Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Particle spectra and the QCD phase transition
A. Jakovác
ELTE, Dept. of Atomic Physics
Budapest, Hungary
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 1 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Contents
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 2 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 3 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 4 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
The QCD equation of state
pressure from MC simulation
(Sz. Borsanyi et al, JHEP 1011 (2010) 077)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 5 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
The QCD equation of state
pressure from MC simulation
Regimes
QGP at high T : 8 gluon + 3 quark dof
(Sz. Borsanyi et al, JHEP 1011 (2010) 077)
hadrons at low T : Hadron Resonance Gas
(HRG)
in between continuous crossover phase
transition (PT) with “Tc ” = 156 MeV
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 5 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At high temperature: Quark Gluon Plasma (QGP)
3-loop HTL calculation
1.0
0.8
æ
æ
æ
æ æ
0.6
æ
ææ
æ æ
æ
æ
0.8
æ
1 loop Αs ; L
Μ B = 0 MeV
æ
æ
æ
ææ
ææ ææ
æææ
æ
æ
NNLO HTLpt
0.2
æ
ææ
æ
æ æ
æ
æ æ
æ
ææ æ
æææ
Wuppertal - Budapest
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
=176 MeV
0.6
0.4
NNLO HTLpt
0.2
Wuppertal - Budapest
HotQCD
400
MS
ææ
æ
æ
æ
æ
æ
æ
200
1.0
= 316 MeV
æ
ææ
0.4
0.0
MS
E E ideal
P P ideal
3 loop Αs ; L
Μ B = 0 MeV
600
800
1000
T @MeVD
0.0
200
400
600
800
1000
T @MeVD
(N. Haque, et.al., e-Print: arXiv:1402.6907 )
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 6 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At high temperature: Quark Gluon Plasma (QGP)
3-loop HTL calculation
1.0
0.8
æ
æ
æ
æ æ
0.6
æ
ææ
æ æ
æ
æ
0.8
æ
1 loop Αs ; L
Μ B = 0 MeV
æ
æ
æ
ææ
ææ ææ
æææ
æ
æ
NNLO HTLpt
0.2
æ
ææ
æ
æ æ
æ
æ æ
æ
ææ æ
æææ
Wuppertal - Budapest
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
=176 MeV
0.6
0.4
NNLO HTLpt
0.2
Wuppertal - Budapest
HotQCD
400
MS
ææ
æ
æ
æ
æ
æ
æ
200
1.0
=176 MeV
æ
ææ
0.4
0.0
MS
E E ideal
P P ideal
1 loop Αs ; L
Μ B = 0 MeV
600
800
1000
T @MeVD
0.0
200
400
600
800
1000
T @MeVD
(N. Haque, et.al., e-Print: arXiv:1402.6907 )
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 6 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At high temperature: Quark Gluon Plasma (QGP)
3-loop HTL calculation
1.0
æ
æ
æ
æ æ
0.6
æ
ææ
æ æ
æ
4
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
3
æ
ææ
æ
æ æ
æ
æ æ
æ
ææ æ
æææ
1
HotQCD
600
800
1 loop Αs ; L
MS
=176 MeV
NNLO HTLp t
æ
WB
æ
æ
æ
Wuppertal - Budapest
400
Μ B = 0 MeV
æ
æ
æ
æ
æ
æ
æ
2
NNLO HTLpt
0.2
200
5
=176 MeV
æ
ææ
0.4
0.0
MS
H E -3PL T 4
0.8
P P ideal
1 loop Αs ; L
Μ B = 0 MeV
1000
T @MeVD
0
æ
æ
æ
æ
æ
200
400
600
æ
æ
800
1000
T @ MeVD
(N. Haque, et.al., e-Print: arXiv:1402.6907 )
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 6 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At high temperature: Quark Gluon Plasma (QGP)
3-loop HTL calculation
1.0
æ
æ
æ
æ æ
0.6
æ
ææ
æ æ
æ
4
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
3
æ
ææ
æ
æ æ
æ
æ æ
æ
ææ æ
æææ
1
HotQCD
600
800
1 loop Αs ; L
MS
=176 MeV
NNLO HTLp t
æ
WB
æ
æ
æ
Wuppertal - Budapest
400
Μ B = 0 MeV
æ
æ
æ
æ
æ
æ
æ
2
NNLO HTLpt
0.2
200
5
=176 MeV
æ
ææ
0.4
0.0
MS
H E -3PL T 4
0.8
P P ideal
1 loop Αs ; L
Μ B = 0 MeV
1000
T @MeVD
0
æ
æ
æ
æ
æ
200
400
600
æ
æ
800
1000
T @ MeVD
(N. Haque, et.al., e-Print: arXiv:1402.6907 )
Lesson
Correct description for temperatures T >
