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Critical lines and Critical lines and pointspoints in the in the
QCD phase QCD phase diagramdiagram
Understanding the phase Understanding the phase diagramdiagram
nuclear matternuclear matterB,isospin (IB,isospin (I33) spontaneously broken, S conserved) spontaneously broken, S conserved
quark matter : superfluidquark matter : superfluidB spontaneously brokenB spontaneously broken
quark-gluon plasmaquark-gluon plasma““deconfinement”deconfinement”
Phase diagram for mPhase diagram for mss > m > mu,du,d
Order parametersOrder parameters
• Nuclear matter and quark matter are Nuclear matter and quark matter are separated from other phases by true separated from other phases by true critical linescritical lines
• Different realizations of Different realizations of globalglobal symmetriessymmetries
• Quark matter: SSB of baryon number BQuark matter: SSB of baryon number B• Nuclear matter: SSB of combination of B Nuclear matter: SSB of combination of B
and isospin Iand isospin I3 3 neutron-neutron condensateneutron-neutron condensate
““minimal” phase diagramminimal” phase diagram for equal nonzero quark massesfor equal nonzero quark masses
Endpoint of critical line ?Endpoint of critical line ?
How to find out ?How to find out ?
MethodsMethods
• Lattice :Lattice : You have to wait until chiral limit You have to wait until chiral limit is properly implemented !is properly implemented !
• Models :Models : Quark meson models cannot Quark meson models cannot workwork
Higgs picture of QCD ?Higgs picture of QCD ?
• Experiment :Experiment : Has THas Tcc been measured ? been measured ? Indications for Indications for first order transition !first order transition !
LatticeLattice
Lattice resultsLattice results
e.g. Karsch,Laermann,Peikerte.g. Karsch,Laermann,Peikert
Critical temperature in chiral limit :Critical temperature in chiral limit :
NNff = 3 : T = 3 : Tcc = ( 154 ± 8 ) MeV = ( 154 ± 8 ) MeVNNff = 2 : T = 2 : Tcc = ( 173 ± 8 ) MeV = ( 173 ± 8 ) MeV
Chiral symmetry restoration and Chiral symmetry restoration and deconfinement at same Tdeconfinement at same Tcc
pressurepressure
realistic QCDrealistic QCD
• precise lattice results not yet availableprecise lattice results not yet available
for first order transition vs. crossoverfor first order transition vs. crossover
• also uncertainties in determination of also uncertainties in determination of critical temperature ( chiral limit …)critical temperature ( chiral limit …)
• extension to nonvanishing baryon extension to nonvanishing baryon number only for QCD with relatively number only for QCD with relatively heavy quarksheavy quarks
ModelsModels
Analytical description of Analytical description of phase transition phase transition
• Needs model that can account Needs model that can account simultaneously for the correct simultaneously for the correct degrees of freedom below and above degrees of freedom below and above the transition temperature.the transition temperature.
• Partial aspects can be described by Partial aspects can be described by more limited models, e.g. chiral more limited models, e.g. chiral properties at small momenta. properties at small momenta.
Chiral quark meson modelChiral quark meson model
• Limitation to chiral behaviorLimitation to chiral behavior
• Small up and down quark massSmall up and down quark mass
- large strange quark mass- large strange quark mass
• Particularly useful for critical behavior Particularly useful for critical behavior of second order phase transition or of second order phase transition or near endpoints of critical linesnear endpoints of critical lines
(see N. Tetradis for possible QCD-endpoint )(see N. Tetradis for possible QCD-endpoint )
Quark descriptions ( NJL-model ) fail to describeQuark descriptions ( NJL-model ) fail to describethe high temperature and high density phasethe high temperature and high density phasetransitions correctlytransitions correctly
High T : chiral aspects could be ok , but High T : chiral aspects could be ok , but glueglue … … (pion gas to quark gas )(pion gas to quark gas )
High density transition : different Fermi surface forHigh density transition : different Fermi surface for quarks and baryons ( T=0) quarks and baryons ( T=0) – – in mean field theory factor 27 for density at in mean field theory factor 27 for density at
given chemical potential –given chemical potential –Confinement is important : baryon enhancementConfinement is important : baryon enhancement Berges,Jungnickel,…Berges,Jungnickel,…
Chiral perturbation theory even less completeChiral perturbation theory even less complete
Universe cools below 170 MeV…Universe cools below 170 MeV…
Both gluons and quarks disappear fromBoth gluons and quarks disappear from thermal equilibrium : mass generationthermal equilibrium : mass generation Chiral symmetry breaking Chiral symmetry breaking mass for fermionsmass for fermions Gluons ?Gluons ?
