PHASEDIAGRAM: FREEZE-OUT IN HEAVY-ION COLLISIONS. Ratios. 10. -2 .....
ud|2 + |∆us|2 + |∆ds|2. 4GD ..... 200 400 600 800 1000 1200 1400. T [MeV]. 0.
T HE
QUEST FOR THE DENSE MATTER PHASE DIAGRAM AND
EOS
David Blaschke Institute for Theoretical Physics, University of Wroclaw, Poland Bogoliubov Laboratory for Theoretical Physics, JINR Dubna, Russia • Current QCD Phase Diagram: – Lattice QCD – HIC: Chem. Freezeout – Astrophysics: Compact Stars • (Nonlocal) PNJL Model • Compact Stars and Supernovae • Freeze-out in Chiral HRG • Outlook HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
HIC: H ADRONIC ( CHEMICAL )
FREEZE - OUT
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
Ratios
P HASEDIAGRAM : F REEZE - OUT
Λ /Λ
p /p
Ξ /Ξ
π-/π+ K /K -
+
K /π-
p /π- K /h K /h *0
-
*0
1
T
f.o.
-1
10
= 174 MeV
µf.o.= 46 MeV STAR PHENIX PHOBOS BRAHMS
-
IN
H EAVY-I ON C OLLISIONS
Statistical model describes composition of hadron yields in Heavy-Ion Collisions with few freeze-out parameters. X gi Z ∞ ln Z[T, V, {µ}] = ±V dp p2 ln[1 ± λi exp(−βεi(p))] 2 2π 0 i λi (T, {µ}) = exp[β(µB Bi + µS Si + µQ Qi)]
Braun-Munzinger, Redlich, Stachel, in QGP III (2003)
RHIC
Brookhaven
-2
10
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
Ratios
P HASEDIAGRAM : F REEZE - OUT
Λ /Λ
p /p
Ξ /Ξ
π-/π+ K /K -
+
K /π-
p /π- K /h K /h *0
-
*0
1
T
f.o.
-1
10
= 174 MeV
µf.o.= 46 MeV STAR PHENIX PHOBOS BRAHMS
-
IN
H EAVY-I ON C OLLISIONS (II)
Statistical model describes composition of hadron yields in Heavy-Ion Collisions with few freeze-out parameters. X gi Z ∞ dp p2 ln[1 ± λi exp(−βεi(p))] ln Z[T, V, {µ}] = ±V 2 2π 0 i λi (T, {µ}) = exp[β(µB Bi + µS Si + µQ Qi)]
Braun-Munzinger, Redlich, Stachel, in QGP III (2003)
RHIC
Brookhaven
-2
10
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P HASEDIAGRAM : F REEZE - OUT
IN
H EAVY-I ON C OLLISIONS (III)
Strange MatterHorn (Pisarski) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
IN
H EAVY-I ON C OLLISIONS (III)
10 9 8
Total
7
3
6
s/T
P HASEDIAGRAM : F REEZE - OUT
5
Mesons
4 3
Baryons
2 1 0
10
20
30
40 ___ 50 60 \/s NN (GeV)
Baryon → Meson Dominance
70
80
90
100
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
IN
H EAVY-I ON C OLLISIONS (IV)
10 9 8
Total
7
3
6
s/T
P HASEDIAGRAM : F REEZE - OUT
5
Mesons
4 3
Baryons
2 1 0
10
20
30
40 ___ 50 60 \/s NN (GeV)
Baryon → Meson Dominance
70
80
90
100
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P HASEDIAGRAM : F REEZE - OUT
IN
H EAVY-I ON C OLLISIONS (V)
10 9 8
Total
7
s/T
3
6 5
Mesons
4 3
Baryons
2 1 0
10
20
30
40 ___ 50 60 \/s NN (GeV)
70
80
90
100
Andronic et al., arxiv:0911.4806; NPA (2010) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