∼ 2Tc ≈ 300 MeV.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 6 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At low temperature: hadrons
HRG: free hadrons with fixed (T = 0) masses from experiments
Thermodynamics
(Sz. Borsanyi, G. Endrodi, Z. Fodor, A.J., S. D. Katz)
( S. Krieg, C. Ratti, K.K. Szabo, JHEP 1011 (2010) 077)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 7 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At low temperature: hadrons
HRG: free hadrons with fixed (T = 0) masses from experiments
Thermodynamics
Chiral susceptibility
1
χS
2
0.8
0.6
0.4
p4
asqtad
0.2
T [MeV]
0
140
(Sz. Borsanyi, G. Endrodi, Z. Fodor, A.J., S. D. Katz)
( S. Krieg, C. Ratti, K.K. Szabo, JHEP 1011 (2010) 077)
Particle spectra and the QCD phase transition
160
180
200
220
240
(P. Huovinen and P. Petreczky, Nucl. Phys. A 837)
(26 (2010) [arXiv:0912.2541 [hep-ph]].)
University of Sussex, March 31, 2014. 7 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
At low temperature: hadrons
HRG: free hadrons with fixed (T = 0) masses from experiments
Thermodynamics
Chiral susceptibility
1
χS
2
0.8
0.6
0.4
p4
asqtad
0.2
T [MeV]
0
140
(Sz. Borsanyi, G. Endrodi, Z. Fodor, A.J., S. D. Katz)
( S. Krieg, C. Ratti, K.K. Szabo, JHEP 1011 (2010) 077)
160
180
200
220
240
(P. Huovinen and P. Petreczky, Nucl. Phys. A 837)
(26 (2010) [arXiv:0912.2541 [hep-ph]].)
Lesson
HRG describes thermodynamics at T < 150 − 180 MeV
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 7 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Phase transition region
Temperature range of 150 MeV <
∼T <
∼ 300 MeV.
Columbia plot
Nf = 2
Pure
Gauge
∞
2nd order
O(4) ?
m
deconfined
2nd order
Z(2)
Mechanisms of the PT
deconfinement: 1st order PT
hadrons become unstable
1st
order parameter: Polyakov-loop
tric
s
valid at mu,d,s → ∞ (quenched)
Nf = 3
phys.
point
ms
1st
crossover
Nf = 1
chiral
2nd order
Z(2)
chiral condensate unstable
order parameter: Ψ̄Ψ
0
0
chiral phase transition: 1st order PT
m u , md
∞
Particle spectra and the QCD phase transition
valid at mu,d,s → 0 (chiral case)
University of Sussex, March 31, 2014. 8 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Phase transition region
Temperature range of 150 MeV <
∼T <
∼ 300 MeV.
Columbia plot
Nf = 2
Pure
Gauge
∞
2nd order
O(4) ?
m
deconfined
2nd order
Z(2)
Mechanisms of the PT
deconfinement: 1st order PT
hadrons become unstable
1st
order parameter: Polyakov-loop
tric
s
valid at mu,d,s → ∞ (quenched)
Nf = 3
phys.
point
ms
1st
crossover
Nf = 1
chiral
2nd order
Z(2)
chiral condensate unstable
order parameter: Ψ̄Ψ
0
0
chiral phase transition: 1st order PT
m u , md
∞
valid at mu,d,s → 0 (chiral case)
Physical point
What happens in the crossover regime?
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 8 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
What happens with the hadrons at Tc ?
HRG contains infinitely many dof ⇒ PSB = ∞
singularity in PSB at TH Hagedorn temperature.
(R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965); W. Broniowski, et.al. PRD 70, 117503 (2004))
⇒ we must get rid of the hadrons before TH .
no change of ground state (1st or 2nd order phase transition)
⇒ hadrons must not disappear at once
(J. Liao, E.V. Shuryak PRD73 (2006) 014509 [hep-ph/0510110])
MC: hadronic states are observable even at T ∼ 1.2-1.5Tc !
(S. Datta et.al. PRD 69, 094507 (2004) [hep-lat/0312037])
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 9 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
What happens with the hadrons at Tc ?
