QCD – from the vacuum toQCD – from the vacuum to high temperature high temperature

an analytical approachan analytical approach

3 lectures3 lectures

•Condensates and phase diagramCondensates and phase diagram

•Higgs picture of QCDHiggs picture of QCD

•Functional renormalization groupFunctional renormalization group

Condensates Condensates andand

Phase DiagramPhase Diagram

Action and functional integralAction and functional integral

Generating functionalsGenerating functionals

Chiral symmetryChiral symmetry

Condensates and symmetry breakingCondensates and symmetry breaking

Phase transitionPhase transition

Has THas Tc c been measured?been measured?

Understanding the phase Understanding the phase diagramdiagram

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 …

Why analytical approach ?Why analytical approach ?

Complementary to numerical Complementary to numerical investigationsinvestigations

(lattice – QCD )(lattice – QCD )

Many questions cannot be addressed Many questions cannot be addressed by lattice - QCDby lattice - QCD

QCDQCD

Quantum numbersQuantum numbers

quarkquark quarkquark gluongluon mesonmeson baryobaryonn

particlparticlee

uu dd gg ππ++ pp

electrielectric c chargchargee

2/32/3 -1/3-1/3 00 11 11

isospiisospinn

1/21/2 -1/2-1/2 00 11 1/21/2

colorcolor 33 33 88 11 11

Light particlesLight particles

mesonsmesons

• pseudoscalar octet: pseudoscalar octet: ((ππ++,,ππ--,,ππ00,K,K++,K,K--,K,K00,K,K00,,ηη))

• vector meson octet + …vector meson octet + …

baryonsbaryons

• octet : (p,n,octet : (p,n,ΛΛ,,ΣΣ++,,ΣΣ--,,ΣΣ00,,ΞΞ00,,ΞΞ--))

• decuplet + …decuplet + …

--

quarks and gluons quarks and gluons at high momenta at high momenta

• Visible as jets at high virtual Visible as jets at high virtual momentamomenta

• Perturbation theory validPerturbation theory valid

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?

confinementconfinement

ActionAction

++

QCD is microscopic theory ofQCD is microscopic theory ofstrong interactionsstrong interactions

Local gauge symmetryLocal gauge symmetry

Running coupling : QCDRunning coupling : QCD

effective gauge coupling effective gauge coupling

depends ondepends on

momentum scale momentum scale μμ

Generating Generating functionals,functionals, Effective action Effective action

Partition function and Partition function and functional integral functional integral

Integration over all configurations of gauge field A and quark fields Integration over all configurations of gauge field A and quark fields ψψ

Fermion fields are anticommuting Grassmann variablesFermion fields are anticommuting Grassmann variables

This is the difficult part ! Needs regularisation !This is the difficult part ! Needs regularisation !

Lattice QCD ; gauge fixing,ghosts,Slavnov Taylor identities …Lattice QCD ; gauge fixing,ghosts,Slavnov Taylor identities …

Correlation functionsCorrelation functions

Only gauge invariant correlation functions can differ Only gauge invariant correlation functions can differ from zero.from zero.

Local gauge symmetries are not spontaneously broken Local gauge symmetries are not spontaneously broken ( exact proof )( exact proof )• Scalar correlation functionsScalar correlation functions

describes propagation of (pseudo)scalar describes propagation of (pseudo)scalar mesonsmesons

in suitable channelin suitable channel

• Vector meson correlation functionVector meson correlation function

Generating functionalsGenerating functionals

• Add to the action a source termAdd to the action a source term

• Partition function becomes a functional of the Partition function becomes a functional of the sources, Z[j]. Can be evaluated for arbitrary j(x)sources, Z[j]. Can be evaluated for arbitrary j(x)

Generating functional and correlation Generating functional and correlation functionsfunctions

Concentrate on scalars – Concentrate on scalars –

other channels can be treated in complete other channels can be treated in complete analogyanalogy

• Order parameterOrder parameter

• Correlation functionCorrelation function

Generating functional for Generating functional for connected Green’s functionsconnected Green’s functions

W[ j ]=ln Z[ j ]W[ j ]=ln Z[ j ]

GGcc : propagator or : propagator or connected two point functionconnected two point function

Effective actionEffective action

Effective action is defined by Legendre Effective action is defined by Legendre transformtransform

Matrix of second functional derivatives Matrix of second functional derivatives ΓΓ(2)(2) is is the inverse propagatorthe inverse propagator

Physical sourcesPhysical sources

With this convention for physical sources the With this convention for physical sources the quark mass term is not included in S quark mass term is not included in S anymore.anymore.