Q UIESCENT L OW-M ASS X- RAY B INARIES ( Q LMXB S )
• When the LMXB is accreting, we mostly observe Xrays from the disc • In some sources (“transients”) accretion can stop - and then we see only the neutron star
System names: LMXB’s, Soft X-ray transients, Neutron star binaries HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
M-R lit us a ca
Ce
• Luminosity measured 4 F = σTeff (R∞ /D)2
13
1
• Spectrum measured (ROSAT, XMM, Chandra)
→ effective temperature Teff :
M
M [Msun]
n
6 85 J1
RX
ω
1.5
M7
• Distance measured
y
2
CONSTRAINTS FROM Q LMXB’ S AND
→ photospheric radius:
p R∞ = R/ 1 − RS /R , RS = 2GM p → redshift z = 1/ 1 − RS /R − 1 ≈ RS /(2R)
0.5 DBHF 6
8
10 12 R [km]
14
Object R∞ [km] RXJ 1856 16.8 ωCen 13.6 ± 0.3 M13 12.8 ± 0.4
Reference Trumper ¨ et al. (2004) Gendre et al. (2003) Gendre et al. (2004)
Lower limit from RXJ 1856 incompatible with ω Cen and M13 ? HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
M-R Probability Distributions 2.4 0.0014 2.2
2
M (M )
1.8 1.6
ω Cen
2
0.0012 2
0.6
0.002 2 0.0018
0.001 1.8
0.5 1.5
0.0016 1.5 0.0014
0.0008 1.6
0.0012
0.4
0.0006 1.4
0.3
1.2
0.0004 1.2
0.2 0.5
1
0.0002 1
0.1
0.8 0
0 0 0
6
8
10
12
14
16
18
6
8
10
R (km)
12
14
16
18
0.001 1
1
1.4
0.8
0.0022 4U 1820-30
0.7
M (M )
M13
2.2
2.5 0.0024
EXO 1745-248
M (M )
2.4
×10-3 2.5 0.8
0.0008 0.0006 0.5 0.0004 0.0002 2
4
6
R (km)
8
10
12
14
16
18
0 0 0
R (km)
2
4
6
8
10
12
14
16
18
R (km)
Steiner, Lattimer & Brown 2010 ×10-3 0.45 2.4 0.4 2.2 0.35 2 0.3
2.2 2
M (M )
1.8 1.6
1.2
1.2 0.1
0.8
6
8
RXJ 1856
0.0016
2 0.0008
0.0014 0.0012
1.5 0.0006
1.6 0.2 1.4 0.15
X7
4U 1608-52
1.8 0.25
1.4
1
2.5 0.001
M (M )
2.4
0.001 0.0008
1 0.0004 0.0006 0.0004
0.5 0.0002
1 0.05 10
12
R (km)
14
16
18
0.8 0
0.0002 6
8
10
12
R (km)
14
16
18
0 0 0
2
4
6
8
10
12
14
16
18
0
R (km)
J.M. Lattimer, NFQCD 2010, 18 January 2010 – p. 28/36
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
Bayesian TOV Inversion • • • • •
n < 0.5ns : EOS is BBP and NV 0.5ns < n < n1 : EOS is determined by K, K ′ , Sv , γ n1 < n < n2 : EOS is polytropic with γ1 ; n > n2 : EOS is polytropic with γ2 M and R probability distributions for 7 neutron stars treated equally (0.8 M⊙ < M < 2.5 M⊙ ; 5 km < R < 18 km) Assume all prior model parameters are uniformly distributed 2.5
10
50000
22000
Steiner, Lattimer & Brown 2010
20000 2
16000 30000
10-1 20000
14000
M (M⊙ )
p (fm−4 )
18000
40000
1
1.5
12000 10000 1
8000 6000
10-2 10000
4000 0.5 2000
-3
10
1
2
3
4
5
ǫ
6 −4 (fm )
7
8
9
10
0
6
8
10
12
14
16
18
0
R (km) J.M. Lattimer, NFQCD 2010, 18 January 2010 – p. 29/36
“Measuring” the EoS with Compact Stars: M-R −→ P(e) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