HRG contains infinitely many dof ⇒ PSB = ∞
singularity in PSB at TH Hagedorn temperature.
(R. Hagedorn, Nuovo Cim. Suppl. 3, 147 (1965); W. Broniowski, et.al. PRD 70, 117503 (2004))
⇒ we must get rid of the hadrons before TH .
no change of ground state (1st or 2nd order phase transition)
⇒ hadrons must not disappear at once
(J. Liao, E.V. Shuryak PRD73 (2006) 014509 [hep-ph/0510110])
MC: hadronic states are observable even at T ∼ 1.2-1.5Tc !
(S. Datta et.al. PRD 69, 094507 (2004) [hep-lat/0312037])
Proposal
150 MeV <
∼T <
∼ 300 − 400 MeV is the melting hadron phase (hadron
fluid phase). Quarks appear gradually with the disappearance of
the hadrons. (ionization-recombination, chemistry, Gribov)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 9 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 10 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 11 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
What is melting?
Heuristically: disappearance of a particle species
Usual approaches in the literature
fast growing (thermal) mass
(J. Liao, E.V. Shuryak PRD73 (2006) 014509 )
Would explain why we do not see quarks at low energy and hadrons
at high energy
⇒ in contradiction with lattice results
FRG: all states are present, but with different wave fct.
renormalization
⇒ but Z drops out from pressure
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 12 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
What is melting?
Heuristically: disappearance of a particle species
Usual approaches in the literature
fast growing (thermal) mass
(J. Liao, E.V. Shuryak PRD73 (2006) 014509 )
Would explain why we do not see quarks at low energy and hadrons
at high energy
⇒ in contradiction with lattice results
FRG: all states are present, but with different wave fct.
renormalization
⇒ but Z drops out from pressure
Question
How can a particle state disappear?
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 12 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
What is a particle? Free systems.
∃ conserved particle number operator: N̂ =
X
ak† ak ,
[Ĥ, N̂] = 0
k
definition
particle: energy (and momentum) eigenstate in N = 1 sector.
Moreover:
one particle spectrum contains a single line at E = E (p)
(dispersion relation)
time evolution |t, E , ni = e −iEt |0, E , ni is unique from any
initial condition
in particular linear response function Gr has the same time
dependence, also at T > 0
particles are also thermodynamical
degrees of freedom, eg.
PSB =
π2 T 4
90
7
Nb + Nf
8
Particle spectra and the QCD phase transition
is the Stefan-Boltzmann limit.
University of Sussex, March 31, 2014. 13 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Identifications
Since these are true in free particle case, we intuitively identify the
following concepts:
particle number operator
spectral line (energy eigenstate)
general time evolution
linear response theory
linear response theory at T > 0
statistical/thermodynamical definition
They all mean “particle”.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 14 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Identifications
Since these are true in free particle case, we intuitively identify the
following concepts:
particle number operator
spectral line (energy eigenstate)
general time evolution
linear response theory
linear response theory at T > 0
statistical/thermodynamical definition
They all mean “particle”.
Warning
These all mean different things in an interacting theory!
. . . and we get mixed up, when these definitions are contradicting
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 14 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 15 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Spectrum at zero temperature
Usually in interacting systems 6 ∃ enough conserved quantities
to fully describe the system
⇒ 6 ∃ particle definition through particle number
Exception: integrable systems
energy levels of different N sectors mix together! At T = 0
1
0.8
0.6
=
ρ
ρ
+
1
0.5
1
0.8
0.4
0.6
.
ρ
2
1.5
0.4
0.2
0.2
0
0
0
0.5
1
E/Ep
1.5
2
0
0.5
1
1.5
2
2.5
E/Ethr
3
3.5
4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
E/m
multiple energy levels, non-unique time dependence
BUT ∃ discrete E-level
⇒ linear response for long times: Ze −iEt + Ct −3/2 e −iEthr t
⇒ define particles as asymptotic particle states
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 16 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
General case
ρ/m2
no clear distinction between particle and continuum states, if
100
10
1
0.1
0.01
0.001
1e-04
1e-05
1e-06
p=0.2443m
zero mass excitation (no gap)
unstable particles
T > 0 environment
0
1
2
3 4
p0/m
5
6
⇒ 6 ∃ asymptotic states
(in practically all realistic cases. . . )
(AJ, PRD76 (2007) 125004 [hep-ph/0612268])
linear response: %(t) = Ze −iEt−γt + fbckg (t) = pole + cut
for large Z and small γ: complex pole dominates long time
evolution ⇒ quasiparticles
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 17 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Quasiparticles and thermodynamics
In QM quasiparticles give fundamental particle-like
contribution to free energy (Beth, Uhlenbeck)
δZ ∼
R∞
0
dω ∂δ −βω
π δω e
∼ e −βE
since δ` (ε) phase shift jumps π-t at pole ω = E
(Landau, Lifsitz V.)
true also for bound states
In QFT this is true only for well separated quasiparticle peaks
(R.F Dashen, R. Rajaraman, PRD10 (1974), 694.)
