Phenomenology I: Phase transition

1
Phenomenology I Phase transition

• The quark-gluon and hadron equations of state
• The energy density of (massless) quarks and
  gluons is derived from Fermi-Dirac statistics and
  Bose-Einstein statistics.
• where m is the quark chemical potential, mq
  - mq and b 1/T.
• Taking into account the number of degrees of
  freedom
• Consider two extremes
• 1. High temperature, low net baryon density (T gt
  0, mB 0).
• 2. Low temperature, high net baryon density (T
  0, mB gt 0).

mB 3 mq
2
Phenomenology II critical parameters

• High temperature, low density limit - the early
  universe
• Two terms contribute to the total energy density
• For a relativistic gas
• For stability
• Low temperature, high density limit - neutron
  stars
• Only one term contributes to the total energy
  density
• By a similar argument

2-8 times normal nuclear matter density given
pFermi 250 MeV and r 2m3/3p2
3
Nuclear Equation of State
4
Phase Diagram of Nuclear Matter
hadrons quarks and gluons hadrons
5
Estimating the critical parameters, Tc and ec

• Mapping out the Nuclear Matter Phase Diagram
• Perturbation theory highly successful in
  applications of QED.
• In QCD, perturbation theory is only applicable
  for very hard processes.
• Two solutions
• 1. Phenomenological models (MIT Bag model)
• 2. Lattice QCD calculations

6
What is the prediction according to Lattice QCD ?
Essay on lattice QCD
F. Karsch, Prog. Theor. Phys. Suppl. 153, 106
(2004)

• Quarks and gluons are studied on a discrete
  space-time lattice
• Solves the problem of divergences in pQCD
  calculations (which arise due to loop diagrams)

Stefan-Boltzman limit for ideal plasma
No ideal plasma
RHIC
Lattice QCD assumes thermal equilibration
Transition point T 170 MeV e 1.0 GeV/fm3
Temperature

• Lattice QCD shows a rapid increase in the entropy
  associated with the deconfinement of quarks and
  gluons.
• ?Critical temperature (phase transition) Tc
  170 MeV
• Ideal plasma limit not reached
• Strong coupling between partonic degrees of
  freedom

7
Phase Diagram for the Strong Interaction
8
Statistical ModelsA.) Chemical
equilibration(Braun-Munzinger, Stachel, Redlich,
Tounsi, Rafelski)B.) Thermal equilibration(Schne
dermann, Heinz)C.) Hydrodynamics(Heinz,
Eskola,Ruuskanen, Teaney,Hirano)
9
Basic Idea of Statistical Hadronic Models

• Assume thermally (constant Tch) and chemically
  (constant ni) equilibrated system
• Given Tch and ? 's ( system size), ni's can be
  calculated in a grand canonical ensemble

• Chemical freeze-out
• (yields ratios)
• inelastic interactions stops
• particle abundances fixed (except maybe
  resonances)
• Thermal freeze-out
• (shapes of pT,mT spectra)
• elastic interactions stops
• particle dynamics fixed

10
Basic Idea of Statistical Hadronic Models

• Assume thermally (constant Tch) and chemically
  (constant ni) equilibrated system

Multiplicity of L(uds) particle
n number of particles J spin V volume
1/GeV3 T temperature m mass mb
baryochemical potential
Particle ratios Volume cancels out
particle/antiparticle ? sensitive to
mb mass difference ?
sensitive to T same
quark content ? depend only on T

11
Particle productionStatistical models do well
We get a chemical freeze-out temperature and a
baryochemical potential out of the fit
12
Ratios that constrain model parameters
13
Statistical Hadronic Models Misconceptions

• Model says nothing about how system reaches
  chemical equilibrium
• Model says nothing about when system reaches
  chemical equilibrium
• Model makes no predictions of dynamical
  quantities
• Some models use a strangeness suppression factor,
  others not
• Model does not make assumptions about a partonic
  phase However the model findings can complement
  other studies of the phase diagram (e.g.
  Lattice-QCD)

14
Thermalization in Elementary Collisions ?
Seems to work rather well ?!
Beccatini, Heinz, Z.Phys. C76 (1997) 269
15
Thermalization in Elementary Collisions ?

