 + a1
+ a2
2
T0
T
+ a3
3
T0
T
.
a0
a1
a2
a3
b3
b4
6.75 -1.95 2.625 -7.44 0.75 7.5
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (II)
Temperature dependence of the Polyakov-loop potential U (Φ, Φ̄; T )
5
4
U( ) / T
4
0
-0.2
-0.4
-0.6
-0.8
-1
-1.2
-1.4
U( ) / T
4
5
4
3
2
1
0
-1
-2
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
3
2
1
0
1.5
1.5
1
1
0.5
0.5
-1.5
-1.5
0
-1
-0.5
-0.5
0
0.5
Re
0
-1
-0.5
Im
1
-0.5
0
0.5
-1
Re
-1.5
1.5
-1
1
-1.5
1.5
T = 1.0 GeV> T0
“Color deconnement”
T = 0.26 GeV< T0
“Color connement”
Critical temperature for pure gauge SUc (3) lattice simulations: T0 = 270 MeV.
Hansen et al., Phys.Rev. D75, 065004 (2007)
01003-p.18
Im
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (III)
Lagrangian for Nf = 2, Nc = 3 quark matter, coupled to the gauge sector
LP NJL = q̄(iγ μDμ − m̂ + γ0μ)q + G1 (q̄q)2 + (q̄iγ5τ q)2 − U Φ[A], Φ̄[A]; T ,
D μ = ∂ μ − iAμ ; Aμ = δ0μ A0 (Polyakov gauge), with A0 = −iA4
>
=
Diagrammatic Hartree equation:
+
= −(p/ − m + γ 0(μ − iA4))−1
Dynamical chiral symmetry breaking σ = m − m0 =
0? Solve Gap Equation! (E = p2 + m2)
+∞
−1
d3p
m − m0 = 2G1T Tr
3
/
p
−
m
+
γ 0(μ − iA4)
(2π)
Λ
n=−∞
3
d p 2m
= 2G1Nf Nc
[1 − fΦ+ (E) − fΦ− (E)]
3
Λ (2π) E
S0 (p) =
= −(p/ − m0 + γ 0(μ − iA4))−1 ; S(p) =
Modied quark distribution functions (Φ = Φ̄ = 0: “poor man’s nucleon”: EN = 3E, μN = 3μ)
Φ + 2Φ̄e−β (Ep∓μ) e−β (Ep∓μ) + e−3β (Ep∓μ)
1
fΦ± (E) =
−→ f0± (E) =
β(EN ∓μN )
−β (Ep ∓μ)
−β (Ep ∓μ)
−3β (Ep ∓μ)
1
+
e
1 + 3 Φ + Φ̄e
e
+e
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (IV)
1
ΨΨΨΨT0
0.8
0.6
0.4
0.2
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
T GeV
1
3
nq/T
0.8
0.6
Grand canonical thermodynamical potential
d3 p
σ2
E θ Λ2 − p 2
− 6Nf
Ω(T, μ; Φ, m) =
2G
(2π)3
d3p
−(E−μ)/T
− 2Nf T
3 Trc ln 1 + L e
(2π)
+ Trc ln 1 + L† e−(E+μ)/T
+ U Φ, Φ̄, T
Appearance of quarks below Tc largely suppressed:
ln det 1 + L e−(E−μ)/T + ln det 1 + L† e−(E+μ)/T
= ln 1 + 3 Φ + Φ̄e−(E−μ)/T e−(E−μ)/T + e−3(E−μ)/T
+ ln 1 + 3 Φ̄ + Φe−(E+μ)/T e−(E+μ)/T + e−3(E+μ)/T .
Accordance with QCD lattice susceptibilities! Example:
0.4
nq (T, μ)
1 ∂Ω (T, μ)
=− 3
,
T3
T
∂μ
0.2
0
0
=0.6 Tc
0.5
1
1.5
2
Ratti, Thaler, Weise, PRD 73 (2006) 014019.