In scattering theory: quasiparticles are included in S-matrix as
Breit-Wigner resonances with complex amplitudes
unitarity ⇒ constraints
(H. Feshbach, Ann. Phys. 43, 110 (1967); L. Rosenfeld, Acta Phys. Polonica A38, 603 (1970); M. Svec,
PRD64, 096003 (2001) [hep-ph/0009275].)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 18 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
The particle concept
concept of free particles can be saved as quasiparticles
spectral definition ⇒ broadened spectral line
linear response theory
⇒ unique long time dependence
thermodynamical degree of freedom
case
Particle spectra and the QCD phase transition
⇒ for well separated
University of Sussex, March 31, 2014. 19 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
The particle concept
concept of free particles can be saved as quasiparticles
spectral definition ⇒ broadened spectral line
linear response theory
⇒ unique long time dependence
thermodynamical degree of freedom
case
⇒ for well separated
there are important differences
Quasiparticles are not energy eigenstates!
collective excitations with environment dependent spectral
weights ⇒ mass, width environment dependent
⇒ they may give not particle-like contribution to P .
...
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 19 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 20 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Condition for unitarity
Local Hamiltonian? Exponential damping
⇒ loss of unitarity!
⇒ Ĥ → Ĥ − iγ
Solution
For consistent description one has to take into account the
complete spectrum, not just the quasiparticle peak!
Physics: quasiparticle 6 ∃ independently of environment.
(Ward, Luttinger, Phys.Rev. 118 (1960) 1417; G. Baym, Phys. Rev. 127 (1962) 1391; Cornwall Jackiw,
Tomboulis, Phys.Rev. D10 (1974) 2428-2445;J. Berges and J. Cox, Phys. Lett. B 517 (2001) 369)
From where can we take the spectrum?
Φ-derivable (2PI) or SD approach: G −1 = G0−1 − Σ(G ).
We can also use experimental inputs for %.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 21 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Typical spectral functions
ρ/m
2
Φ4 model 2 loop 2PI, T = m
100
10
1
0.1
0.01
0.001
1e-04
1e-05
1e-06
p=0.2443m
0
1
2
3 4
p0/m
5
6
(AJ, PRD76 (2007) 125004 [hep-ph/0612268])
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 22 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Typical spectral functions
Φ4 model 2 loop 2PI, T = m
meson
100 model FRG
p=0.2443m
10
1
0.1
ΡΠ
1000.01
0.001
1
1e-04
1e-05
0.01
1e-06
T=100 MeV
ρ/m
2
@L-2
UVD
ΡΣ
0
1
2
3 4
p0/m
5
10-4
0
100
200
300
400
500
(AJ, PRD76 (2007) 125004 [hep-ph/0612268])
(R.-A. Tripolt et.al. PRD 89 (2014) 034010)
Particle spectra and the QCD phase transition
6
600
700
Ω @MeVD
University of Sussex, March 31, 2014. 22 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Typical spectral functions
Φ4 model 2 loop 2PI, T = m
meson
FRG
100 model
Bloch-Nordsieck
model at finite T
p=0.2443m
10
1
0.1
ΡΠ
1000.01
0.001
1
1e-04
1e-05
0.01
1e-06
T=100 MeV
0.5
0.45
0.4 ΡΣ
0.35
0.3
0.25
0.2
0.15
0 1 2 0.13 4 5 6
0.05
p0/m
10-4
0
100
200
300
400
500
600
0
(AJ, PRD76 (2007) 125004 [hep-ph/0612268])
-2 034010)
-1
0
(R.-A. Tripolt et.al. PRD 89 (2014)
α=0.2
α=0.5
α=1.0
α=3.0
Tρ
ρ/m
2
@L-2
UVD
700
Ω @MeVD
1
2
3
w/T
(A.J, P. Mati, PRD 87 (2013) 125007 [arXiv:1301.1803])
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 22 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Typical spectral functions
Φ4 model 2 loop 2PI, T = m
meson
FRG J/Ψ spectrum MC simulation, MEM
100 model
Bloch-Nordsieck
model at finite T
