QCD from the vacuum to high temperature

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QCD from the vacuum to high temperature

• an analytical approach

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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.

3
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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Many pictures

• of the QCD vacuum have been proposed
• monopoles, instantons, vortices, spaghetti vacuum
  
• in principle, no contradiction there may be
  more than one valid picture
• most proposals say essentially nothing about the
  low mass excitations in real QCD, i.e mesons and
  baryons
• different for Higgs picture !

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Electroweak phase diagram
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Masses of excitations (d3)
small MH
large MH
O.Philipsen,M.Teper,H.Wittig 97
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Continuity
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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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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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Analogy between weak and strong interactions
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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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Electric charge

• lt octet gt ? 0
• Spontaneous breaking of electromagnetic U(1)
  symmetry
• (some components of octet carry electric charge
  similar to Higgs mechanism for hypercharge in
  electroweak theory)
• Combined U(1) symmetry survives
• (cf. QI3 ½ Y in e.w. standard model)

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Electric charge of quarks
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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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Related earlier ideas K.Bardakci,M.Halpern
I.Bars 72 R.Mohapatra,J.Pati,A.Salam
76 A.De Rujula,R.Giles,R.Jaffe
78 T.Banks,E.Rabinovici 79 E.Fradkin,S.Shenker
79 G. tHooft 80 S.Dimopoulos,S.Raby,L.Suss
kind 80 T.Matsumoto 80 B.Iijima,R.Jaffe
81 M.Yasue 90 M.Alford,K.Rajagopal,F.Wilczek
99 T.Schaefer,F.Wilczek 99
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Color-flavor-locking

• Chiral symmetry breaking
• SU(3)L x SU(3)R SU(3)V
• Color symmetry breaking
• SU(3)c x SU(3)V
  SU(3)diagonal
• Quarks 3 x 3 8 1
• Gluons 8 x 1 8
• Similar to high density QCD
• Alford,Rajagopal,Wilczek
  Schaefer,Wilczek

_
color
flavor
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Octet condensate

• Color symmetry breaking
• SU(3)c x SU(3)V
  SU(3)diagonal

8 x 8 1
lt ? gt
color
flavor
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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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Duality
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Quantum numbers match !

• Of course , there are many more excitations
• (resonances ).
• Strong interactions bound states

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Higgs description seems possible - is it simple ?
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Effective low energy model for QCD

• Composite scalars
• ( quark-antiquark- bound states )
• Gauge invariance
• Approximation
• renormalizable interactions
• for QCD with scalars
• Comparison with observation?

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Low energy effective action
?f?
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Simplicity

• This simple effective action will yield the
  masses and couplings of the baryons,
  pseudoscalars and vector mesons, ( including
  electromagnetic couplings by covariant
  derivatives ) !
• ( five parameters , to be later determined by
  QCD )

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New scalar interactions

• Gauge covariant kinetic term
• Effective potential
• Yukawa coupling to quarks

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Calculability

• Remember no fundamental scalars
• Effective couplings should be calculable from QCD
  i.e. gauge coupling or confinement scale

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Effective octet potential
simple instanton computation
?0 150 MeV
U
M? 850 MeV
?
Chiral anomaly !
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Masses of physical particles
determine three phenomenological parameters
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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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First conclusions

• Spontaneous color symmetry breaking plausible in
  QCD
• QCD - computation of effective vector mass needed
• Simple effective action can account for mass
  spectrum of light baryons and mesons as well as
  their couplings
• Gluon - Meson duality
• Quark - Baryon duality

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Nonlinear formulation

• Use of nonlinear fields makes physical content of
  the effective action more transparent.
• Similar to nonlinear fields for pions
• Selection of nonlinear fields follows symmetry
  content of the theory

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Gauge invariance

• Higgs picture is a guide for ideas and a way to
  compute gauge invariant quantities at the end
• Intuition can be misleading for certain questions
  
• Effective action, U( f,? ) gauge invariant
• Nonlinear fields gauge singlets
• Only assumptions
• A) minimum of U preserves global SU(3)
• B) minimum not for ?0
• ( for appropriate gauge and normalization of
  ? )

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Nonlinear fields p,K,?, ?
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Nonlinear fields diquark cloud

• The product Wv transforms as an antidiquark
• B-2/3
• v color triplet

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How quarks get dressed as baryons
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Gauge bosons/vector mesons
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All fields except v are gauge singlets
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Effective action in terms of physical fields
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Effective action in terms of physical fields
linear fields
nonlinear fields
Insert expressions for ?,A,?,f
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Nonlinear local symmetry

• Has been investigated since long ago in the
• context of chiral theories, describes ? - bosons
• Here
• Not postulated
• Consequence of local color symmetry SSB
• Gauge bosons gluons ? - bosons
• Predictions correct !

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Reparameterization symmetry

• Decomposition into nonlinear
• fields is not unique. E.g.
• N can be multiplied by unitary
• transformation from left, and
• W from right.
• local U(3)
• reparameterization symmetry

infinitesimal transformation
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Baryons
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Pion nucleon coupling
Two more successful predictions
F,D are not fixed by chiral symmetry !
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Pseudoscalar mesons

• meson decay constant

• Kinetic term for pseudoscalar mesons as in
  chiral perturbation theory

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Vector mesons
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Electromagnetic interactions

• include by
• covariant
• derivative

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? - couplings
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? - couplings
prediction
experiment
Vector dominance is realized by Higgs picture of
QCD
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Connection to gauge invariant formulation for
linear fields

• Vector channel use singlet fields
• (in addition to A,f,? fermions
  omitted here )
• Solve field equations for colored bosons
• Gf,? contains directly the information for
  gauge invariant correlation functions

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A - ? mixing
Insert solution A?
Mixing produces mass shift
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Conclusion (2)

• Phenomenology
• works well for
• simple effective action

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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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Temperature dependent effective potential
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Temperature corrections to effective octet
potential
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Vacuum effective potential ( T0 )
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Interesting relation between Tc and ? properties
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A simple mean field calculation
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Conclusions ( 3 )

• Coherent picture for phase diagram of QCD
• is emerging
• Gluon meson duality allows for analytical
  calculations
• Quark-baryon duality
• Direct contact to quantities of nuclear
  physics

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Questions ?
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Lattice tests

• a) Continuity
• Add fundamental scalar octets and start in
  perturbative Higgs phase
• ( large negative mass term ).
• Remove scalars continuously by increasing the
  mass term to large positive values
• Phase transition or analytical crossover ?

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Challenges

• Instanton computation of U(f,?)
• (improve by nonperturbative flow equation )
• Check continuity between Higgs and confinement
  description by lattice simulation
• Explicit construction of a local diquark operator
  with transformation Wv
• (nonvanishing expectation value )

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