Thermodynamics of relativistic gases and the QGP

Thermodynamics of relativistic gases and the QGP

Thermodynamics of relativistic gases
and the QGP
Daniel Müller
Seminar WS 2009/2010
Relativistische Schwerionenphysik
October 22, 2009 | TU Darmstadt | D. Müller | 1
Outline
Hadrons, Quarks and Gluons
Thermodynamics
Critical temperature
Critical density / chemical potential
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Matter at different scales
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Quarks and Gluons
Quarks
◮
fermions
◮
color charge (r/g/b)
Gluons
◮
gauge bosons
◮
spin 1
◮
mass: 0 GeV
◮
electric charge 0
◮
color charge
◮
Nc2 − 1 = 8 types
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Confinement
◮
Quarks confined in colorless
mesons (quark - antiquark) and
baryons (3 quarks)
◮
linear rising potential V (r ) ∝ r
◮
large energies: string breaking
Quark - antiquark potential
[S. Necco, R. Sommer (2002)]
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Running Coupling and Asymptotic Freedom
◮
running coupling constant of QCD
αs (q 2 ) ≈
◮
4π
11 −
2
N
3 f
1
ln(q 2 /Λ2QCD )
large q: αs → 0
⇒ asymptotic freedom
[D. J. Gross, F. Wilczek (1973), D.
Politzer (1973)]
[S. Bethke (2006)]
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QCD Phase Diagram
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Basic thermodynamic relations
Density operator
ρ̂ =
1 −(Ĥ −µN̂)/T
e
,
Z
hÔ i = Tr (ρ̂Ô)
Partition function
Z (T , V , µ) = Tr e−(Ĥ −µN̂)/T ≡ e−Ω(T ,V ,µ)/T
Grand potential
Ω(T , V , µ) = −T ln Z (T , V , µ) = E − TS − µN = −pV
Energy density
ǫ =
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T 2 ∂ (Ω/T )
µ ∂Ω
E
= −
−
V
V
∂T
V ∂µ
Noninteracting Bose gas
Partition function
ZB =
Y X
k
e
−l(E(k)±µ)/T
l
!d
=
Y
k
1
1−
e−(E(k)−µ)/T
d
,
E(k ) =
√
k 2 + m2
Thermodynamic potential
Ω = −T ln ZB = VTd
2
◮
◮
Z
d3k
π2 4
−(E(k)−µ)/T m=0,µ=0
=
−
Vd
ln
1
−
e
T
(2π )3
90
2
Pressure: P = d π90 T 4 , energy density ǫ = 3d π90 T 4 = 3P
Stefan-Boltzmann law for massless Bosons with degeneracy d: P ∝ T 4
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Noninteracting Fermi gas
Partition function
ZF =
Y X
k
e−l(E(k)±µ)/T
l=0,1
!d
=
Y
1 + e−(E(k)±µ)/T
k
d
Thermodynamic potential at µ = 0 and m = 0
Ω = −T ln ZF = −VTd
◮
◮
Z
7 π2 4
d 3k
−(E(k)±µ)/T m=0,µ=0
=
−
Vd
ln
1
+
e
T
(2π )3
8 90
Stefan-Boltzmann law for Fermions
difference to Bosons: factor 87
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Ideal gas of massless pions
Number density
nπ (T ) =
1 ∂
Ω = 3
V ∂µ
Z
d 3k
1
3
(E(k)
−µ
)/T − 1
(2π ) e
Pion volume
Vπ =
4π 3
R ,
3 π
Rπ ≃ 0.65 fm
Critical temperature
close-packing: nπ (Tc ) = V1π → Tc ≃ 263 MeV
→ Tc ≃ 186 MeV
percolation theory: nπ (Tc ) = 0.35
Vπ
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m=0,µ=0
=
3ζ (3)
π2
T3
Bag model
◮
◮
free (perturbative) quarks trapped
inside bags → Hadrons
Bag constant B: positive
contribution to energy +B and
negative contribution to pressure
−B inside the bag
◮
equivalent: negative contribution to
energy −B and positive contribution
to pressure +B outside the bag
◮
nontrivial vacuum: ǫvac = −B,
pvac = B
[Chodos et al. (1974)]
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Simple estimate of the bag constant
Hadron mass in the bag model
EBM (R) =
4π 3
3x
z0
R B+
−
+ ...
3
R
R
Minimizing hadron mass
0 =
◮
◮
◮
∂ EBM (R)
z0
3x
= 4π R 2 B − 2 + 2
∂R
R
R
x /R: kinetic energy of quarks inside the bag, x = 2.04
z0 /R: correction term [T. A. DeGrand et al. (1975)]
assume mN = 938 MeV, R = 0.877 fm → B 1/4 ≃ 160 MeV
October 22, 2009 | TU Darmstadt | D. Müller | 13
Simple estimate of the bag constant
Hadron mass in the bag model
EBM (R) =
4π 3
3x
z0
R B+
−
+ ...