p=0.2443m
10
1
0.1
ΡΠ
1000.01
0.001
1
1e-04
1e-05
0.01
1e-06
T=100 MeV
0.14
0.5
0.12
0.45
0.1
0.4 ΡΣ
0.35
0.08
0.3
0.06
0.25
0.2
0.04
0.15
0.02
0 1 2 0.13 4 5 6
0
0.05
p0/m
0
5
10-4
0
100
200
300
400
500
600
0
(AJ, PRD76 (2007) 125004 [hep-ph/0612268])
-2 034010)
-1
0
(R.-A. Tripolt et.al. PRD 89 (2014)
α=0.2
α=0.5 T=0, Ndata=20
α=1.0
1.2Tc
α=3.0
σ(ω)/ω2
Tρ
ρ/m
2
@L-2
UVD
700
1
Ω10
@MeVD
ω[GeV]
2
15
20
3
(AJ., P. Petreczky, K. Petrov, A. Velytsky, PRD75 (2007) 014506)
w/T
(A.J, P. Mati, PRD 87 (2013) 125007 [arXiv:1301.1803])
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 22 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Lagrangian representation of general spectral functions
L = 12 Φ∗ (p)K(p)Φ(p)
unique % → K relation:
Gret (p) =
R
dω %(ω,p)
2π p0 −ω+iε ,
K = Re GR−1
defines a consistent nonlocal field theory:
unitary, causal, Lorentz-invariant, E , p conserving
(just like in 2PI case)
(AJ. Phys.Rev. D86 (2012) 085007 [arXiv:1206.0865])
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 23 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics from the spectral function
Technically:
→ energy-momentum tensor from Noether currents
→ energy density ε = Z1 Tr e −β Ĥ T̂00
→ averaging with KMS relations
→ free energy, pressure from thermodynamical relations
Result:
Z
ε=
d 4p
E (p)n(p0 )%(p),
(2π)4
E (p) = p0
∂K
−K
∂p0
plausible: sum up n(p) weighted energy values
classical mechanical analogy: K quadratic kernel
”Lagrangian” with p0 ∼ q̇ ⇒ E (p) energy.
but: energy values depend on K and so on %
ε is a nonlinear functional of %!
ε does not depend on the normalization of %.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 24 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 25 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 26 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Spectral function in gauge theories
Gauge theories are complicated – solvable simplification:
resummation of all photon contribution in 1-component QED
(Bloch-Nordsieck resummation)
One can compute the spectral function at finite temperature. In
comoving frame:(A.J, P. Mati, Phys.Rev. D87(2013) 125007 [arXiv:1301.1803])
Tρ
Nα β sin α e βw /2
%(w ) =
cosh βw − cos α
−2
Γ 1 + α + i βw ,
2π
2π α = e 2 /(4π) structure consant
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
α=0.2
α=0.5
α=1.0
α=3.0
function of w = p0 − m
Near the peak: Lorentzian with width
γ = αT
−1−α/π
p0 m power law: ∼ p0
-2
-1
0
1
2
3
p0 m exponential: ∼ e 2βw
w/T
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 27 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Real time dependence
Fourier transform of the result: %(t) = e −imt %̄(t)
0.8
0.7
0.5
log|Tρ(t)|
log|Tρ(t)|
0.6
0.5
0.4
0.3
data
1-Atα/π
-αt
Be
0.6
0.4
η=0.00
η=2.77
0.2
0.3
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
0.5
1
1.5
2
Tt
2.5 3
tT/u0
3.5
4
4.5
5
for long times Tt 1: ∼ e −αeff (u)Tt quasiparticle behaviour
for short times Tt 1: ∼ 1 − c(Tt)α/π not quasiparticle-like!
formation time of the quasiparticle: t ∼ β !
at T → 0 %(t) → e −imt , but we have to wait long to see the
QP behaviour.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 28 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 29 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
The Gibbs paradox
(J.W.Gibbs, 1875-1878; E.T.Jaynes, 1996)
take two containers with (ideal) gases:
initially n1 , V1 n2 , V2 , p1 = p2 , T1 = T2
mix them: V = V1 + V2 , n = n1 + n2
entropy difference (f = n1 /n2 )
∆S = nR log V − R(n1 log V1 − n2 log V2 )
= −nR(f log f + (1 − f ) log(1 − f ))
⇒ −nR log 2, for n1 = n2 , V1 = V2 .
