Critical lines and points in the QCD phase diagram

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Critical lines and points in
the QCD phase diagram
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Understanding the phase diagram
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Phase diagram for ms gt mu,d
quark-gluon plasma deconfinement
quark matter superfluid B spontaneously broken
nuclear matter B,isospin (I3) spontaneously
broken, S conserved
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Order parameters

• Nuclear matter and quark matter are separated
  from other phases by true critical lines
• Different realizations of global symmetries
• Quark matter SSB of baryon number B
• Nuclear matter SSB of combination of B and
  isospin I3
• neutron-neutron condensate

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minimal phase diagram for
equal nonzero quark masses
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Endpoint of critical line ?
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How to find out ?
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Methods

• Lattice You have to wait until chiral
  limit
• is properly implemented !
• Models Quark meson models cannot work
• Higgs picture of QCD ?
• Experiment Has Tc been measured ?
• Indications for
• first order transition !

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Lattice
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Lattice results

• e.g. Karsch,Laermann,Peikert
• Critical temperature in chiral limit
• Nf 3 Tc ( 154 8 ) MeV
• Nf 2 Tc ( 173 8 ) MeV
• Chiral symmetry restoration and deconfinement at
  same Tc

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pressure
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realistic QCD

• precise lattice results not yet available
• for first order transition vs. crossover
• also uncertainties in determination of critical
  temperature ( chiral limit )
• extension to nonvanishing baryon number only for
  QCD with relatively heavy quarks

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Models
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Analytical description of phase
transition

• Needs model that can account simultaneously for
  the correct degrees of freedom below and above
  the transition temperature.
• Partial aspects can be described by more limited
  models, e.g. chiral properties at small momenta.

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Chiral quark meson model

• Limitation to chiral behavior
• Small up and down quark mass
• - large strange quark mass
• Particularly useful for critical behavior of
  second order phase transition or near endpoints
  of critical lines
• (see N. Tetradis for possible
  QCD-endpoint )

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• Quark descriptions ( NJL-model ) fail to describe
• the high temperature and high density phase
• transitions correctly
• High T chiral aspects could be ok , but glue
• (pion gas to quark gas )
• High density transition different Fermi surface
  for
• quarks and baryons ( T0)
• in mean field theory factor 27 for density at
  given chemical potential
• Confinement is important baryon enhancement
• 
  Berges,Jungnickel,
• Chiral perturbation theory even less complete

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Universe cools below 170 MeV

• Both gluons and quarks disappear from
• thermal equilibrium mass generation
• Chiral symmetry breaking
• mass for fermions
• Gluons ?
• Analogous situation in electroweak phase
  transition understood by Higgs mechanism
• Higgs description of QCD vacuum ?

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Higgs picture of QCD

• spontaneous breaking of color
• in the QCD vacuum
• octet condensate
• for Nf 3 ( u,d,s
  )

C.Wetterich, Phys.Rev.D64,036003(2001),hep-ph/0008
150
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Higgs phase and confinement

• can be equivalent
• then simply two different descriptions
  (pictures) of the same physical situation
• Is this realized for QCD ?
• Necessary condition spectrum of excitations
  with the same quantum numbers in both pictures
• - known for QCD mesons baryons -

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Quark antiquark condensate
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Octet condensate

• lt octet gt ? 0
• Spontaneous breaking of color
• Higgs mechanism
• Massive Gluons all masses equal
• Eight octets have vev
• Infrared regulator for QCD

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Flavor symmetry

• for equal quark masses
• octet preserves global SU(3)-symmetry
• diagonal in color and flavor
• color-flavor-locking
• (cf. Alford,Rajagopal,Wilc
  zek Schaefer,Wilczek)
• All particles fall into representations of
• the eightfold way
• quarks 8 1 , gluons 8

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Quarks and gluons carry the observed quantum
numbers of isospin and strangenessof the baryon
and vector meson octets !They are integer
charged!
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Low energy effective action
?f?
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accounts for masses and couplings of light
pseudoscalars, vector-mesons and baryons !
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Phenomenological parameters

• 5 undetermined parameters

• predictions

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Chiral perturbation theory

• all predictions of chiral perturbation theory
• determination of parameters

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Chiral phase transition at high temperature

• High temperature phase transition in QCD
• Melting of octet condensate
• Lattice simulations
• Deconfinement temperature critical temperature
  for restoration of chiral symmetry
• Why ?