Analogous situation in electroweak phase Analogous situation in electroweak phase transition understood by Higgs mechanismtransition understood by Higgs mechanism
Higgs description of QCD vacuum ?Higgs description of QCD vacuum ?
Higgs picture of QCDHiggs picture of QCD
““spontaneous breaking of color “spontaneous breaking of color “
in the QCD – vacuumin the QCD – vacuum
octet condensateoctet condensate
for Nfor Nf f = 3 ( u,d,s )= 3 ( u,d,s )
C.Wetterich, Phys.Rev.D64,036003(2001),hep-ph/0008150
Higgs phase and Higgs phase and confinementconfinementcan be equivalent –can be equivalent –
then simply two different descriptions then simply two different descriptions (pictures) of the same physical situation(pictures) of the same physical situation
Is this realized for QCD ?Is this realized for QCD ?
Necessary condition : spectrum of Necessary condition : spectrum of excitations with the same quantum excitations with the same quantum numbers in both picturesnumbers in both pictures
- known for QCD : mesons + baryons -- known for QCD : mesons + baryons -
Quark –antiquark Quark –antiquark condensatecondensate
Octet condensateOctet condensate
< octet > ≠ 0 :< octet > ≠ 0 :
•““Spontaneous breaking of color”Spontaneous breaking of color”
•Higgs mechanismHiggs mechanism
•Massive Gluons – all masses equalMassive Gluons – all masses equal
•Eight octets have vevEight octets have vev
• Infrared regulator for QCDInfrared regulator for QCD
Flavor symmetryFlavor symmetry
for equal quark masses :for equal quark masses :
octet preserves global SU(3)-symmetryoctet preserves global SU(3)-symmetry “ “diagonal in color and flavor”diagonal in color and flavor” “ “color-flavor-locking”color-flavor-locking”
(cf. Alford,Rajagopal,Wilczek ; Schaefer,Wilczek)(cf. Alford,Rajagopal,Wilczek ; Schaefer,Wilczek)
All particles fall into representations ofAll particles fall into representations of the “eightfold way”the “eightfold way”
quarks : 8 + 1 , gluons : 8quarks : 8 + 1 , gluons : 8
Quarks and gluons carry the Quarks and gluons carry the observed quantum numbers of observed quantum numbers of isospin and strangenessisospin and strangenessof the baryon and of the baryon and vector meson octets !vector meson octets !
They are integer charged!They are integer charged!
Low energy effective actionLow energy effective action
γ=φ+χ
……accounts for accounts for massesmasses and and couplingscouplings of light of light pseudoscalars, vector-pseudoscalars, vector-mesons and baryons !mesons and baryons !
Phenomenological Phenomenological parametersparameters• 5 undetermined 5 undetermined
parametersparameters• predictionspredictions
Chiral perturbation theoryChiral perturbation theory
+ all predictions of chiral perturbation + all predictions of chiral perturbation theorytheory
+ determination of parameters+ determination of parameters
Chiral phase transition Chiral phase transition at high temperatureat high temperatureHigh temperature phase transition in QCD :High temperature phase transition in QCD :
Melting of octet condensateMelting of octet condensate
Lattice simulations :Lattice simulations :
Deconfinement temperature = critical Deconfinement temperature = critical temperature for restoration of chiral temperature for restoration of chiral symmetrysymmetry
Why ?Why ?
Simple explanation :Simple explanation :
Higgs picture of the QCD-phase Higgs picture of the QCD-phase transitiontransition
A simple mean field calculation A simple mean field calculation gives roughly reasonable description gives roughly reasonable description that should be improved.that should be improved.
TTcc =170 MeV =170 MeV First order transitionFirst order transition
ExperimentExperiment
Has the Has the critical temperature of critical temperature of the QCD phase the QCD phase transition been transition been measured ?measured ?
Heavy ion collisionHeavy ion collision
Chemical freeze-out Chemical freeze-out temperaturetemperature T Tch ch =176 MeV=176 MeV
hadron abundancieshadron abundancies
Exclusion argumentExclusion argument
hadronic phasehadronic phasewith sufficient with sufficient production of production of ΩΩ : : excluded !!excluded !!