J.W.Gibbs (1839-1903)
Independent of the gas properties,
provided they are different
e.g. let the two gases have the
same quantum numbers, but different
masses
⇒ discontinuity at ∆m = 0
Particle spectra and the QCD phase transition
discuntinuous entropy
S
∆S
∆m
University of Sussex, March 31, 2014. 30 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Understanding Gibbs paradox
indistinguishability from Fock-space construction:
|0i vacuum, ap† p-momentum particle creation operator
⇒ |p1 , n1 , . . . pi , ni . . .i = ap†n1 1 . . . ap†ni i . . . |0i
multiparticle state ⇒ single state, permutation ± sign
several gases: ap(1)† , ap(2)† , . . . we assign new creation operators
for all species
we have to fix the number of species in advance!
But in Gibbs paradox ∆m is the control parameter
should be able to compute S(∆m)
⇒ we
To describe Gibbs paradox number of particle species must be
a dynamical parameter
(integer number?)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 31 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Gibbs paradox in interacting systems
Without interaction the energy levels (spectral lines) are infinitely
thin lines. In interacting gases the spectral lines broaden.
spectrum in interacting gases
1st plot: 2 lines
4th plot: one broad peak
Gibbs: particles are distinguishable,
if a mixed gas can be separated by
some means. Going from case 1 to
4 this is harder and harder!
E
E
Γ width sets resolution ⇒ in
case Γ >
∼ ∆m we do not see
separate peaks!
E
Particle spectra and the QCD phase transition
E
real question is quantitative: how
does it appear in thermodynamics?
University of Sussex, March 31, 2014. 32 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
⇒
ρ
ρ
Change of spectrum:
0
0.5
1
1.5
2
2.5
E
Particle spectra and the QCD phase transition
3
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0.96 0.98
1
1.02 1.04 1.06 1.08
1.1
E
University of Sussex, March 31, 2014. 33 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
⇒
ρ
ρ
Change of spectrum:
energy density
0
0.5
1
1.5
0.7
2
0.96 0.98
1
1.02 1.04 1.06 1.08
1.1
E
2SB
Curves
i
m1 = 1, m2 = 2, Γ = 0 or 0.2
ii
0.5
ε/T
3
E
0.6
4
2.5
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
i. Γ = 0
0.4
iii
ii. independent, finite Γ
SB
0.3
iv
0.2
iii. Γ/∆m = 0.2
0.1
iv. one free particle
0
0
0.5
1
1.5 2
T/m1
Particle spectra and the QCD phase transition
2.5
3
University of Sussex, March 31, 2014. 33 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
⇒
ρ
ρ
Change of spectrum:
energy density
0
0.5
1
1.5
0.7
2
0.96 0.98
1
1.02 1.04 1.06 1.08
1.1
E
2SB
Curves
i
m1 = 1, m2 = 2, Γ = 0 or 0.2
ii
0.5
ε/T
3
E
0.6
4
2.5
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
i. Γ = 0
0.4
iii
ii. independent, finite Γ
SB
0.3
iv
0.2
iii. Γ/∆m = 0.2
0.1
iv. one free particle
0
0
0.5
1
1.5 2
T/m1
Particle spectra and the QCD phase transition
2.5
3
degrees of freedom disappear
continuously!
University of Sussex, March 31, 2014. 33 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.5
finite width
Gibbs paradox
ρ
⇒
1
log(Neff)/log(2)
ρ
Change
of spectrum:
number
of species
energy density
0.5
0
0.5
1
1.5
0.7
2
ε/T
0.96 0.98
1
0
2SB
1.1
Curves
i
m1 = 1, m2 = 2, Γ = 0 or 0.2
ii
-0.5
0
10
SB
0.3
1.02 1.04 1.06 1.08
E
0.5
4
3
E
0.6
0.4
2.5
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
20
30
iii
∆m/Γ
40
50
i. Γ = 0
ii. independent, finite Γ
iv
0.2
iii. Γ/∆m = 0.2
0.1
iv. one free particle
0
0
0.5
1
1.5 2
T/m1
Particle spectra and the QCD phase transition
2.5
3
degrees of freedom disappear
continuously!
University of Sussex, March 31, 2014. 33 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 34 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Temperature dependence of a typical spectral function
Spectrum in QFT: QP peak(s) and multiparticle continuum.
