QCD Green Functions:

Transcription

1 QCD Green Functions: What their Infrared Behaviour tells us about Confinement! Reinhard Alkofer FWF-funded Doctoral Program Hadrons in Vacuum, Nuclei and Stars Institute of Physics Theoretical Physics University Graz Oct. 17, 2006 R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

2 Outline 1 Theories of Confinement 2 Basic Concepts Ghosts & BRST Gribov horizon & Zwanziger condition Kugo Ojima confinement criterion QCD Green functions & Positivity 3 Infrared Exponents for Gluons and Ghosts An exact inequality Infrared Expansion Scheme 4 Running Coupling 5 Positivity violation for the gluon propagator 6 Dynamically induced scalar quark confinement 7 Partial gluon confinement at any T 8 Quark confinement in the Coulomb gauge 9 Summary R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

3 Confinement Alive Theories of Confinement: see e.g. R.A. and J. Greensite, Quark Confinement: The Hard Problem of Hadron Physics, Hadron Physics Focus Issue of J. Phys. G chromomagnetic monopoles t Hooft, digiacomo,... center vortices Greensite, Olejnik,... Coulomb confinement Gribov, Zwanziger,... Landau gauge Green Functions Smekal, Fischer,... AdS 5 / QCD correspondence Maldacena, Brodsky,... R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

4 Confinement monop. vort. Coulomb Green Funct. AdS 5 Casimir scaling x x x?? N-ality? x?? x Gribov Zwanziger? x x x? Kugo Ojima (BRST quartet)?? x x? Positivity violation?? x x? Conformal YM sector?? x x x DχSB? x x x? U A (1) x x x x? R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

5 Covariant Gauge Theory Covariant Gauge theory: Unphysical degrees of freedom! QED: Physical states obey Lorentz condition. µ A µ Ψ = 0 Two physical massless photons. (Gupta Bleuler). Time-like photon (i.e. negative norm state!) cancels longitudinal photon in S matrix elements! Time like photon confines longitudinal photon! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

6 Covariant Gauge Theory Covariant Gauge theory: Unphysical degrees of freedom! QED: Physical states obey Lorentz condition. µ A µ Ψ = 0 Two physical massless photons. (Gupta Bleuler). Time-like photon (i.e. negative norm state!) cancels longitudinal photon in S matrix elements! Time like photon confines longitudinal photon! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

7 Covariant Gauge Theory Covariant Gauge theory: Unphysical degrees of freedom! QED: Physical states obey Lorentz condition. µ A µ Ψ = 0 Two physical massless photons. (Gupta Bleuler). Time-like photon (i.e. negative norm state!) cancels longitudinal photon in S matrix elements! Time like photon confines longitudinal photon! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

8 Covariant Gauge Theory Covariant Gauge theory: Unphysical degrees of freedom! QED: Physical states obey Lorentz condition. µ A µ Ψ = 0 Two physical massless photons. (Gupta Bleuler). Time-like photon (i.e. negative norm state!) cancels longitudinal photon in S matrix elements! Time like photon confines longitudinal photon! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

9 Covariant Gauge Theory In S-Matrix A A T 0 T L A A + = 0 A T A 0 A L A T = / 0 R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

10 Ghosts & BRST Quantization of QCD: Selfinteraction of gluons, cancelation process much more complicated! Faddeev Popov ghosts = anticomm. scalar fields. Ghosts are unphysical (anti-commuting scalar) Yang Mills degrees of freedom! Important in quantum fluct., but no associated particles! Global ghost field as gauge parameter : BRST symmetry of the gauge-fixed action! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

11 Ghosts & BRST Quantization of QCD: Selfinteraction of gluons, cancelation process much more complicated! Faddeev Popov ghosts = anticomm. scalar fields. Ghosts are unphysical (anti-commuting scalar) Yang Mills degrees of freedom! Important in quantum fluct., but no associated particles! Global ghost field as gauge parameter : BRST symmetry of the gauge-fixed action! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