The Proposed International X-Ray Observatory (IXO)
• Neutron Star Masses and Radii can be measured from H-atmosphere neutron stars with ∼ few % accuracy with IXO • Field sources are most promising targets due to their brightness, but they must be identified first • Path: simultaneous M-R measurements determine the cold dense matter EoS • Launch 2021 (NASA)
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
E QUATION
OF
S TATE
AND
S TABILITY
OF
C OMPACT S TARS
Tolman-Oppenheimer-Volkoff Equations 1. Stability: General Relativistic Hydrostatic Equilibrium −1 m(r)ε(r) 4πr3P (r) dP (r) P (r) 2Gm(r) = −G 1+ 1+ 1− dr r2 ε(r) m(r) r
M [Msun]
3
0.4
0.6 0.5 -3 nB [fm ]
0.7
0.8
GD/GS=
1.20 1.25 1.30 1.35 DBHF
0.5
0.9
6
6
0.3
85
0.2
1
13
ηD=1.20 ηD=1.25 ηD=1.30 ηD=1.35 DBHF (Bonn A)
J1
en
1.5
10
1
C
Flow constraint Danielewicz et al. (2002)
M
P [MeV/fm ]
2
ω
100
ε(r) 4π r2 dr
RX
ηV=0.1
0
G ENERAL R EL . T HEORY
ity
m(R) =
RR
C ORRECTIONS
ca us
2. Mass Distribution:
E INSTEIN
al
N EWTON
8
10 12 R [km]
14
DB, T. Kl¨ahn, R. Łastowiecki, F. Sandin, arxiv:1002.1299 [nucl-th]; J. Phys. G 37 (2010) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
C HIRAL M ODEL F IELD T HEORY
FOR
Q UARK M ATTER
• Partition function as a Path Integral (imaginary time τ = i t) Z β Z Z ¯ ¯ µ∂µ − m − γ 0(µ + λ8µ8 + iλ3φ3]ψ − Lint + U (Φ)] Z[T, V, µ] = D ψDψ exp − dτ d3x[ψ[iγ V
Polyakov loop: Φ = Nc−1Trc[exp(iβλ3φ3)]
Order parameter for deconfinement
• Current-current interaction (4-Fermion coupling) P ¯ M ψ)2 + P GD (ψ¯C ΓD ψ)2 Lint = M=π,σ,... GM (ψΓ D
• Bosonization (Hubbard-Stratonovich Transformation) Z X M2 |∆D |2 1 † −1 M Z[T, V, µ] = DMM D∆D D∆D exp − − + Tr lnS [{MM }, {∆D }, Φ] + U (Φ) 4GM 4GD 2 M,D
• Collective quark fields: Mesons (MM ) and Diquarks (∆D ); Gluon mean field: Φ
• Systematic evaluation: Mean fields + Fluctuations – Mean-field approximation: order parameters for phase transitions (gap equations) – Lowest order fluctuations: hadronic correlations (bound & scattering states) – Higher order fluctuations: hadron-hadron interactions HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
NJL
MODEL FOR NEUTRAL
3- FLAVOR
QUARK MATTER
Thermodynamic Potential Ω(T, µ) = −T ln Z[T, µ]
X Z d3p 1 φ2u + φ2d + φ2s |∆ud|2 + |∆us|2 + |∆ds|2 1 −1 Tr ln S (iωn, p~) + Ωe − Ω0. Ω(T, µ) = + −T 32 8GS 4GD (2π) T n InverseNambu − GorkovPropagator
S −1 (iωn, p~) =
"
b p) γµp − M (~p) + µγ ∆(~ b †(~p) ∆ γµ pµ − M (~p) − µγ 0 µ
0
#
C b p) = iγ5ǫαβγ ǫijk ∆kγ g(~p) ; ∆kγ = 2GD h¯ ∆(~ qiα iγ5ǫαβγ ǫijk g(~q)qjβ i. P18 −1 Fermion Determinant (Tr ln D = ln det D): lndet[β S (iωn , p~)] = 2 a=1 ln{β 2[ωn2 + λa(~p)2]} .