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Simple explanation
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Higgs picture of the QCD-phase transition

• A simple mean field calculation gives roughly
  reasonable description that should be improved.
• Tc 170 MeV
• First order transition

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Experiment
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Has the critical temperature of the QCD phase
transition been measured ?
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Heavy ion collision
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Chemical freeze-out temperature
Tch 176 MeV
hadron abundancies
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Exclusion argument
hadronic phase with sufficient production of O
excluded !!
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Exclusion argument

• Assume T is a meaningful concept -
• complex issue, to be discussed later
• Tch lt Tc hadrochemical equilibrium
• Exclude Tch much smaller than Tc
• say Tch gt 0.95 Tc
• 0.95 lt Tch /Tc lt 1

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Has Tc been measured ?

• Observation statistical distribution of hadron
  species with chemical freeze out temperature
  Tch176 MeV
• Tch cannot be much smaller than Tc hadronic
  rates for
• Tlt Tc are too small to produce multistrange
  hadrons (O,..)
• Only near Tc multiparticle scattering becomes
  important
• ( collective excitations ) proportional to
  high power of density

TchTc
P.Braun-Munzinger,J.Stachel,CW
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Tch Tc
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Phase diagram
ltfgt0
ltfgt s ? 0
R.Pisarski
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Temperature dependence of chiral
order parameter

• Does experiment indicate a first order phase
  transition for µ 0 ?

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Second order phase transition
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Second order phase transition

• for T only somewhat below Tc
• the order parameter s is expected to
• be close to zero and
• deviate substantially from its vacuum value
• This seems to be disfavored by observation of
  chemical freeze out !

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Temperature dependent masses

• Chiral order parameter s depends on T
• Particle masses depend on s
• Chemical freeze out measures m/T for many species
• Mass ratios at T just below Tc are
• close to vacuum ratios

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Ratios of particle masses and
chemical freeze out

• at chemical freeze out
• ratios of hadron masses seem to be close to
  vacuum values
• nucleon and meson masses have different
  characteristic dependence on s
• mnucleon s , mp s -1/2
• ?s/s lt 0.1 ( conservative )

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first order phase transition seems to be
favored by chemical freeze out
or extremely rapid crossover
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How far has first order line been measured?
quarks and gluons
hadrons
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Exclusion argument for large density
hadronic phase with sufficient production of O
excluded !!
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First order phase transition line
quarks and gluons
µ923MeV transition to nuclear matter
hadrons
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Phase diagram for ms gt mu,d
quark-gluon plasma deconfinement
quark matter superfluid B spontaneously broken
nuclear matter B,isospin (I3) spontaneously
broken, S conserved
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Is temperature defined ?Does comparison with
equilibrium critical temperature make sense ?
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Prethermalization
J.Berges,Sz.Borsanyi,CW
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Vastly different time scales

• for thermalization of different quantities
• here scalar with mass m coupled to fermions
• ( linear quark-meson-model )
• method two particle irreducible non-
  equilibrium effective action ( J.Berges et al )

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Prethermalization equation
of state p/e
similar for kinetic temperature
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different temperatures
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Mode temperature
np occupation number for momentum p late
time Bose-Einstein or Fermi-Dirac distribution
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Kinetic equilibration before
chemical equilibration
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Once a temperature becomes stationary it takes
the value of the equilibrium temperature.Once
chemical equilibration has been reached the
chemical temperature equals the kinetic
temperature and can be associated with the
overall equilibrium temperature.Comparison of
chemical freeze out temperature with critical
temperature of phase transition makes sense
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Short and long distance degrees of freedom are
different ! Short distances quarks and
gluons Long distances baryons and
mesons How to make the transition?

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How to come from quarks and gluons to baryons and
mesons ?

• Find effective description where relevant
  degrees of freedom depend on momentum scale or
  resolution in space.
• Microscope with variable resolution
• High resolution , small piece of volume
• quarks and gluons
• Low resolution, large volume hadrons

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Functional Renormalization Group

• from small to large scales

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Exact renormalization group equation
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Infrared cutoff
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Nambu Jona-Lasinio model
and more general quark meson models
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Chiral condensate
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Scalingformofequationof state
Berges, Tetradis,
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temperature dependent
masses

• pion mass
• sigma mass

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conclusion

• Experimental determination of critical
  temperature may be more precise than lattice
  results
• Rather simple phase structure is suggested
• Analytical understanding is only at beginning

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end
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Cosmological phase transition

• when the universe cools below 175 MeV
• 10-5 seconds after the big bang

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QCD at high density

• Nuclear matter
• Heavy nuclei
• Neutron stars
• Quark stars

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QCD at high temperature

• Quark gluon plasma
• Chiral symmetry restored
• Deconfinement ( no linear heavy quark potential
  at large distances )
• Lattice simulations both effects happen at the
  same temperature