12 Ghosts & BRST Quantization of QCD: Selfinteraction of gluons, cancelation process much more complicated! Faddeev Popov ghosts = anticomm. scalar fields. Ghosts are unphysical (anti-commuting scalar) Yang Mills degrees of freedom! Important in quantum fluct., but no associated particles! Global ghost field as gauge parameter : BRST symmetry of the gauge-fixed action! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

13 Ghosts & BRST Quantization of QCD: Selfinteraction of gluons, cancelation process much more complicated! Faddeev Popov ghosts = anticomm. scalar fields. Ghosts are unphysical (anti-commuting scalar) Yang Mills degrees of freedom! Important in quantum fluct., but no associated particles! Global ghost field as gauge parameter : BRST symmetry of the gauge-fixed action! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

14 Ghosts & BRST Symmetry of the gauge-fixed generating functional: δ B A a µ = Dµ ab c b λ, δ B q = igt a c a q λ, δ B c a = g 2 f abc c b c c λ, δ B c a = 1 ξ µa a µ λ, Becchi Rouet Stora & Tyutin (BRST), 1975 Parameter λ Grassmann algebra of the ghost fields λ carries ghost number N FP = 1 Via Noether theorem: BRST charge operator Q B generates ghost # graded algebra δ B Φ = {iq B,Φ} R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

15 Ghosts & BRST BRST algebra: Q 2 B = 0, [iq c, Q B ] = Q B, complete in indefinite metric state space V. generates ghost # graded δ B Φ = {iq B,Φ}. L GF = δ B ( c ( µ A µ + α 2 B)) BRST exact. Positive definite subspace V pos = Ker(Q B ) (i.e. all states ψ V with Q B ψ = 0) contains ImQ B (i.e. all states Q B φ ), c.f. exterior derivative in differential geometry. Hilbert space: cohomology H = KerQ B ImQ B V s BRST singlet longitudinal & timelike gluons, ghosts : BRST quartet (c.f. Gupta Bleuler mechanism in QED) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

16 Ghosts & BRST BRST algebra: Q 2 B = 0, [iq c, Q B ] = Q B, complete in indefinite metric state space V. generates ghost # graded δ B Φ = {iq B,Φ}. L GF = δ B ( c ( µ A µ + α 2 B)) BRST exact. Positive definite subspace V pos = Ker(Q B ) (i.e. all states ψ V with Q B ψ = 0) contains ImQ B (i.e. all states Q B φ ), c.f. exterior derivative in differential geometry. Hilbert space: cohomology H = KerQ B ImQ B V s BRST singlet longitudinal & timelike gluons, ghosts : BRST quartet (c.f. Gupta Bleuler mechanism in QED) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

17 Ghosts & BRST BRST algebra: Q 2 B = 0, [iq c, Q B ] = Q B, complete in indefinite metric state space V. generates ghost # graded δ B Φ = {iq B,Φ}. L GF = δ B ( c ( µ A µ + α 2 B)) BRST exact. Positive definite subspace V pos = Ker(Q B ) (i.e. all states ψ V with Q B ψ = 0) contains ImQ B (i.e. all states Q B φ ), c.f. exterior derivative in differential geometry. Hilbert space: cohomology H = KerQ B ImQ B V s BRST singlet longitudinal & timelike gluons, ghosts : BRST quartet (c.f. Gupta Bleuler mechanism in QED) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

18 Gribov horizon & Zwanziger condition Gauge fixing in YM theories never completely unique: A l A tr A tr [A] Ω Λ A=0 [A] Λ topologically non-trivial as complete config. space also is! Landau gauge: Γ = {A : A = 0} Minimal Landau gauge: Ω = {A : A 2 minimal} First Gribov region: Ω = {A : A = 0, D(A) 0} Fundam. Modular Region: Λ = {A : global extrema} R. Alkofer (Graz) NO GRIBOV QCD GreenCOPIES Functions Oct. 17, / 32