T/Tc
01003-p.19
EPJ Web of Conferences
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (V)
Mesonic currents
JPa (x) = q̄(x)iγ5τ a q(x)
... and correlation functions
PP 2
Cab
(q ) ≡ i
(pion) ; JS (x) = q̄(x)q(x) − q̄(x)q(x)
(sigma)
d4xeiq.x 0|T JPa (x)JPb† (0) |0 = C P P (q 2)δab
SS 2
Cab
(q ) ≡ i
Schwinger-Dyson Equations, T = μ = 0
d4xeiq.x0|T JS (x)JS† (0) |0
C MM (q 2) = ΠMM (q 2) +
′
′
ΠMM (q 2)(2G1)C M M (q 2)
M′
One-loop polarization functions
′
ΠMM (q 2) ≡
d4p
Tr (ΓM S(p + q)ΓM ′ S(q))
4
Λ (2π)
Hartree quark propagator S(p)
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (VI)
Example of the pion channel:
m2 − p2 + q 2 /4
d4p
ΠP P (q 2) = −4iNcNf
= 4iNcNf I1 − 2iNcNf q 2I2(q 2)
4
2
2
2
2
Λ (2π) [(p + q/2) − m ][(p − q/2) − m ]
Loop Integrals:
I1 =
1
d4p
;
4 2
2
Λ (2π) p − m
I2(q 2) =
d4p
1
4
2
2
2
2
Λ (2π) [(p + q) − m ] [p − m ]
With pseudoscalar decay constant (fP ) and gap equation for I1
fP2 (q 2) = −4iNcm2I2(q 2) ; I1 =
2
2 2 q
SS 2
0
One obtains ΠP P (q 2) = m−m
2G1 m + fP (q ) m2 ; Π (q ) =
(m0 = 0), the correlation functions
C MM (q 2) = ΠMM (q 2 ) + ΠMM (q 2)(2G1)C MM (q 2) =
m − m0
,
8iG1mNcNf
m−m0
2G1 m
+ fP2 (q 2 ) q
2 −4m2
m2
. In the chiral limit
ΠMM (q 2)
, M = P, S ,
1 − 2G1ΠMM (q 2)
have poles at q 2 = MP2 = 0 (Pion) and q 2 = MS2 = (2m)2 (Sigma meson) =⇒ Check !
01003-p.20
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (VII)
d4 p
d3 p
1
Finite T, μ: p = (p0, p) → (iωn + μ − iA4, p) ; i Λ (2π)
4 → −T Nc Trc
n Λ (2π)3
d3p 1 − f (Ep − μ) − f (Ep + μ)
I1 = −i
(2π)3
2Ep
Λ3
dp
1
f (Ep + μ) + f (Ep − μ) − f (Ep+q + μ) − f (Ep+q − μ)
I2(ω, q) = i
(2π)3 2Ep2Ep+q
ω − Ep+q + Ep
Λ3
1
d p 1 − f (Ep − μ) − f (Ep+q + μ)
1
−
+i
3
2Ep2Ep+q
ω + Ep+q + Ep ω − Ep+q − Ep
Λ (2π)
(1)
For a meson at rest in the medium (q = 0)
d3p 1 − f (Ep + μ) − f (Ep − μ)
I2 ω, 0 = −i
3
Ep ω 2 − 4Ep2
Λ (2π)
which develops an imaginary part
ω
ω
ω 2 − 4m2
1
1−f
×Θ(ω 2 −4m2)Θ(4(Λ2+m2)−ω 2)
−μ −f
+μ
16π
2
2
ω2
with the Pauli-blocking factor: N (ω, μ) = 1 − f ω2 − μ − f ω2 + μ
ℑm (−iI2(ω, 0)) =
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (VIII)
Spectral function
ΠMM (ω + iη, q)
.
1 − 2G1ΠMM (ω + iη, q)
π 1
2G1ℑm ΠMM (ω + iη)
.
F MM (ω) =
MM
2G1 π (1 − 2G1ℜe Π (ω))2 + (2G1ℑm ΠMM (ω + iη))2
F MM (ω, q) ≡ ℑm C MM (ω + iη, q) = ℑm
For ω < 2m(T, μ), ℑm Π = 0: decay channel closed → bound state!
π
π
!
!
F MM (ω) =
δ 1 − 2G1ℜe ΠMM (ω) =
! ∂ℜe ΠM M !
2G1
2
4G !
!
1
The meson mass mM is the solution of
1 − 2G1ℜe ΠMM (mM ) = 0
The decay width (inverse lifetime) is
ΓM = 2G1ℑm ΠMM (mM )
01003-p.21
∂ω
ω=mM
δ(ω − mM ) .