,
Result for the thermodynamic Potential (Meanfield approximation) Z 18 i φ2u + φ2d + φ2s |∆ud|2 + |∆us|2 + |∆ds|2 d3p X h −λa /T Ω(T, µ) = + − λa + 2T ln 1 + e + Ωe − Ω0 . 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. HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (I) SU (Nc ) pure gauge sector: Polyakov line Z β dτ A4 (~x, τ ) ; A4 = iA0 = λ3φ3 + λ8φ8 L (~x) ≡ P exp i 0
Polyakov loop
1 TrL(~x) , hl(~x)i = e−β∆FQ (~x). Nc ZNc symmetric phase: hl(~x)i = 0 =⇒ ∆FQ → ∞: Confinement ! Polyakov loop field: 1 Trc hhL(~x)ii Φ(~x) ≡ hhl(~x)ii = Nc Potential for the PL-meanfield Φ(~x) =const., which fits quenched QCD lattice thermodynamics ¯ U Φ, Φ; T b3 3 ¯ 3 b4 ¯ 2 b2 (T ) ¯ Φ +Φ + ΦΦ − ΦΦ , = − T4 2 6 4 l(~x) =
b2 (T ) = a0 + a1
T0 T
+ a2
T0 T
2
+ a3
T0 T
3
.
a0 a1 a2 a3 b3 b4 6.75 -1.95 2.625 -7.44 0.75 7.5 HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (II) ¯ T) Temperature dependence of the Polyakov-loop potential U (Φ, Φ;
5
U( Φ ) / T4
U( Φ ) / T4
0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4
4 5 4 3 2 1 0 -1 -2
3 2 1 0
0.5 0 -0.5 -1 -1.5 -2 -2.5 -3
1.5
1.5
1
1
0.5 -1.5
0.5 -1.5
0
-1 -0.5
-0.5
0 Re Φ
0.5
Im Φ
-1 1
-1.5 1.5
T = 0.26 GeV< T0 “Color confinement”
0
-1 -0.5
-0.5
0 Re Φ
0.5
Im Φ
-1 1
-1.5 1.5
T = 1.0 GeV> T0 “Color deconfinement”
Critical temperature for pure gauge SUc (3) lattice simulations: T0 = 270 MeV. Hansen et al., Phys.Rev. D75, 065004 (2007) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (III) Lagrangian for Nf = 2, Nc = 3 quark matter, coupled to the gauge sector h i 2 2 µ ¯ LP NJL = q¯(iγ Dµ − m ˆ + γ0µ)q + G1 (¯ q q) + (¯ q iγ5~τ q) − U Φ[A], Φ[A]; T ,
D µ = ∂ µ − iAµ ; Aµ = δ0µ A0 (Polyakov gauge), with A0 = −iA4 >
Diagrammatic Hartree equation:
=
+
= −(p/ − m + γ 0(µ − iA4))−1 p Dynamical chiral symmetry breaking σ = m − m0 = 6 0? Solve Gap Equation! (E = p2 + m2) +∞ Z X d3p −1 m − m0 = 2G1T Tr 3 /p − m + γ 0(µ − iA4) n=−∞ Λ (2π) Z d3 p 2m + − = 2G1Nf Nc [1 − f (E) − f (E)] Φ Φ 3 E Λ (2π) S0(p) =
= −(p/ − m0 + γ 0(µ − iA4 ))−1 ; S(p) =
¯ = 0: “poor man’s nucleon”: EN = 3E, µN = 3µ) Modified quark distribution functions (Φ = Φ −β E ∓µ ( ) p ¯ e−β (Ep∓µ) + e−3β (Ep∓µ) Φ + 2Φe 1 −→ f0± (E) = fΦ± (E) = 1 + eβ(EN ∓µN ) ¯ −β (Ep∓µ) e−β (Ep ∓µ) + e−3β (Ep ∓µ) 1 + 3 Φ + Φe HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P OLYAKOV- LOOP NAMBU –J ONA -L ASINIO M ODEL (IV) 1