Exclusion argumentExclusion argument
Assume T is a meaningful concept -Assume T is a meaningful concept - complex issue, to be discussed latercomplex issue, to be discussed later
TTchch < T < Tc c : : hadrohadrochemical equilibriumchemical equilibrium
Exclude TExclude Tch ch much smaller than Tmuch smaller than Tcc ::
say Tsay Tchch > 0.95 T > 0.95 Tcc
0.95 < T0.95 < Tchch /T /Tcc < 1 < 1
Has THas Tc c been measured ?been measured ?
• Observation : statistical distribution of hadron species Observation : statistical distribution of hadron species with “chemical freeze out temperature “ Twith “chemical freeze out temperature “ Tchch=176 MeV=176 MeV
• TTch ch cannot be much smaller than Tcannot be much smaller than Tc c : hadronic rates for : hadronic rates for T< TT< Tc c are too small to produce multistrange hadrons (are too small to produce multistrange hadrons (ΩΩ,..),..)
• Only near TOnly near Tc c multiparticle scattering becomes important multiparticle scattering becomes important ( collective excitations …) – proportional to high power of ( collective excitations …) – proportional to high power of
densitydensity
P.Braun-Munzinger,J.Stachel,CWP.Braun-Munzinger,J.Stachel,CW
TTchch≈T≈Tcc
TTch ch ≈ T ≈ Tcc
Phase diagramPhase diagram
R.PisarskiR.Pisarski
<<φφ>= >= σσ ≠≠ 0 0
<<φφ>>≈≈00
Temperature Temperature dependence of dependence of chiral order parameter chiral order parameter
Does experiment Does experiment indicate a first order indicate a first order phase transition for phase transition for μμ = 0 = 0 ??
Second order phase Second order phase transitiontransition
Second order phase Second order phase transitiontransitionfor T only somewhat below Tfor T only somewhat below Tc c ::
the order parameter the order parameter σσ is expected to is expected to
be close to zero andbe close to zero and
deviate substantially from its deviate substantially from its vacuum valuevacuum value
This seems to be disfavored by This seems to be disfavored by observation of chemical freeze out !observation of chemical freeze out !
Temperature dependent Temperature dependent massesmasses
• Chiral order parameter Chiral order parameter σσ depends depends on Ton T
• Particle masses depend on Particle masses depend on σσ
• Chemical freeze out measures m/T Chemical freeze out measures m/T for many speciesfor many species
• Mass ratios at T just below TMass ratios at T just below Tcc are are
close to vacuum ratiosclose to vacuum ratios
RatiosRatios of particle masses and of particle masses and chemical freeze out chemical freeze out
at chemical freeze out :at chemical freeze out :
• ratios of hadron masses seem to be close to ratios of hadron masses seem to be close to vacuum valuesvacuum values
• nucleon and meson masses have different nucleon and meson masses have different characteristic dependence on characteristic dependence on σσ
• mmnucleon nucleon ~ ~ σσ , m , mππ ~ ~ σσ -1/2-1/2
• ΔσΔσ/σ < 0.1 ( conservative ) /σ < 0.1 ( conservative )
first order phase first order phase transitiontransition seems to be favored by seems to be favored by chemical freeze out chemical freeze out
……or extremely rapid crossoveror extremely rapid crossover
How How farfar has first order line been has first order line been measured?measured?
hadrons hadrons
quarks and gluonsquarks and gluons
hadronic phasehadronic phasewith sufficient with sufficient production of production of ΩΩ :: excluded !!excluded !!
Exclusion argument for large Exclusion argument for large densitydensity
First order phase transition lineFirst order phase transition line
hadrons hadrons
quarks and gluonsquarks and gluons
μμ=923MeV=923MeVtransition totransition to nuclearnuclear matter matter
nuclear matternuclear matterB,isospin (IB,isospin (I33) spontaneously broken, S conserved) spontaneously broken, S conserved
quark matter : superfluidquark matter : superfluidB spontaneously brokenB spontaneously broken
quark-gluon plasmaquark-gluon plasma““deconfinement”deconfinement”
Phase diagram for mPhase diagram for mss > m > mu,du,d
Is temperature defined ?Is temperature defined ?
Does comparison with Does comparison with equilibrium critical equilibrium critical temperature make sense ?temperature make sense ?