19 Gribov horizon & Zwanziger condition Landau gauge: Γ = {A : A = 0} Minimal Landau gauge: Ω = {A : A 2 minimal} First Gribov region: Ω = {A : A = 0, D(A) 0} Fundam. Modular Region: Λ = {A : global extrema} NO GRIBOV COPIES Relevant configuration space: Λ/SU(N c ) Gribov: Cut off integral at boundary Ω Zwanziger: Ambiguities resolved due to additional IR boundary condition on ghost prop. lim k 2 0 ( k 2 D Ghost (k 2 ) ) 1 = 0. R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

20 Gribov horizon & Zwanziger condition Landau gauge: Γ = {A : A = 0} Minimal Landau gauge: Ω = {A : A 2 minimal} First Gribov region: Ω = {A : A = 0, D(A) 0} Fundam. Modular Region: Λ = {A : global extrema} NO GRIBOV COPIES Relevant configuration space: Λ/SU(N c ) Gribov: Cut off integral at boundary Ω Zwanziger: Ambiguities resolved due to additional IR boundary condition on ghost prop. lim k 2 0 ( k 2 D Ghost (k 2 ) ) 1 = 0. R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

21 Kugo Ojima confinement criterion Physical states are BRST singlets! (BRST cohomology: Hilbert space H = Ker Q BRST Im Q BRST.) Time like and longitudinal gluons (BRST quartet) removed from asymptotic states as in QED, but: Transverse gluons and quarks also BRST quartets, i.e. confined, if ghost propagator is highly infrared singular! ( Kugo Ojima confinement criterion) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

22 Kugo Ojima confinement criterion Physical states are BRST singlets! (BRST cohomology: Hilbert space H = Ker Q BRST Im Q BRST.) Time like and longitudinal gluons (BRST quartet) removed from asymptotic states as in QED, but: Transverse gluons and quarks also BRST quartets, i.e. confined, if ghost propagator is highly infrared singular! ( Kugo Ojima confinement criterion) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

23 QCD Green functions & Positivity Assume: States with transverse gluons violate positivity! These states do not belong to KerQ B! BRST quartet: transverse gluon, gluon-ghost; gluon-antighost, 2-gluon = cancelation in all amplitudes (Feynman diagrams) if asymptotic states (external lines)! = Gluon Confinement as Unobservability of Gluons! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

24 QCD Green functions & Positivity Assume: States with transverse gluons violate positivity! These states do not belong to KerQ B! BRST quartet: transverse gluon, gluon-ghost; gluon-antighost, 2-gluon = cancelation in all amplitudes (Feynman diagrams) if asymptotic states (external lines)! = Gluon Confinement as Unobservability of Gluons! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

25 QCD Green functions & Positivity Assume: States with transverse gluons violate positivity! These states do not belong to KerQ B! BRST quartet: transverse gluon, gluon-ghost; gluon-antighost, 2-gluon = cancelation in all amplitudes (Feynman diagrams) if asymptotic states (external lines)! = Gluon Confinement as Unobservability of Gluons! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

26 QCD Green functions & Positivity Assume: States with transverse gluons violate positivity! These states do not belong to KerQ B! BRST quartet: transverse gluon, gluon-ghost; gluon-antighost, 2-gluon = cancelation in all amplitudes (Feynman diagrams) if asymptotic states (external lines)! = Gluon Confinement as Unobservability of Gluons! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

27 Infrared Exponents for Gluons and Ghosts -1 = -1-1/2-1/2-1/6-1/ = -1 Dyson-Schwinger equations for propagators -1 = -1 R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

28 Infrared Exponents for Gluons and Ghosts: Inequality P. Watson and R. A., Phys. Rev. Lett. 86 (2001) 5239 µ k=p-q Landau gauge Ghost-Gluon vertex : q p Not renormalized! ( Z 1 = 1) Bare for p µ 0! (Γ µ (q, p = 0, k) = 1) It cannot be singular for vanishing ghost momenta! (finite positive constant) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

29 Infrared Exponents for Gluons and Ghosts: Inequality Assume that Green s functions can be expanded in asymptotic series (e.g. G(p 2 ;µ 2 ) = ( ) n d p 2 δn) n µ 2 and analyse ghost equation -1 = -1 IR integral too complicated to treat analytically :-( UV integral does not contribute directly to IR ;-( Renorm. constant Z 3 is related to constant! ;-) R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