XΨΨ\XΨΨ\T=0
F
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 @GeVD
1
3
nq/T
0.8 0.6
Grand canonical thermodynamical potential Z σ2 d3p 2 2 E θ Λ − p ~ Ω(T, µ; Φ, m) = − 6Nf 2G (2π)3 Z h i d3p −(E−µ)/T − 2Nf T 3 Trc ln 1 + L e (2π) h i † −(E+µ)/T ¯ + U Φ, Φ, T + Trc ln 1 + L e
Appearance of quarks below Tc largely suppressed: h i h i −(E−µ)/T † −(E+µ)/T ln det 1 + L e + ln det 1 + L e h i −(E−µ)/T −(E−µ)/T −3(E−µ)/T ¯ = ln 1 + 3 Φ + Φe e +e h i −(E+µ)/T −(E+µ)/T −3(E+µ)/T ¯ + ln 1 + 3 Φ + Φe e +e .
Accordance with QCD lattice susceptibilities! Example: 0.4
nq (T, µ) 1 ∂Ω (T, µ) , = − T3 T 3 ∂µ
0.2 0 0
µ=0.6 Tc 0.5
1
1.5
2
Ratti, Thaler, Weise, PRD 73 (2006) 014019.
T/Tc
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P HASES
OF
QCD @ E XTREMES :
NO
C OLOR N EUTRALITY
χSB
Confinement
2SC
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
N ONLOCAL P OLYAKOV L OOP C HIRAL Q UARK M ODEL 3-flavor, rank-2, 4D separable susceptibilities:
2-flavor, rank-1, 4D separable order parameters: 1.0
40
0.6
qq
T
/
qq
,
susceptibilities
0.8
0.4
0.2
dΦ(Τ)/dT dmu,d(T)/dT
Nf=2+1
Nf=2
dms(T)/dT
30 20 10
0.0 0
50
100
150
200
250
300
T (MeV)
D.B., Buballa, Radzhabov, Volkov, Yad. Fiz. 71 (2008); arXiv:0705.0384
350
0.18
0.19
0.20
temperature T[GeV]
0.19
0.2
temperature T[GeV]
D.B., Horvatic, Klabucar, in prep. HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
Hadronic Matter
P HASE D IAGRAM
120 80 40 0 0
Liquid-Gas Transition
T [MeV]
160
FOR
Normal Quark 4 3
S YMMETRIC M ATTER (HIC)
0.5
Matter
Color Superc (2SC) Quark onducting Matter 4 0.25
3 Φ = 0.1
2 1
2 0.5
0.3
CFL
s/n = 0.25
0.6
0.9 1.2 -3 n [fm ]
• Almost crossover (masquerade!), i.e. small density jump, small latent heat/ time delay in heavy-ion coll.! • High Tc ≈ 0.9Td for 2SC phase due to Polyakov loop.
0.75
1
• Critical density for chiral restoration nχ ≥ 1.5 n0 increasing (!) with low T
1.5
1.8
DB, Sandin, Skokov, NICA WhitePaper (2009)
• 2SC - CFL phase transition at n ≥ 6 n0 with density jump and latent heat/ time delay! Provided the temperature can be kept low T ≤ 100 MeV HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
S UPERNOVA
POSTBOUNCE EVOLUTION IN THE PHASE DIAGRAM
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
BAG
VS .