PrethermalizationPrethermalization
J.Berges,Sz.Borsanyi,CWJ.Berges,Sz.Borsanyi,CW
Vastly different time scalesVastly different time scales
for “thermalization” of different for “thermalization” of different quantitiesquantities
here : scalar with mass m coupled to fermions here : scalar with mass m coupled to fermions
( linear quark-meson-model )( linear quark-meson-model )
method : two particle irreducible non- method : two particle irreducible non- equilibrium effective action ( equilibrium effective action ( J.Berges et alJ.Berges et al ) )
PrethermalizationPrethermalization equation of state p/ equation of state p/εε
similar for kinetic temperaturesimilar for kinetic temperature
different “temperatures”different “temperatures”
Mode temperatureMode temperature
nnpp :occupation number :occupation number for momentum pfor momentum p
late time:late time:Bose-Einstein orBose-Einstein orFermi-Dirac distributionFermi-Dirac distribution
Kinetic equilibration beforeKinetic equilibration before chemical equilibration chemical equilibration
Once a temperature becomes Once a temperature becomes stationary it takes the value of the stationary it takes the value of the equilibrium temperature.equilibrium temperature.
Once chemical equilibration has been Once chemical equilibration has been reached the chemical temperature reached the chemical temperature equals the kinetic temperature and equals the kinetic temperature and can be associated with the overall can be associated with the overall equilibrium temperature.equilibrium temperature.
Comparison of chemical freeze out Comparison of chemical freeze out temperature with critical temperaturetemperature with critical temperature of phase transition makes senseof phase transition makes sense
Short and long distance Short and long distance
degrees of freedom degrees of freedom
are different !are different !
Short distances : quarks and gluonsShort distances : quarks and gluons
Long distances : baryons and mesonsLong distances : baryons and mesons
How to make the transition?How to make the transition?
How to come from quarks How to come from quarks and gluons to baryons and and gluons to baryons and mesons ?mesons ?
Find effective description where Find effective description where relevant degrees of freedom depend on relevant degrees of freedom depend on momentum scale or resolution in spacemomentum scale or resolution in space..
Microscope with variable resolution:Microscope with variable resolution:• High resolution , small piece of volume:High resolution , small piece of volume: quarks and gluonsquarks and gluons• Low resolution, large volume : hadronsLow resolution, large volume : hadrons
Functional Renormalization Functional Renormalization GroupGroup
from small to large scalesfrom small to large scales
Exact renormalization group Exact renormalization group equationequation
Infrared cutoffInfrared cutoff
Nambu Jona-Lasinio modelNambu Jona-Lasinio model
……and more general quark meson modelsand more general quark meson models
Chiral condensateChiral condensate
ScalingScalingformformofofequationequationof stateof state
Berges,Tetradis,…
temperature temperature dependent dependent massesmasses
pion masspion mass
sigma masssigma mass
conclusionconclusion
• Experimental determination of critical Experimental determination of critical temperature may be more precise than temperature may be more precise than lattice resultslattice results
• Rather simple phase structure is suggestedRather simple phase structure is suggested
• Analytical understanding is only at Analytical understanding is only at beginningbeginning
endend
Cosmological phase Cosmological phase transition…transition…
……when the universe cools below 175 when the universe cools below 175 MeVMeV
1010-5 -5 seconds after the big bangseconds after the big bang
QCD at high densityQCD at high density
Nuclear matterNuclear matter
Heavy nucleiHeavy nuclei
Neutron starsNeutron stars
Quark stars …Quark stars …
QCD at high temperatureQCD at high temperature
• Quark – gluon plasmaQuark – gluon plasma
• Chiral symmetry restoredChiral symmetry restored
• Deconfinement ( no linear heavy Deconfinement ( no linear heavy quark potential at large distances )quark potential at large distances )
• Lattice simulations : both effects Lattice simulations : both effects happen at the same temperaturehappen at the same temperature
““Solution” of QCDSolution” of QCD
Effective action ( for suitable fields ) contains all Effective action ( for suitable fields ) contains all the relevant information of the solution of QCDthe relevant information of the solution of QCD
Gauge singlet fields, low momenta:Gauge singlet fields, low momenta:
Order parameters, meson-( baryon- ) Order parameters, meson-( baryon- ) propagatorspropagators
Gluon and quark fields, high momenta:Gluon and quark fields, high momenta:
Perturbative QCDPerturbative QCD
Aim: Computation of effective actionAim: Computation of effective action
QCD – phase transitionQCD – phase transition
Quark –gluon plasmaQuark –gluon plasma
• Gluons : 8 x 2 = 16Gluons : 8 x 2 = 16
• Quarks : 9 x 7/2 =12.5Quarks : 9 x 7/2 =12.5
• Dof : 28.5Dof : 28.5
Chiral symmetryChiral symmetry
Hadron gasHadron gas
• Light mesons : 8Light mesons : 8
• (pions : 3 )(pions : 3 )
• Dof : 8Dof : 8
Chiral sym. brokenChiral sym. broken
Large difference in number of degrees of freedom !Large difference in number of degrees of freedom !Strong increase of density and energy density at TStrong increase of density and energy density at Tcc !!
