QCD – from the vacuum to high temperature an analytical approach an analytical approach

Analytical description of phase transition Needs model that can account simultaneously for the correct degrees of freedom below and above the transition temperature. 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. Partial aspects can be described by more limited models, e.g. chiral properties at small momenta.

Higgs picture of QCD “spontaneous breaking of color “ “spontaneous breaking of color “ in the QCD – vacuum in the QCD – vacuum octet condensate octet condensate for N f = 3 ( u,d,s ) for N f = 3 ( u,d,s ) C.Wetterich, Phys.Rev.D64,036003(2001),hep-ph/

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 !

Electroweak phase diagram

Masses of excitations (d=3) O.Philipsen,M.Teper,H.Wittig ‘97 small M H large M H

Continuity

Higgs phase and confinement can be equivalent – then simply two different descriptions (pictures) of the same physical situation 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 - - known for QCD : mesons + baryons -

Spontaneous breaking of color Condensate of colored scalar field Condensate of colored scalar field Equivalence of Higgs and confinement description in real (N f =3) QCD vacuum Equivalence of Higgs and confinement description in real (N f =3) QCD vacuum Gauge symmetries not spontaneously broken in formal sense ( only for fixed gauge ) Gauge symmetries not spontaneously broken in formal sense ( only for fixed gauge ) Similar situation as in electroweak theory Similar situation as in electroweak theory No “fundamental” scalars No “fundamental” scalars Symmetry breaking by quark-antiquark- condensate Symmetry breaking by quark-antiquark- condensate

Analogy between weak and strong interactions

Quark –antiquark condensate

Octet condensate ≠ 0 : ≠ 0 : “Spontaneous breaking of color” “Spontaneous breaking of color” Higgs mechanism Higgs mechanism Massive Gluons – all masses equal Massive Gluons – all masses equal Eight octets have vev Eight octets have vev Infrared regulator for QCD Infrared regulator for QCD

Electric charge ≠ 0 : ≠ 0 : Spontaneous breaking of electromagnetic U(1) symmetry Spontaneous breaking of electromagnetic U(1) symmetry (some components of octet carry electric charge – similar to Higgs mechanism for hypercharge in electroweak theory) (some components of octet carry electric charge – similar to Higgs mechanism for hypercharge in electroweak theory) Combined U(1) symmetry survives Combined U(1) symmetry survives (cf. Q=I 3 + ½ Y in e.w. standard model) (cf. Q=I 3 + ½ Y in e.w. standard model)

Electric charge of “quarks”

Flavor symmetry for equal quark masses : for equal quark masses : octet preserves global SU(3)-symmetry octet 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 of All particles fall into representations of the “eightfold way” the “eightfold way” quarks : 8 + 1, gluons : 8 quarks : 8 + 1, gluons : 8

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. t’Hooft ’80 S.Dimopoulos,S.Raby,L.Susskind ’80 T.Matsumoto ’80 B.Iijima,R.Jaffe ’81 M.Yasue ’90 M.Alford,K.Rajagopal,F.Wilczek ’99 T.Schaefer,F.Wilczek ‘99

Color-flavor-locking Chiral symmetry breaking : SU(3) L x SU(3) R SU(3) V SU(3) L x SU(3) R SU(3) V Color symmetry breaking : SU(3) c x SU(3) V SU(3) diagonal SU(3) c x SU(3) V SU(3) diagonal Quarks : 3 x Gluons : 8 x 1 8 Similar to high density QCD : Alford,Rajagopal,Wilczek ; Schaefer,Wilczek Alford,Rajagopal,Wilczek ; Schaefer,Wilczek color~flavor _

Octet condensate Color symmetry breaking : SU(3) c x SU(3) V SU(3) diagonal SU(3) c x SU(3) V SU(3) diagonal color~flavor 8 x … :

Quarks and gluons carry the observed quantum numbers of isospin and strangeness of the baryon and vector meson octets ! They are integer charged!

Duality

Quantum numbers match ! Of course, there are many more excitations (resonances ). Strong interactions bound states

Higgs description seems possible - is it simple ?

