qcd – from the vacuum to high temperature an analytical approach an analytical approach
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QCD – from the QCD – from the vacuum to high vacuum to high
temperaturetemperature
an analytical an analytical approachapproach
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.
Higgs picture of Higgs picture of QCDQCD
““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
Many pictures …Many pictures …
… … of the QCD vacuum have been proposedof the QCD vacuum have been proposed
monopoles, instantons, vortices, spaghetti vacuum monopoles, instantons, vortices, spaghetti vacuum ……
in principle, no contradiction – there may be more in principle, no contradiction – there may be more than one valid picturethan one valid picture
most proposals say essentially nothing about the most proposals say essentially nothing about the low mass excitations in low mass excitations in realreal QCD, i.e mesons QCD, i.e mesons and baryonsand baryons
different for Higgs picture !different for Higgs picture !
Masses of excitations Masses of excitations (d=3)(d=3)
O.Philipsen,M.Teper,H.Wittig ‘97
small MH large MH
Higgs phase and Higgs phase and confinementconfinement
can 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 -
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 Gauge symmetries not spontaneously
broken in formal sense ( only for fixed broken in formal sense ( only for fixed gauge ) 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
Octet condensateOctet condensate
< octet > ≠ 0 :< octet > ≠ 0 : ““Spontaneous breaking of color”Spontaneous breaking of color” Higgs mechanismHiggs mechanism Massive Gluons – all masses Massive Gluons – all masses
equalequal Eight octets have vevEight octets have vev Infrared regulator for QCDInfrared regulator for QCD
Electric chargeElectric charge
< octet > ≠ 0 :< octet > ≠ 0 : Spontaneous breaking of Spontaneous breaking of
electromagnetic U(1) symmetry electromagnetic U(1) symmetry (some components of octet carry (some components of octet carry
electric charge – similar to Higgs electric charge – similar to Higgs mechanism for hypercharge in mechanism for hypercharge in electroweak theory)electroweak theory)
Combined U(1) symmetry survives Combined U(1) symmetry survives (cf. Q=I(cf. Q=I33 + ½ Y in e.w. standard model) + ½ Y in e.w. standard model)
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
Related earlier ideas:
K.Bardakci,M.Halpern; I.Bars ’72R.Mohapatra,J.Pati,A.Salam ’76A.De Rujula,R.Giles,R.Jaffe ‘78T.Banks,E.Rabinovici ’79E.Fradkin,S.Shenker ’79G. t’Hooft ’80S.Dimopoulos,S.Raby,L.Susskind ’80T.Matsumoto ’80B.Iijima,R.Jaffe ’81M.Yasue ’90M.Alford,K.Rajagopal,F.Wilczek ’99T.Schaefer,F.Wilczek ‘99
Color-flavor-lockingColor-flavor-locking
Chiral symmetry breaking :Chiral symmetry breaking :
SU(3)SU(3)LL x SU(3) x SU(3)RR SU(3) SU(3)VV
Color symmetry breaking :Color symmetry breaking :
SU(3)SU(3)cc x SU(3) x SU(3)VV SU(3) SU(3)diagonaldiagonal
Quarks :Quarks : 3 x 3 8 + 13 x 3 8 + 1Gluons :Gluons : 8 x 1 8 8 x 1 8
Similar to high density QCD :Similar to high density QCD :
Alford,Rajagopal,Wilczek ; Schaefer,WilczekAlford,Rajagopal,Wilczek ; Schaefer,Wilczek
color~ flavor
_
Octet condensateOctet condensate
Color symmetry breaking :Color symmetry breaking :
SU(3)SU(3)cc x SU(3) x SU(3)VV SU(3)SU(3)diagonaldiagonal
color~ flavor
8 x 8 1 + …< χ > :
Quarks and gluons carry Quarks and gluons carry the observed quantum the observed quantum numbers of isospin and numbers of isospin and
strangenessstrangenessof the baryon and of the baryon and
vector meson octets !vector meson octets !
They are integer charged!They are integer charged!
Quantum numbers match Quantum numbers match !!
Of course , there are many more Of course , there are many more excitations excitations
(resonances ).(resonances ).
Strong interactions bound statesStrong interactions bound states
EffectiveEffective low energy model low energy model for QCDfor QCD
Composite scalarsComposite scalars
( quark-antiquark- bound states )( quark-antiquark- bound states ) Gauge invarianceGauge invariance ApproximationApproximation::
renormalizable interactions renormalizable interactions
for QCD with scalarsfor QCD with scalars Comparison with observation?Comparison with observation?
SimplicitySimplicity
This simple effective action will yield This simple effective action will yield the the massesmasses and and couplingscouplings of the of the baryons, pseudoscalarsbaryons, pseudoscalars and and vector vector mesonsmesons, ( including electromagnetic , ( including electromagnetic couplings by covariant derivatives ) !couplings by covariant derivatives ) !
( five parameters , to be later determined by ( five parameters , to be later determined by QCD )QCD )
New scalar interactionsNew scalar interactions
Gauge covariant kinetic termGauge covariant kinetic term Effective potentialEffective potential Yukawa coupling to quarksYukawa coupling to quarks
CalculabilityCalculability
Remember : no fundamental Remember : no fundamental scalarsscalars
Effective couplings should be Effective couplings should be calculable from QCD – i.e. gauge calculable from QCD – i.e. gauge coupling or confinement scalecoupling or confinement scale
Effective octet potentialEffective octet potential
simple instanton computation
χ0 = 150 MeV
Mρ = 850 MeV
Chiral anomaly !