PNJL
MODEL IN THE PHASE DIAGRAM
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
E VOLUTION
OF CENTRAL MASS ELEMENTS IN
S HEN /PNJL P H D
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
P HASE
DIAGRAM
/
COMPACT STARS FOR
S HEN /BAG
MODEL
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
A DDITIONAL
NEUTRINO BURST FROM QUARK - HADRON TRANSITION
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
PNJL
BEYOND
MF: PION (q q¯) AND
NUCLEON
(qqq) MEDIUM
Idea: melting h¯ q qi → swelling hadrons → flavor kinetics = quark percolation → freeze-out h¯ q qi(T, µ) =
∂ Ω(T, µ) , Ω(T, µ) = ΩPNJL,MF (T, µ) + Ωmeson (T, µ) + Ωbaryon(T, µ) ∂m0 Z
Z o dω d3k n ω −βω AM (ω, k) , Ωmeson (T, µ) = dM + T ln 1 − e π (2π)3 2 M=π,... Z Z h i o X d3k n ω dω −β(ω−µB ) + T ln 1 + e + (µB ↔ −µB ) AB (ω, k) , Ωbaryon(T, µ) = − dB π (2π)3 2 X
B=N,...
AM (ω, k) = πδ(ω − EM (k)) + continuum , AB (ω, k) . . . analoguous
Remove vacuum terms; neglect continuum (for the freeze-out); q qi and σN = m0(∂mN /∂m0) = 45 MeV, use GMOR: Mπ2 fπ2 = −m0h¯ Enforce Mπ (T, µ) = const by setting fπ2(T , µ) = −m0h¯ q qi(T , µ)/Mπ2, (“BRST”, arxiv:1005.4610) Mπ2T 2 σN + ns,N (T , µ) q qiPNJL,MF (T , µ) + −h¯ q qi(T , µ) = −h¯ 8m0 m0 R∞ 2 with the scalar nucleon density ns,N (T, µ) = π2 0 dp p2 EmN(p) {fN (T, µ) + fN (T, −µ)} N
J. Berdermann, D.B., J. Cleymans, K. Redlich, in progress (2010)
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
PNJL M ODEL −h¯ q qi = −h¯ q qiPNJL,MF
BEYOND
MF -
RESULTS
Mπ2 T 2 σN + ns,N (T, µ)+. . . −h¯ qqi = −h¯ qqiPNJL,MF + 8m0 m0
J. Berdermann, D.B., J. Cleymans, K. Redlich, in progress (2010) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
PNJL M ODEL
BEYOND
−h¯ q qi = −h¯ q qiPNJL,MF Mπ2T 2 σN +κM + κB ns,N (T, µ) 8m0 m0
MF -
RESULTS
−h¯ q qi = −h¯ q qiPNJL,MF Mπ2T 2 σN + + ns,N (T, µ) + . . . 8m0 m0
J. Berdermann, D.B., J. Cleymans, K. Redlich, in progress (2010) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
PNJL M ODEL
BEYOND
nb = n(T, µ) + n ¯ (T, µ) =
X
i=N,∆,xB
MF
di
Z
VS .
P HENOMENOLOGICAL F IT
dp p2 1 + (µ ↔ −µ) 2π 2 exp(β[Ei(p) − µ]) + 1 0.5 0.45
250 NICA energies
0.4 0.35
150
−3
0.3
n+n [fm ]
T [MeV]
200
0.25 0.2
100 0.15 0.1
50
0.05 0
200
400
600 800 1000 1200 1400 µ [MeV] B
J. Berdermann, D.B., J. Cleymans, K. Redlich, in progress (2010) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
εR(T, {µj }) =
L ATTICE QCD E O S
AND
Z
Z
X
i=π,K,...