(2)
EPJ Web of Conferences
P ION C ORRELATIONS
IN THE
P HASE D IAGRAM
300
300
250
) [arb. units]
250
>50 MeV
200
SB
150
(
T [MeV]
m
=0
5
D = 1.0
4
m
3
2
200
1
0
150
0
200 400 600 800 1000
T=0
T = 200 MeV
T = 215 MeV
T = 250 MeV
100
=0
100
50
50
D= 1.00
0
0
0
0
2SC
100
200
[MeV]
300
200
400
600
[MeV]
400
800
1000
Zablocki, D.B., Anglani, arXiv:0805.2687 [hep-ph]
C OLOR N EUTRALITY
IN THE
PNJL P HASE D IAGRAM
Color neutrality constraint: μ̃ = μ1 + μ8λ8 + iφ3λ3 ; ∂ΩMF /∂μ8 = n8 = nr + ng − 2nb = 0
Gap equations: ∂ΩMF /(∂σ, ∂Δ, ∂φ3) = 0
First order transition
Second order transition
Smooth crossover
250
Without color
neutrality
250
200
150
With color
neutrality
100
T=0
T = 50 MeV
T = 100 MeV
200
NQM
[ MeV ]
200
T [ MeV ]
T [ MeV ]
250
150
SB
100
150
100
EP
50
50
SB - 2SC
2SC
200
400
50
0
0
0
50
100
150
200
250
[ MeV ]
300
350
400
0
0
100
300
[ MeV ]
500
0
100
200
Gomez-Dumm, D.B., Grunfeld, Scoccola, PRD 78, 114021 (2008) [arXiv:0807.1660]
01003-p.22
300
[ MeV ]
400
500
Dense Matter In Heavy Ion Collisions and Astrophysics (DM2008)
N ONLOCAL P OLYAKOV L OOP C HIRAL Q UARK M ODEL
3-avor, rank-2, 4D separable
susceptibilities:
2-avor, rank-1, 4D separable
order parameters:
susceptibilities
40
d()/dT
dmu,d(T)/dT
Nf=2+1
Nf=2
dms(T)/dT
30
20
10
0.18
0.19
0.20
temperature T[GeV]
D.B., Buballa, Radzhabov, Volkov,
Yad. Fiz. 71 (2008); arXiv:0705.0384
C OMPLEX
0.2
D.B., Horvatic, Klabucar, in prep.
MASS POLE FIT TO
L ATTICE
PROPAGATOR
S(p) sum of N pairs of complex conj. mass poles
N
1
zi
zi∗
S(p) =
+
= −i/
pσV (p2) + σS (p2)
∗
Z
i/
p
+
m
i/
p
+
m
2
i
i
i=1
1.0
0.8
Representation of the scalar amplitude
N
zi∗ m∗i
zi mi
σS (p2) =
+
Z2−1
p2 + m2i p2 + m∗i 2
i=1
0.6
0.4
0.2
0.0
0.0
0.19
temperature T[GeV]
1.0
2.0
p (GeV)
3.0
4.0
B HAGWAT, P ICHOWSKY, ROBERTS ,
TANDY, P HYS . R EV. C68 (2003)
015203
S(p)−1 = i/
pA(p2)+ B(p2) ,
M (p2) = B(p2)/A(p2)
Z(p2) = 1/A(p2)
“Derivation” of the equivalent separable model (in
Feynman-like gauge) Dμν (p − q) = δμν D(p, q) and
D(p, q) = f0(p2) f0(q 2) + f1(p2) p · q f1 (q 2)
A(p2) − 1
B(p2) − mc
f1(p2) =
; f0(p2) =
a
b
16 Λ
[B(q 2) − mc]σs(q 2 )
3 q
q2
8 Λ
a2 =
[A(q 2 ) − 1] σv (q 2 )
3 q
4
b2 =
01003-p.23
EPJ Web of Conferences
N UCLEONS
IN THE
N ONLOCAL C HIRAL Q UARK M ODEL
|Δ|2
− T r lnS −1 [Δ, Δ†]}
4GD
Cahill, Roberts, Prashifka: Aust. J. Phys. 42 (1989) 129, 161
Cahill, ibid, 171; Reinhardt: PLB 244 (1990) 316; Buck, Alkofer, Reinhardt: PLB 286 (1992) 29
Zfluct =
DΔ†DΔ exp{−
Quark sextett (diquark triplett): bound by exchange forces? sextett condensate?
S UMMARY
• Compressed nuclear matter: quarkyonic phase (QP)! Coexisting chiral symm. + conf.
• Similarities: Mott-Hagedorn picture, string-ip model, conning DSE
• Here: PNJL model as microscopic formulation of the QP
• Color singlet quark triplets in chiral phase for μ > μc (approx. massless baryons)
• Color neutrality by singlet projection = sum over color hexagon
• Prospects for CBM & NICA: dilepton enhancement (peak?) from diquark-antidiquark annih.
• Preparatory step to compact stars: single avor CSL phase - OK with structure & cooling
O UTLOOK :
NEXT STEPS
...
• Walecka model as limit of PNJL model: chiral transition effects in nuclear EoS
• Beyond meaneld: mesons and baryons in the PNJL, higher clusters: sextetting
• Astrophysics: Maximum mass & cooling of quarkyonic stars; quarkyonic supernovae
01003-p.24
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MARKUS MUELLER
Paul Scherrer Institute
Maksym Strikha
Taras Shevchenko National University of Kyiv
Virginie Boucher
CSIC (Consejo Superior de Investigaciones Científicas-Spanish National Research Council)
Miguel Angel Rubio
Universidad Nacional de Educación a Distancia