U
χ
Masses of physical Masses of physical particlesparticles
determine three phenomenological parameters
Phenomenological Phenomenological parametersparameters
5 undetermined 5 undetermined parametersparameters
predictionspredictions
Chiral perturbation Chiral perturbation theorytheory
+ all predictions of chiral perturbation + all predictions of chiral perturbation theorytheory
+ determination of parameters+ determination of parameters
First conclusionsFirst conclusions
Spontaneous color symmetry Spontaneous color symmetry breaking plausible in QCDbreaking plausible in QCD
QCD - computation of effective vector QCD - computation of effective vector mass neededmass needed
Simple effective action can account Simple effective action can account for mass spectrum of light baryons for mass spectrum of light baryons and mesons as well as their couplingsand mesons as well as their couplings
Gluon - Meson dualityGluon - Meson duality Quark - Baryon dualityQuark - Baryon duality
Nonlinear formulationNonlinear formulation
Use of nonlinear fields makes Use of nonlinear fields makes physical content of the effective physical content of the effective action more transparent.action more transparent.
Similar to nonlinear fields for pionsSimilar to nonlinear fields for pions Selection of nonlinear fields follows Selection of nonlinear fields follows
symmetry content of the theorysymmetry content of the theory
Gauge invarianceGauge invariance Higgs picture is a guide for ideas and a way to Higgs picture is a guide for ideas and a way to
compute gauge invariant quantities at the endcompute gauge invariant quantities at the end Intuition can be misleading for certain Intuition can be misleading for certain
questions questions Effective action, U( Effective action, U( φφ,,χχ ) : gauge invariant ) : gauge invariant Nonlinear fields : gauge singlets Nonlinear fields : gauge singlets
Only assumptions :Only assumptions : A) minimum of U preserves global SU(3)A) minimum of U preserves global SU(3) B) minimum not for B) minimum not for χχ=0=0 ( for appropriate gauge and normalization of ( for appropriate gauge and normalization of
χχ ) )
Nonlinear fields : Nonlinear fields : diquark cloud diquark cloud
The product The product WW••vv transforms as transforms as an antidiquarkan antidiquark
B=-2/3B=-2/3 vv : color triplet : color triplet
Effective action in terms of Effective action in terms of physical fieldsphysical fields
linear fields nonlinear fields
Insert expressions for ψ,A,χ,φ
Nonlinear local Nonlinear local symmetrysymmetry
Has been investigated since long ago in Has been investigated since long ago in thethe
context of chiral theories, describes context of chiral theories, describes ρρ - - bosonsbosons
Here :Here : Not postulatedNot postulated Consequence of local color Consequence of local color
symmetry + “SSB”symmetry + “SSB” Gauge bosons = gluons = Gauge bosons = gluons = ρρ - bosons - bosons
Predictions correct !Predictions correct !
Reparameterization Reparameterization symmetrysymmetry
Decomposition into nonlinearDecomposition into nonlinear
fields is not unique. E.g.fields is not unique. E.g.
N can be multiplied by N can be multiplied by unitaryunitary
transformation from left, andtransformation from left, and
W from right.W from right.
local U(3)local U(3)
reparameterization symmetryreparameterization symmetry
infinitesimal transformation
Pion nucleon couplingPion nucleon coupling
Two moresuccessfulpredictions
F,D are not fixed by chiral symmetry !
Pseudoscalar mesonsPseudoscalar mesons
Kinetic term for Kinetic term for pseudoscalar pseudoscalar mesons as in chiral mesons as in chiral perturbation perturbation theorytheory
meson decay meson decay constantconstant
Electromagnetic Electromagnetic interactionsinteractions
include byinclude by
covariant covariant
derivativederivative
ρρ - couplings - couplings
prediction :experiment :
Vector dominance is realized by Higgs picture of QCD
Connection to gauge Connection to gauge invariant formulation for invariant formulation for
linear fieldslinear fields Vector channel : use singlet fieldsVector channel : use singlet fields
(in addition to A,(in addition to A,φφ,,χχ ; fermions omitted here ) ; fermions omitted here )
Solve field equations for colored bosonsSolve field equations for colored bosons
ΓΓ[[φφ,,ρρ] contains directly the information ] contains directly the information for gauge invariant correlation functionsfor gauge invariant correlation functions
Conclusion (2)Conclusion (2)
Phenomenology Phenomenology
works well for works well for
simple effective actionsimple effective action
Chiral phase transition Chiral phase transition at high temperatureat high temperature
High temperature phase transition in High temperature phase transition in QCD :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 ?
Temperature corrections to Temperature corrections to effective octet potentialeffective octet potential
Conclusions ( 3 )Conclusions ( 3 )
Coherent picture for phase diagram of Coherent picture for phase diagram of QCD QCD
is emergingis emerging Gluon meson duality allows for Gluon meson duality allows for
analytical calculationsanalytical calculations Quark-baryon duality :Quark-baryon duality :
Direct contact to quantities of nuclear Direct contact to quantities of nuclear physicsphysics
Lattice testsLattice tests
a) Continuitya) Continuity
- Add “fundamental” scalar octets and start Add “fundamental” scalar octets and start in perturbative Higgs phase in perturbative Higgs phase
( large negative mass term ).( large negative mass term ).- Remove scalars continuously by Remove scalars continuously by
increasing the mass term to large positive increasing the mass term to large positive valuesvalues
Phase transition or analytical crossover ?Phase transition or analytical crossover ?
ChallengesChallenges
Instanton computation of U(Instanton computation of U(φφ,,χχ))
(improve by nonperturbative flow (improve by nonperturbative flow equation )equation )
Check continuity between Higgs and Check continuity between Higgs and confinement description by lattice confinement description by lattice simulationsimulation
Explicit construction of a local diquark Explicit construction of a local diquark operator with transformation Wv operator with transformation Wv
(nonvanishing expectation value )(nonvanishing expectation value )