εi(T, {µi})+
X
r=M,B
gr
mr
dm
M OTT-H AGEDORN G AS ds ρ(m)A(s, m; T )
Z
3
dp (2π)3
p
p2 + s √
exp Hagedorn mass spectrum: ρ(m)
p2 +s−µr T
+ δr
Spectral function for heavy resonances:
TH=1.2 Tc
A(s, m; T ) = Ns
resonances
10
ε/T
4
15
quarks + gluons
5
0
mΓ(T ) (s − m2)2 + m2Γ2(T )
Ansatz with Mott effect at T = TH = 180 MeV: 2.5 6 m m T Γ(T ) = BΘ(T − TH ) exp TH TH TH No width below TH : Hagedorn resonance gas Apparent phase transition at Tc ∼ 150 MeV Blaschke & Bugaev, Fizika B13, 491 (2004)
0.5
1
1.5
2 T/Tc
2.5
3
3.5
Prog. Part. Nucl. Phys. 53, 197 (2004) Blaschke & Yudichev, in preparation Bugaev, Petrov, Zinovjev, arXiv:0812.2189 HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
εR(T, {µj }) =
L ATTICE QCD E O S
AND
Z
Z
X
i=π,K,...
εi(T, {µi})+
X
gr
r=M,B
mr
dm
M OTT-H AGEDORN G AS ds ρ(m)A(s, m; T )
Z
3
dp (2π)3
p
p2 + s √
exp Hagedorn mass spectrum: ρ(m)
p2 +s−µr T
+ δr
Spectral function for heavy resonances:
20 p/T
ε/T 2 cs *30
mΓ(T ) (s − m2)2 + m2Γ2(T )
Ansatz with Mott effect at T = TH = 180 MeV: 2.5 6 m m T Γ(T ) = BΘ(T − TH ) exp TH TH TH
4
p[T ], ε[T ], cs
2
4
15
A(s, m; T ) = Ns
Hadron resonance gas with broadened resonances (Blaschke, Prorok, ...)
4
10
4
No width below TH : Hagedorn resonance gas Apparent phase transition at Tc ∼ 150 MeV
5
Blaschke & Bugaev, Fizika B13, 491 (2004)
0 0
Prog. Part. Nucl. Phys. 53, 197 (2004)
100
200
300 400 T [MeV]
500
600
Bugaev, Petrov, Zinovjev, arXiv:0812.2189 Blaschke & Prorok, in preparation HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
H ADRON εR(T, {µj }) =
DECAY WIDTHS : ANALYTICAL CONFINEMENT MODEL
X
i=π,K,...
εi(T, {µi})+
X
r=M,B
gr
Z
mr
dm
Z
ds ρ(m)A(s, m; T )
Z
3
dp (2π)3
exp
p
p2 + s √ p2 +s−µr T
+ δr
Hagedorn mass spectrum: ρ(m) Spectral function for heavy resonances: A(s, m; T ) = Ns
mΓ(T ) (s − m2)2 + m2Γ2(T )
Breit-Wigner analysis of bound state widths in a confining model: Approximate exponential dependence on mass ! Kalloniatis, Nedelko, von Smekal, PRD 70, 094037 (2004)
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
W HAT
HAPPENS ON
“H APPY I SLAND ”?
“beach”: later “cliff”: • (unmodified) vacuum bound state energies • fast chemical equilibration Explanation:
" ch
"H
ea ic "b on d" ky an ar Isl qu py ap
Strong medium dependence of rates for flavor (quark) exchange processes Reason:
lif
"c f"
• lowering of thresholds • increase of hadron size (Pauli principle) → geometrical overlap (percolation) HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
C OLLABORATIONS Thanks to: Deborah Aguilera, Jens Berdermann, Michael Buballa, Fiorella Burgio, Christian Fuchs, Daniel Gomez Dumm, Hovik Grigorian, Gabriela Grunfeld, Davor Horvatic, Igor Iosilevsky, Dubravko Klabucar, ¨ Thomas Klahn, Gevorg Poghosyan, Sergey ¨ Popov, Krzysztof Redlich, Gerd Ropke, Fredrik Sandin, Norberto Scoccola, Joachim Trumper, ¨ Stefan Typel, Dima Voskresensky, a.m.m.
ESF Research Networking Programme “CompStar”, 2008 - 2013 http://www.compstar-esf.org HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
T HANKS
FOR YOUR ATTENTION !
HIC10.THI NSTITUTE @ CERN . CH , 10.09.2010