At finite T
continuum height increases, and so
QP width grows
quasiparticle peaks merge into the continuum
relative height of quasiparticle peak decreases (sum rule)
Lorentz-invariance is broken
T -variation of a spectrum
thermal mass
2
original
decr. height
incr. width
T -dependent couplings
Strategy:
compute thermodynamics for
generic %Q (p0 , |p|; T , µ).
ρ
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
p0
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 35 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Increasing continuum, fixed mass
Trial spectral functions with 3 QP peaks and continuum
⇒ typical for bound states
Spectra
1000
S31
S32
100
S33
lnρ
10
1
0.1
0.01
0
1
2
3
4
5
ω
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 36 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Increasing continuum, fixed mass
Trial spectral functions with 3 QP peaks and continuum
⇒ typical for bound states
Spectra
thermodynamics
1000
0.45
S31
100
S32
0.4
S33
0.35
Dirac-deltas
S34
S35
lnρ
p/T4
10
1
3SB
S36
0.3
S37
S38
0.25
0.2
0.15
0.1
0.1
0.05
0.01
0
0
1
2
3
ω
Particle spectra and the QCD phase transition
4
5
0
1
2
3
4
5
T/m
University of Sussex, March 31, 2014. 36 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Increasing continuum, fixed mass
Trial spectral functions with 3 QP peaks and continuum
⇒ typical for bound states
Spectra
thermodynamics
1000
0.45
S31
100
S32
0.4
S33
0.35
Dirac-deltas
S34
S35
lnρ
p/T4
10
1
3SB
S36
0.3
S37
S38
0.25
0.2
0.15
0.1
0.1
0.05
0.01
0
0
1
2
3
ω
4
5
0
1
2
3
4
5
T/m
General behaviour
Pressure decreases for increasing continuum height; for pure
continuum the pressure is very small!
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 36 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Effective number of degrees of freedom
Characterization: pressure is roughly proportional to the free gas
pressure
⇒ Neff (T ) =
P(T )
is appr. T -independent
P0 (T )
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 37 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Effective number of degrees of freedom
Characterization: pressure is roughly proportional to the free gas
pressure
⇒ Neff (T ) =
P(T )
is appr. T -independent
P0 (T )
effective ndof
T -variation: green band
3
-bhc
3e
data
2.5
Neff
2
1.5
fit a streched exponential
c
e −(γ/γ0 )
where γ0 = 0.38, c = 1.6.
1
0.5
0
0
0.02 0.04 0.06 0.08
γ1
0.1
Particle spectra and the QCD phase transition
0.12 0.14
University of Sussex, March 31, 2014. 37 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Effective number of degrees of freedom
Characterization: pressure is roughly proportional to the free gas
pressure
⇒ Neff (T ) =
P(T )
is appr. T -independent
P0 (T )
effective ndof
T -variation: green band
3
-bhc
3e
data
2.5
Neff
2
1.5
fit a streched exponential
c
e −(γ/γ0 )
where γ0 = 0.38, c = 1.6.
1
0.5
0
0
0.02 0.04 0.06 0.08
γ1
0.1
0.12 0.14
Physics
We describe vanishing particle species!
Particle spectra and the QCD phase transition
⇒
melting
University of Sussex, March 31, 2014. 37 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Momentum dependence of the spectral function
Extreme case of spatial momentum dependence:
for small momenta: Dirac-delta (free particle)
for large momenta: very broad spectral function
Model
simplified model for hadrons
for quarks (asymptotic freedom) we expect inverse behaviour
broad spectral function gives no contribution to P
⇒ effective cutoff of spatial integration
for simplicity we choose Λeff = gT
(g can be T -dependent)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 38 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Effective number of dof
Pressure
0.12
g=∞
SB
g=7
0.1
g=5
g=4
0.08
g=3
p/T4
g=2
g=1
0.06
0.04
0.02
0
-2
-1
0
1
2
3
4
5
6
T
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 39 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Effective number of dof
Neff
Pressure
0.12
g=∞
SB
1
g=7
0.1
g=4
0.08
0.8
g=3
Neff
g=2
p/T4
fit
data
g=5
g=1
0.06
0.6
0.04
0.4
0.02
0.2
0
0
-2
-1
0
1
2
3
4
5
6
0
1
2
3
T
fit function:
1
1+
4
5
6
7
8
g
x −2 e −(bx)a
Particle spectra and the QCD phase transition
(a = 1.79,
b = 0.58)
University of Sussex, March 31, 2014. 39 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 40 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 41 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Statistical description
HRG: huge # of hadronic contributions, each small!