• Is a process which leads to multiparticle
  production thermal?
• Any mechanism for producing hadrons which evenly
  populates the free particle phase space will
  mimic a microcanonical ensemble.
• Relative probability to find a given number of
  particles is given by the ratio of the
  phase-space volumes Pn/Pn fn(E)/fn(E) ?
  given by statistics only. Difference between MCE
  and CE vanishes as the size of the system N
  increases.

This type of thermal behavior requires no
rescattering and no interactions. The collisions
simply serve as a mechanism to populate phase
space without ever reaching thermal or chemical
equilibrium In RHI we are looking for large
collective effects.
16
Statistics ? Thermodynamics
pp
Ensemble of events constitutes a statistical
ensemble T and µ are simply Lagrange multipliers
Phase Space Dominance
AA

• We can talk about pressure
• T and µ are more than Lagrange multipliers

17
Does the thermal model always work ?
Data Fit (s) Ratio

• Particle ratios well described by Tch 160?10
  MeV, mB 24 ?5 MeV
• Resonance ratios change from pp to AuAu ?
  Hadronic Re-scatterings!

18
T systematics
Essay on Hagedorn Temperature
Satz Nucl.Phys. A715 (2003) 3c
filled AA open elementary

• it looks like Hagedorn was right!
• if the resonance mass spectrum grows
  exponentially (and this seems to be the case),
  there is a maximum possible temperature for a
  system of hadrons
• indeed, we dont seem to be able to get a system
  of hadrons with a temperature beyond Tmax 170
  MeV!

19
Phase Diagram for the Strong Interaction
20
T Contributions Tch T(transverse radial flow)
21
Radial flow
ltpTgt prediction with Tth and ltbgt obtained from
blastwave fit (green line)
STAR
ltpTgt prediction for Tch 170 MeV and ltbgt0 pp
no rescattering, no flow no thermal equilibrium
preliminary F. Wang
22
Thermal Spectra
Invariant spectrum of particles radiated by a
thermal source
where mT (m2pT2)½ transverse mass (Note
requires knowledge of mass) m b mb s
ms grand canonical chem. potential T temperature
of source Neglect quantum statistics (small
effect) and integrating over rapidity gives
R. Hagedorn, Supplemento al Nuovo Cimento Vol.
III, No.2 (1965)
At mid-rapidity E mT cosh y mT and hence
Boltzmann
23
Resonance pT Spectra in pp at 200 GeV at mid
Rapidity
K(892)
S(1385)
a.u.
pT-coverage (yield) ?pT?
(integrated) K(892) 95
680 ? 30 ? 30 MeV S(1385) 81
1100 ? 20 ? 100 MeV ?(1520) 91
1080 ? 90 ? 110 MeV
?(1520)
dN/dy at y0 K(892) 0.059 ? 0.002 ? 0.004
?(1520) 0.0037 ? 0.004 ? 0.006
24
Thermal Spectra (flow aside)

• Describes many spectra well over several orders
  of magnitude with almost uniform slope 1/T
• usually fails at low-pT
• (? flow)
• most certainly will fail
• at high-pT
• (? power-law)

N.B. Constituent quark and parton recombination
models yield exponential spectra with partons
following a pQCD power-law distribution. (Biro,
Müller, hep-ph/0309052) ? T is not related to
actual temperature but reflects pQCD parameter
p0 and n.
25
Hydrodynamical Models(reading assignments for
next week)
Hydrodynamic Approaches to Relativistic Heavy
Ion Collisions Authors Tetsufumi Hirano
Comments 8 pages, 3 figures, invited talk given
at XXXIV International Symposium on
Multiparticle Dynamics, Sonoma, USA, July 26 -
August 1, 2004Journal-ref Acta Phys.Polon. B36
(2005) 187-194 nucl-th/0410017 Hydrodynamic
Models for Heavy Ion Collisions Authors P.
Huovien, P.V. Ruuskanen Comments 42 pages, 15
figures, An invited review for Nov. 2006 edition
of Annual Review of Nuclear and Particle
Physicsnucl-th/0605008
26
Microscopic Models
Essay on Microscopic models

• all decay
• - measured

Marcus Bleicher and Jörg Aichelin Phys. Lett.
B530 (2002) 81-87. M. Bleicher and Horst Stöcker
.Phys.G30 (2004) 111.
chemical freeze-out 5fm/c kinetic freeze-out
20-30 fm/c (long life time !)