Spontaneous breaking of Spontaneous breaking of colorcolor• Condensate of colored scalar fieldCondensate of colored scalar field• Equivalence of Higgs and confinement Equivalence of Higgs and confinement
description in description in realreal (N (Nff=3) QCD =3) QCD vacuumvacuum• Gauge symmetries not spontaneously broken Gauge symmetries not spontaneously broken
in formal sense ( only for fixed gauge ) in formal sense ( only for fixed gauge ) Similar situation as in electroweak theorySimilar situation as in electroweak theory• No “fundamental” scalarsNo “fundamental” scalars• Symmetry breaking by quark-antiquark-Symmetry breaking by quark-antiquark-
condensatecondensate
A simple mean field calculationA simple mean field calculation
Hadron abundanciesHadron abundancies
Bound for critical Bound for critical temperaturetemperature
0.95 T0.95 Tcc< T< Tchch < T < Tcc
• not : not : “ I have a model where T“ I have a model where Tcc≈ T≈ Tch ch ““
• not :not : “ I use T “ I use Tcc as a free parameter and as a free parameter and find that in a model simulation it is find that in a model simulation it is
close to the lattice value ( or Tclose to the lattice value ( or Tchch ) “ ) “
TTch ch ≈ 176 MeV≈ 176 MeV (?) (?)
Estimate of critical Estimate of critical temperaturetemperature
For TFor Tch ch ≈ 176 MeV :≈ 176 MeV :
0.95 < T0.95 < Tchch /T /Tcc
• 176 MeV < T176 MeV < Tc c < 185 MeV< 185 MeV
0.75 < T0.75 < Tchch /T /Tcc
• 176 MeV < T176 MeV < Tc c < 235 MeV< 235 MeV
Quantitative issue matters!Quantitative issue matters!
Key argumentKey argument
• Two particle scattering rates not Two particle scattering rates not sufficient to produce sufficient to produce ΩΩ
• ““multiparticle scattering for multiparticle scattering for ΩΩ--production “ : dominant only in production “ : dominant only in immediateimmediate vicinity of T vicinity of Tcc
needed :needed :
lowerlower bound bound onon TTchch/ T/ Tcc
Exclude Exclude the hypothesis of a hadronic the hypothesis of a hadronic phase where multistrange particles phase where multistrange particles are produced at T substantially are produced at T substantially smaller than Tsmaller than Tcc
Mechanisms for production of Mechanisms for production of multistrange hadronsmultistrange hadrons
Many proposalsMany proposals
• HadronizationHadronization• Quark-hadron equilibriumQuark-hadron equilibrium• Decay of collective excitation (Decay of collective excitation (σσ – –
field )field )• Multi-hadron-scatteringMulti-hadron-scattering
Different pictures !Different pictures !