S: action for NS: action for Nf f massless quarksmassless quarks

ΓΓ has the full chiral symmetry has the full chiral symmetry

““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

SymmetriesSymmetries

• If functional measure preserves the If functional measure preserves the symmetries of S :symmetries of S :

ΓΓ has the same symmetries as S has the same symmetries as S

• If not :If not : anomalyanomaly

Examples for symmetries of Examples for symmetries of ΓΓ : :

- gauge symmetry, - gauge symmetry,

- global chiral flavor symmetry SU(N- global chiral flavor symmetry SU(Nff)xSU(N)xSU(Nff))

Example for anomaly: Example for anomaly: global axial U(1)global axial U(1)

Chiral symmetry Chiral symmetry breakingbreaking

Chiral symmetryChiral symmetry

For For masslessmassless quarks the action of QCD is quarks the action of QCD isinvariant under independent phase invariant under independent phase

rotationsrotationsacting on left handed and right handedacting on left handed and right handedquarks.quarks.

SU(NSU(Nff))L L x SU(Nx SU(Nff))RR

Quark massesQuark masses

• Quark mass terms mix left handed and right Quark mass terms mix left handed and right handed quarkshanded quarks

• They break the axial symmetryThey break the axial symmetry

• Equal quark masses leave vectorlike Equal quark masses leave vectorlike “diagonal” SU(N) unbroken.“diagonal” SU(N) unbroken.

• Quark mass terms can be treated as Quark mass terms can be treated as sources for scalar quark-antiquark bilinears.sources for scalar quark-antiquark bilinears.

Effective potentialEffective potential

Evaluate effective action Evaluate effective action

for homogenous field for homogenous field φφ

Spontaneous symmetry Spontaneous symmetry breakingbreaking

SYM SYM

<<φφ>=0>=0

SSBSSB

<<φφ>=>=φφ0 0

≠≠ 0 0

PionsPions

Two-flavor QCD , Two-flavor QCD , σσ : 2x2 matrix : 2x2 matrix

Effective potentialEffective potential

Spontaneous chiral symmetry Spontaneous chiral symmetry breakingbreaking

For sufficiently small m² and positive For sufficiently small m² and positive λλ1,21,2 : :

( even for vanishing quark masses )( even for vanishing quark masses )

Goldstone bosonsGoldstone bosons

Flat directions in potential, Flat directions in potential,

dictated by symmetry breakingdictated by symmetry breaking

massless (pseudo)scalar fields : massless (pseudo)scalar fields : pionspions

Presence of quark mass term shifts Presence of quark mass term shifts minimum and gives mass to pionsminimum and gives mass to pions

Nonlinear Nonlinear σσ - model - model

Nonlinear descriptionNonlinear description(neglects the scalar excitations beyond (neglects the scalar excitations beyond ππ,,ηη))

Kinetic termKinetic term

Chiral perturbation theoryChiral perturbation theory

Meson fluctuationsMeson fluctuations

• Insert factor unityInsert factor unity

• New action and functional integral involves also New action and functional integral involves also σσ - field- field

Hubbard,StratonoviHubbard,Stratonovichch

σσ - interactions - interactions

• Four fermion interaction can cancel Four fermion interaction can cancel corresponding piece induced by loops in corresponding piece induced by loops in QCDQCD

• Typical formTypical form

• Mass and kinetic term for Mass and kinetic term for σσ • Yukawa interaction between Yukawa interaction between σσ and quarks and quarks • Source now for Source now for σσ

Connection with quark Connection with quark bilinearsbilinears

Effective action with scalar Effective action with scalar fluctuationsfluctuations

QCD phase QCD phase transitiontransition

Phase diagramPhase diagram

R.PisarskiR.Pisarski

<<φφ>=>=φφ00 ≠≠ 0 0

<<φφ>=>=00

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

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

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

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 !!

PressurePressure

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.

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 ?

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

Second order phase Second order phase transitiontransition

First order phase First order phase transitiontransition

Matsubara formalismMatsubara formalism

• Only change for T ≠ 0 :Only change for T ≠ 0 :

• Euclidean time on torus with Euclidean time on torus with circumference circumference ββ =1/T =1/T

Matsubara formalism for Matsubara formalism for fermionsfermions

Thermodynamic potentials Thermodynamic potentials and observables and observables

Evaluate effective potential U at minimumEvaluate effective potential U at minimum

• free energyfree energy

• pressurepressure

• energy densityenergy density

vanishing chemical potential and sourcesvanishing chemical potential and sources

ThermodynamicsThermodynamics

Effective action , when evaluated on torus Effective action , when evaluated on torus for euclidean time with circumference for euclidean time with circumference 1/T, (and in presence of chemical 1/T, (and in presence of chemical potential ) contains full information about potential ) contains full information about thermodynamic behavior of QCD.thermodynamic behavior of QCD.

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

Heavy ion collisionHeavy ion collision

Hadron abundanciesHadron abundancies

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

( proporional to very high power of ( proporional to very high power of density )density )

Energy densityEnergy density

• Lattice Lattice simulationssimulations

Karsch et alKarsch et al

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 densitypion density

P.Braun-Munzinger,J.Stachel,CWP.Braun-Munzinger,J.Stachel,CW

TTch ch ≈ T ≈ Tcc

Phase diagramPhase diagram

hadrons hadrons

quarks and gluonsquarks and gluons

Running coupling : QEDRunning coupling : QED

Chiral perturbation theoryChiral perturbation theory

Matsubara formalismMatsubara formalism