Effective low energy model for QCD Composite scalars Composite scalars ( quark-antiquark- bound states ) ( quark-antiquark- bound states ) Gauge invariance Gauge invariance Approximation: Approximation: renormalizable interactions renormalizable interactions for QCD with scalars for QCD with scalars Comparison with observation? Comparison with observation?

Low energy effective action γ=φ+χγ=φ+χ

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

New scalar interactions Gauge covariant kinetic term Gauge covariant kinetic term Effective potential Effective potential Yukawa coupling to quarks Yukawa coupling to quarks

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

Effective octet potential simple instanton computation χ 0 = 150 MeV M ρ = 850 MeV Chiral anomaly ! U χ

Masses of physical particles determine three phenomenological parameters

Phenomenological parameters 5 undetermined parameters 5 undetermined parameters predictions

Chiral perturbation theory + all predictions of chiral perturbation theory + determination of parameters

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

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

Gauge invariance Higgs picture is a guide for ideas and a way to compute gauge invariant quantities at the end 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 Intuition can be misleading for certain questions Effective action, U( φ,χ ) : gauge invariant Effective action, U( φ,χ ) : 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 χ=0 B) minimum not for χ=0 ( for appropriate gauge and normalization of χ ) ( for appropriate gauge and normalization of χ )

Nonlinear fields : π,K,η, η’

Nonlinear fields : diquark cloud The product Wv transforms as an antidiquark The product Wv transforms as an antidiquark B=-2/3 B=-2/3 v : color triplet v : color triplet

How quarks get dressed as baryons

Gauge bosons/vector mesons

All fields except v are gauge singlets

Effective action in terms of physical fields

linear fields nonlinear fields Insert expressions for ψ,A,χ,φ

Nonlinear local symmetry Has been investigated since long ago in the context of chiral theories, describes ρ - bosons Here : Here : Not postulated Not postulated Consequence of local color symmetry + “SSB” Consequence of local color symmetry + “SSB” Gauge bosons = gluons = ρ - bosons Gauge bosons = gluons = ρ - bosons Predictions correct ! Predictions correct !

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) local U(3) reparameterization symmetry infinitesimal transformation

Baryons

Pion nucleon coupling Two more successful predictions F,D are not fixed by chiral symmetry !

Pseudoscalar mesons Kinetic term for pseudoscalar mesons as in chiral perturbation theory Kinetic term for pseudoscalar mesons as in chiral perturbation theory meson decay constant

Vector mesons

Electromagnetic interactions include by include by covariant covariant derivative derivative

ρ - couplings

prediction :experiment : Vector dominance is realized by Higgs picture of QCD

Connection to gauge invariant formulation for linear fields Vector channel : use singlet fields Vector channel : use singlet fields (in addition to A,φ,χ ; fermions omitted here ) (in addition to A,φ,χ ; fermions omitted here ) Solve field equations for colored bosons Solve field equations for colored bosons Γ[φ,ρ] contains directly the information for gauge invariant correlation functions Γ[φ,ρ] contains directly the information for gauge invariant correlation functions

A - ρ mixing Insert solution A[ρ] Mixing produces mass shift

Conclusion (2) Phenomenology Phenomenology works well for works well for simple effective action simple effective action

Chiral phase transition at high temperature High temperature phase transition in QCD : Melting of octet condensate Melting of octet condensate Lattice simulations : Deconfinement temperature = critical temperature for restoration of chiral symmetry Why ? Why ?

Simple explanation : Simple explanation :

Temperature dependent effective potential

Temperature corrections to effective octet potential

Vacuum effective potential ( T=0 )

Interesting relation between T c and η’ properties

A simple mean field calculation

Conclusions ( 3 ) Coherent picture for phase diagram of QCD Coherent picture for phase diagram of QCD is emerging is emerging Gluon meson duality allows for analytical calculations Gluon meson duality allows for analytical calculations Quark-baryon duality : Quark-baryon duality : Direct contact to quantities of nuclear physics Direct contact to quantities of nuclear physics

Questions ?

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

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

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