Hadronic picture of Hadronic picture of ΩΩ - - productionproductionShould exist, at least semi-quantitatively, if Should exist, at least semi-quantitatively, if TTchch < T < Tcc
( for T( for Tchch = T = Tc c : T: Tchch>0.95 T>0.95 Tc c is fulfilled anyhow )is fulfilled anyhow )
e.g. collective excitations ≈ multi-hadron-scatteringe.g. collective excitations ≈ multi-hadron-scattering (not necessarily the best and simplest picture )(not necessarily the best and simplest picture )
multihadron -> multihadron -> ΩΩ + X should have sufficient rate + X should have sufficient rate
Check of consistency for many modelsCheck of consistency for many models
Necessary if Necessary if TTchch≠ T≠ Tcc and temperature is defined and temperature is defined
Way to give Way to give quantitativequantitative bound on T bound on Tch ch / T/ Tcc
Rates for multiparticle Rates for multiparticle scatteringscattering
2 pions + 3 kaons -> 2 pions + 3 kaons -> ΩΩ + antiproton + antiproton
Very rapid density increaseVery rapid density increase
……in vicinity of critical temperaturein vicinity of critical temperature
Extremely rapid increase of rate of Extremely rapid increase of rate of multiparticle scattering processesmultiparticle scattering processes
( proportional to very high power of ( proportional to very high power of density )density )
Energy densityEnergy density
Lattice simulationsLattice simulations
Karsch et alKarsch et al
even more even more dramaticdramatic
for first orderfor first order
transitiontransition
Phase spacePhase space
• increases very rapidly with energy and increases very rapidly with energy and therefore with temperaturetherefore with temperature
• effective dependence of time needed to effective dependence of time needed to produce produce ΩΩ
ττΩΩ ~ T ~ T -60-60 ! !
This will even be more dramatic if transition is This will even be more dramatic if transition is closer to first order phase transitioncloser to first order phase transition
Production time for Production time for ΩΩ
multi-meson multi-meson scatteringscattering
ππ++ππ++ππ+K+K ->+K+K -> ΩΩ+p+p
strong strong dependence on dependence on “pion” density“pion” density
P.Braun-Munzinger,J.Stachel,CWP.Braun-Munzinger,J.Stachel,CW
extremely rapid changeextremely rapid change
lowering T by 5 MeV below critical lowering T by 5 MeV below critical temperature :temperature :
rate of rate of ΩΩ – production decreases by – production decreases by
factor 10factor 10
This restricts chemical freeze out to close This restricts chemical freeze out to close vicinity of critical temperaturevicinity of critical temperature
0.95 < T0.95 < Tchch /T /Tcc < 1 < 1
enough time for enough time for ΩΩ - - productionproduction
at T=176 MeV :at T=176 MeV :
ττΩΩ ~ 2.3 fm~ 2.3 fm
consistency !consistency !
Relevant time scale in hadronic Relevant time scale in hadronic phasephase
rates needed for equilibration of rates needed for equilibration of ΩΩ and kaons: and kaons:
ΔΔT = 5 MeV, T = 5 MeV, FFΩΩK K = 1.13 ,= 1.13 ,ττT T =8 fm=8 fm
(0.02-0.2)/fm(0.02-0.2)/fm
two –particle – scattering :two –particle – scattering :
A possible source of error : A possible source of error : temperature-dependent particle temperature-dependent particle massesmasses
Chiral order parameter Chiral order parameter σσ depends on T depends on T
chemical chemical freeze freeze outoutmeasuremeasuressT/m !T/m !
uncertainty in m(T)uncertainty in m(T)
uncertainty in critical uncertainty in critical temperaturetemperature
systematic uncertainty :systematic uncertainty :
ΔσΔσ//σσ==ΔΔTTcc/T/Tcc
ΔσΔσ is negativeis negative
conclusionconclusion
• experimental determination of experimental determination of critical temperature may be critical temperature may be more precise than lattice resultsmore precise than lattice results
• error estimate becomes crucialerror estimate becomes crucial
Thermal equilibration :Thermal equilibration : occupation numbers occupation numbers
Chiral symmetry restoration Chiral symmetry restoration at high temperature at high temperature
High THigh TSYM SYM <<φφ>=0>=0
Low TLow T
SSBSSB
<<φφ>=>=φφ0 0
≠≠ 0 0at high T :at high T :
less orderless order
more symmetrymore symmetry
examples:examples:
magnets, crystalsmagnets, crystals
Order of the phase transition is Order of the phase transition is crucial ingredient for crucial ingredient for experiments experiments ( heavy ion collisions )( heavy ion collisions )and cosmological phase and cosmological phase transitiontransition
Order ofOrder ofthethephasephasetransitiontransition
First order phase First order phase transitiontransition
Simple one loop structure –Simple one loop structure –nevertheless (almost) exactnevertheless (almost) exact
Flow equation for Flow equation for average potentialaverage potential
Critical temperature , NCritical temperature , Nf f = 2= 2
J.Berges,D.Jungnickel,…
Lattice simulation