3
R
R
Minimizing hadron mass
0 =
◮
◮
◮
◮
◮
∂ EBM (R)
z0
3x
= 4π R 2 B − 2 + 2
∂R
R
R
x /R: kinetic energy of quarks inside the bag, x = 2.04
z0 /R: correction term [T. A. DeGrand et al. (1975)]
assume mN = 938 MeV, R = 0.877 fm → B 1/4 ≃ 160 MeV
fit to hadron spectrum: B 1/4 = 146 MeV [DeGrand]
QCD sum rules: B 1/4 = 220 MeV [Shifman et al. (1979)]
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Critical temperature in the bag model
Low T: free pion gas
Pπ = dπ
π2
90
T4
High T: free quark - gluon gas
dQGP
π2
T4 − B
90
7
= 2 × (Nc2 − 1) + × 4 × Nc × Nf = 37
8
PQGP = dQGP
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Critical temperature in the bag model
For B 1/4 = 220 MeV
Low T: free pion gas
Pπ = dπ
π2
90
T4
90
Tc4 =
dQGP
π2
T4 − B
90
7
= 2 × (Nc2 − 1) + × 4 × Nc × Nf = 37
8
P / MeV
4
High T: free quark - gluon gas
PQGP = dQGP
B
→ Tc = 158 MeV
π 2 dQGP − dπ
5e+09
4e+09
3e+09
2e+09
1e+09
0
-1e+09
-2e+09
-3e+09
Pion gas
QGP
0
50
100
T / MeV
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150
200
Latent heat
◮
◮
50
L = ǫQGP (Tc ) − ǫπ (Tc ) = 4B
30
critical energy density:
ǫQGP (Tc ) ≈ L ≈ 1.2 GeV fm−3
nuclear matter:
ǫNM ≈ 0.15 GeV fm−3
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Pion gas
QGP
40
4
◮
steep rise of energy density at the
phase transition
ε/T
◮
20
10
0
100
L
120
140
160
T / MeV
180
200
Lattice QCD
◮
non-perturbative method of solving
QCD
◮
difficulties: discretization, finite
volume, fermions
◮
only at vanishing µ
◮
critical temperature: Tc = 170 − 200
MeV
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[Karsch et al. (2009)]
Hagedorn temperature
◮
◮
exponentially increasing number
density of hadrons with increasing
energy
state density for large M:
c M /Tc
e
Ma
[R. Hagedorn (1965)]
ρ(M) =
◮
fit to data: Tc ≈ 160MeV
[R. Hagedorn (1969)]
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Simple estimate of critical density
Nucleon volume
R = 0.877 fm,
VN =
4π 3
R ≈ 2.8 fm3
3 N
Dense nucleon matter
ρc ≈
1
= 0.35 fm3 ≈ 2.4ρNM
VN
◮
but heavily dependent on radius
◮
for R = 0.5 fm: ρc = 1.9 fm3
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Finite chemical potential
Massless quarks at chemical potential µ
Pq = −T
= dq
dq
2
π2
90
Z
T4
d 3k ln 1 + e−(k −µ)/T + ln 1 + e−(k+µ)/T
3
(2π )
15 µ4
7 15 µ2
+
+
2
8
4 (π T )
8 (π T )4
Phase transition at T = 0 MeV
Pπ = PQGP → 0 = dq
µ4c
−B
48π 2
µc = 463MeV, µB,c = 3µc = 1389MeV
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Phase diagram
Quark gluon plasma vs. pion gas
PQGP = Pπ → dq
π2
90
T
4
7 15 µ2
15 µ4
+
+
8
4 (π T )2
8 (π T )4
+ dg
π2
90
200
phase diagram
T / MeV
150
100
50
0
0
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100
200
300
µ / MeV
400
500
T 4 − B = dπ
π2
90
T4
Summary & Outlook
Summary
◮
critical temperature in different
models: 160 - 200 MeV
◮
critical chemical potential?
µB ∼ 1200 MeV
Outlook (later seminars)
◮
lattice QCD
◮
color superconductivity
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Literature
◮
J. Letessier, J. Rafelski, Hadrons and Quark-Gluon Plasma, Cambridge
University Press 2004
◮
K. Yagi, T. Hatsuda, Y. Miake, Quark-Gluon Plasma, Cambridge University
Press 2005
◮
M. Buballa, Phys. Rep. 407, 205 (2005)
Thanks for your attention
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