⇒ statistical description is needed
we need spectra. . . hard to obtain
⇒ idealized, simplified picture for hadron masses and widths.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 42 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Hadron masses: Coulomb spectrum of QCD
QCD bound state dynamics cannot be solved. . .
experimental evidence: exponentially rising energy level density
statistical description:
%hadr (m) ∼ (m2 + m02 )a e −m/TH
Hagedorn spectrum
several fits (also a = 0) possible
( W. Broniowski, W. Florkowski and L. Y. .Glozman,
Phys. Rev. D 70, 117503 (2004) [hep-ph/0407290].)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 43 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Thermodynamics with free hadrons
3.5
5
MC data
fit
3
MC data
fit
4
3
2
I/T4
p/T4
2.5
1.5
2
1
1
0.5
0
0
0
50
100
150
200
250
300
350
0
50
100
150
T (MeV)
200
250
300
350
T (MeV)
MC data from BMW collaboration
(Sz. Borsanyi et al, JHEP 1011 (2010) 077)
Hagedorn fit: 5000 hadronic resonances,
m1 = 120 MeV, TH = 240 MeV, a = 0
for infinitely many resonances: divergent at T > TH
overestimates pressure above ≈ 200 MeV.
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 44 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Melting: number of hadronic/partonic excitations
Pressure
(hadr )
Phadr (T ) = e
−Geff
PQGP (T ) = e
−Geff
N
X
P0 (T , mn ),
(hadr )
Geff
= aT b ,
n∈hadrons
(part)
X
P0 (T , mn ),
(part)
Geff
(hadr )
= G0 + ce −dGeff
.
n∈partons
We use oversimplified description
continuum height increases with # of decay channels
⇒ effective cut-off in hadron mass
(J. Cleymans, D. Worku, Mod. Phys. Lett. A 26, 1197 (2011).)
assumed same width, height for all hadronic/partonic channels
most simple choice for hadronic Geff ∼ γ b (stretched
exponential) and γ ∼ T
for partons: take into account the number of hadronic modes
correlated parton-hadron description
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 45 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Full QCD pressure
Fit the model parameters to MC data
5
hadrons+partons
hadrons
partons
lattice MC data
4
p/T4
⇒ good agreement
3
2
1
0
0
100
200
300
400
500
600
700
800
T (MeV)
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 46 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Full QCD pressure
Different fits
5
4
4
3
p/T4
p/T4
5
hadrons+partons
hadrons
partons
lattice MC data
3
2
2
1
1
0
0
0
100
200
300
400
500
600
700
800
0
100
200
T (MeV)
300
400
500
600
700
800
T (MeV)
properties
T <
∼ Tc : HRG fully describes thermodynamics
[Tc , 2Tc ], [2Tc , 3Tc ]: hadron or parton dominated QCD
thermodynamics; both dof are present
T >
∼ 3Tc : QGP
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 47 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Full QCD pressure
Different fits
5
4
4
3
p/T4
p/T4
5
hadrons+partons
hadrons
partons
lattice MC data
3
2
2
1
1
0
0
0
100
200
300
400
500
T (MeV)
600
700
800
0
100
200
300
400
500
600
700
800
T (MeV)
Corollary
Tc is not a hadron – QGP transition temperature: partons
just start to appear there
full QGP only for T >
∼ 2.5 − 3Tc : 6 ∃ mechanism which could
do it faster
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 47 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Outlines
1
Introduction
Phases of QCD
2
The particle concept
Particles in free systems
Particles in interacting systems
Mathematical treatment of quasiparticles
3
Physical applications
Formation time of a quasiparticle
Gibbs paradox: indistinguishability of particles
Particle melting
4
QCD thermodynamics
Statistical model of QCD excitations
5
Conclusions
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 48 / 49
Introduction
The particle concept
Physical applications
QCD thermodynamics
Conclusions
Conclusions
excitations can be characterized by their spectra
not necessarily particle-like:
non-exponential time dependence
Gibbs paradox, melting: (continuous) disappearance of species
QCD thermodynamics at physical point at µ = 0
at Tc ≈ 156 MeV partons start to appear
T <
∼ Tc : hadrons
T ∈ [Tc , 3Tc ]: mixed phase
T >
∼ 3Tc : QGP
hadron phyiscs + melting + QGP
⇒ perturbative QCD thermodynamics?
Particle spectra and the QCD phase transition
University of Sussex, March 31, 2014. 49 / 49