30 Infrared Exponents for Gluons and Ghosts: Inequality Results: IR behaviour of gluons and ghosts uniquely related! δ ghost 0 = 1 2 ǫgluon 0 Infrared fixed point for running coupling! IR suppressed gluon propagator! IR enhanced ghost propagator: 2 > ǫ gluon 0 > 0 1 < δ ghost 0 < 0 Kugo Ojima confinement criterion and Gribov Zwanziger horizon condition fulfilled! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

31 Infrared Exponents for Gluons and Ghosts: Inequality Note: No truncation! No specialized ansatz! One basic assumption: asymptotic expansions for Green s functions! Used: General structure of ghost DS equation, multiplicative renormalizibility, only finite renormalization of ghost-gluon vertex. Remark: non-renormalization = no UV infinity! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

32 Infrared Exponents for Gluons and Ghosts: Expansion Scheme R. A., C. S. Fischer, F. Llanes-Estrada, Phys. Lett. B611 (2005) 279. Apply asymptotic expansion to all primitively divergent Green functions: Example: DSE for 3-gluon-vertex Ghost propagator IR divergent Gluon propagator IR suppressed R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

33 Infrared Exponents for Gluons and Ghosts: Expansion Scheme R. A., C. S. Fischer, F. Llanes-Estrada, Phys. Lett. B611 (2005) 279. Apply asymptotic expansion to all primitively divergent Green functions: Skeleton expansion & generalized formulas (neg. dim.) for Feynman integrals: R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

34 Infrared Exponents for Gluons and Ghosts: Expansion Scheme R. A., C. S. Fischer, F. Llanes-Estrada, Phys. Lett. B611 (2005) 279. Apply asymptotic expansion to all primitively divergent Green functions: Ghost propagator IR divergent Gluon propagator IR suppressed Ghost-Gluon vertex IR finite 3- & 4- Gluon vertex IR divergent if and only if all external momenta vanish IR fixed point for the coupling from each vertex Conformal nature of Infrared Yang-Mills theory! Solution unique! [C. S. Fischer and J. M. Pawlowski, hep-th/ ] R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

35 Running Coupling Ghost-Gluon-Vertex UV finite: α S (µ 2 ) = g2 (µ 2 ) 4π = 1 4πβ 0 g 2 0 Z(µ2 )G 2 (µ 2 ) With known IR behavior of gluon (Z ) and ghost (G) renormalization function: IR fix point α c = α S (k 2 0) α S (0) = 4π 6N c Γ(3 2κ)Γ(3+κ)Γ(1+κ) Γ 2 (2 κ)γ(2κ) Infrared fix point also in Coulomb gauge! [C.S. Fischer and D. Zwanziger, PR D72 (2005) ] R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

36 Running Coupling Ghost-Gluon-Vertex UV finite: α S (µ 2 ) = g2 (µ 2 ) 4π = 1 4πβ 0 g 2 0 Z(µ2 )G 2 (µ 2 ) With known IR behavior of gluon (Z ) and ghost (G) renormalization function: IR fix point α c = α S (k 2 0) α S (0) = 4π 6N c Γ(3 2κ)Γ(3+κ)Γ(1+κ) Γ 2 (2 κ)γ(2κ) Infrared fix point also in Coulomb gauge! [C.S. Fischer and D. Zwanziger, PR D72 (2005) ] R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

37 Running Coupling α(p 2 ) 4 3 quenched N f = p 2 [GeV 2 ] α s (MZ 2) =! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

38 Positivity violation for the gluon propagator Simple argument [Zwanziger]: IR vanishing gluon propagator implies 0 = D gluon (k 2 = 0) = d 4 xd gluon (x) = D gluon (x) has to be negative for some values of x. R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

39 Positivity violation for the gluon propagator Fourier transform of DSE result: D Gluon (x) x Gluons unobservable = Gluon Confinement! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