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Solution of QCD

• Effective action ( for suitable fields ) contains
  all the relevant information of the solution of
  QCD
• Gauge singlet fields, low momenta
• Order parameters, meson-( baryon- ) propagators
• Gluon and quark fields, high momenta
• Perturbative QCD
• Aim Computation of effective action

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QCD phase transition

• Quark gluon plasma
• Gluons 8 x 2 16
• Quarks 9 x 7/2 12.5
• Dof 28.5
• Chiral symmetry

• Hadron gas
• Light mesons 8
• (pions 3 )
• Dof 8
• Chiral sym. broken

Large difference in number of degrees of freedom
! Strong increase of density and energy density
at Tc !
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Spontaneous breaking of color

• Condensate of colored scalar field
• Equivalence of Higgs and confinement description
  in real (Nf3) QCD vacuum
• Gauge symmetries not spontaneously broken in
  formal sense ( only for fixed gauge )
• Similar situation as in electroweak theory
• No fundamental scalars
• Symmetry breaking by quark-antiquark-condensate

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A simple mean field calculation
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Hadron abundancies
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Bound for critical temperature

• 0.95 Tclt Tch lt Tc
• not I have a model where Tc Tch
• not I use Tc as a free parameter and
• find that in a model simulation it
  is
• close to the lattice value ( or Tch
  )
• Tch 176 MeV (?)

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Estimate of critical temperature

• For Tch 176 MeV
• 0.95 lt Tch /Tc
• 176 MeV lt Tc lt 185 MeV
• 0.75 lt Tch /Tc
• 176 MeV lt Tc lt 235 MeV
• Quantitative issue matters!

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Key argument

• Two particle scattering rates not sufficient to
  produce O
• multiparticle scattering for O-production
  dominant only in immediate vicinity of Tc

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needed lower bound on Tch/ Tc
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Exclude the hypothesis of a hadronic phase where
multistrange particles are produced at T
substantially smaller than Tc
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Mechanisms for production of multistrange hadrons

• Many proposals
• Hadronization
• Quark-hadron equilibrium
• Decay of collective excitation (s field )
• Multi-hadron-scattering
• Different pictures !

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Hadronic picture of O - production

• Should exist, at least semi-quantitatively, if
  Tch lt Tc
• ( for Tch Tc Tchgt0.95 Tc is fulfilled
  anyhow )
• e.g. collective excitations multi-hadron-scatter
  ing
• (not necessarily the best and simplest
  picture )
• multihadron -gt O X should have sufficient rate
• Check of consistency for many models
• Necessary if Tch? Tc and temperature is defined
• Way to give quantitative bound on Tch / Tc

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Rates for multiparticle scattering
2 pions 3 kaons -gt O antiproton
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Very rapid density increase

• in vicinity of critical temperature
• Extremely rapid increase of rate of multiparticle
  scattering processes
• ( proportional to very high power of density )

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Energy density

• Lattice simulations
• Karsch et al
• even more dramatic
• for first order
• transition

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Phase space

• increases very rapidly with energy and therefore
  with temperature
• effective dependence of time needed to produce O
• tO T -60 !
• This will even be more dramatic if transition is
  closer to first order phase transition

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Production time for O

• multi-meson scattering
• pppKK -gt
• Op
• strong dependence on pion density

P.Braun-Munzinger,J.Stachel,CW
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extremely rapid change

• lowering T by 5 MeV below critical temperature
  
• rate of O production decreases by
• factor 10
• This restricts chemical freeze out to close
  vicinity of critical temperature
• 0.95 lt Tch /Tc lt 1

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enough time for O - production

• at T176 MeV
• tO 2.3 fm
• consistency !

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Relevant time scale in hadronic phase
rates needed for equilibration of O and kaons
?T 5 MeV, FOK 1.13 , tT 8 fm
two particle scattering
(0.02-0.2)/fm
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A possible source of error temperature-dependent
particle masses
Chiral order parameter s depends on T
chemical freeze out measures T/m !
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uncertainty in m(T)uncertainty in critical
temperature
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systematic uncertainty
?s/s?Tc/Tc
?s is negative
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conclusion

• experimental determination of critical
  temperature may be more precise than lattice
  results
• error estimate becomes crucial

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Thermal equilibration
occupation numbers
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Chiral symmetry restoration at
high temperature
Low T SSB ltfgtf0 ? 0
High T SYM ltfgt0
at high T less order more symmetry examples
magnets, crystals
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Order of the phase transition is crucial
ingredient for experiments ( heavy ion
collisions )and cosmological phase transition
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Order ofthephasetransition
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First order phase transition
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Simple one loop structure nevertheless (almost)
exact
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Flow equation for average potential
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Critical temperature , Nf 2
Lattice simulation
J.Berges,D.Jungnickel,