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