40 Positivity violation for the gluon propagator Analytic structure of running coupling: R.A., W. Detmold, C.S. Fischer and P. Maris, PRD70 (2004) α fit (p 2 ) = α S (0) 1 + p 2 /Λ 2 QCD + 4π β 0 p 2 Λ 2 QCD + p 2 with β 0 = (11N c 2N f )/3 Landau pole subtracted ( ) 1 ln(p 2 /Λ 2 QCD) 1 p 2 /Λ 2 QCD 1 analytic in complex p 2 plane except real timelike axis logarithm produces cut for real p 2 < 0 Cutkosky s rule obeyed R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

41 Positivity violation for the gluon propagator Analytic structure of running coupling: R.A., W. Detmold, C.S. Fischer and P. Maris, PRD70 (2004) α fit (p 2 ) = α S (0) 1 + p 2 /Λ 2 QCD + 4π β 0 p 2 Λ 2 QCD + p 2 with β 0 = (11N c 2N f )/3 Landau pole subtracted ( ) 1 ln(p 2 /Λ 2 QCD) 1 p 2 /Λ 2 QCD 1 analytic in complex p 2 plane except real timelike axis logarithm produces cut for real p 2 < 0 Cutkosky s rule obeyed R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

42 Positivity violation for the gluon propagator D fit gluon (p2 ) = w 1 p 2 ( p 2 Λ 2 QCD + p2 ) 2κ ( ) γ α fit (p 2 ) IR part: cut for Λ 2 QCD < p 2 < 0 Dgluon fit : cut along negative, i.e. timelike, half-axis! Wick rotation possible! w arbitrary normalization parameter κ = fixed from IR analysis γ = 13Nc+4N f 22N c 4N f from perturbation theory Effectively one parameter : Λ QCD =520 MeV! from fits to lattice data: Λ QCD 380 MeV R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

43 Positivity violation for the gluon propagator R. Alkofer (Graz) 2 QCD 2 Green Functions Oct. 17, / 32 D fit gluon (p2 ) = w 1 p 2 ( p 2 Λ 2 QCD + p2 ) 2κ ( ) γ α fit (p 2 ) 10 0 Z(p 2 ) 10-1 lattice, N f =0 DSE, N f =0 DSE, N f =3 Fit to DSE, N f = p 2 [GeV 2 ]

44 Positivity violation for the gluon propagator D fit gluon (p2 ) = w 1 p 2 ( p 2 Λ 2 QCD + p2 ) 2κ ( ) γ α fit (p 2 ) 10 0 g (t) DSE, N f =3 Fit to DSE IR-part of Fit t [GeV -1 ] R. Alkofer (Graz) 2 QCD 2 Green Functions Oct. 17, / 32

45 Positivity violation for the gluon propagator D fit gluon (p2 ) = w 1 p 2 ( p 2 Λ 2 QCD + p2 ) 2κ ( ) γ α fit (p 2 ) IR part: cut for Λ 2 QCD < p 2 < 0 Dgluon fit : cut along negative, i.e. timelike, half-axis! Wick rotation possible! w arbitrary normalization parameter κ = fixed from IR analysis γ = 13Nc+4N f 22N c 4N f from perturbation theory Effectively one parameter : Λ QCD =520 MeV! from fits to lattice data: Λ QCD 380 MeV R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

46 Positivity violation for the gluon propagator D fit gluon (p2 ) = w 1 p 2 ( p 2 Λ 2 QCD + p2 ) 2κ ( ) γ α fit (p 2 ) IR part: cut for Λ 2 QCD < p 2 < 0 Dgluon fit : cut along negative, i.e. timelike, half-axis! Wick rotation possible! w arbitrary normalization parameter κ = fixed from IR analysis γ = 13Nc+4N f 22N c 4N f from perturbation theory Effectively one parameter : Λ QCD =520 MeV! from fits to lattice data: Λ QCD 380 MeV R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

47 Dynamically induced scalar quark confinement R.A., C. S. Fischer, F. Lllanes-Estrada, hep-ph/ see talk by Christian S. Fischer... chiral symmetry dynamically or explicitely broken: Quark-Gluon vertex IR divergent! = + + (..) V(r) = d 3 p (2π) 3 H(p0 = 0, p)e ipr r R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

48 Partial gluon confinement at any T 2 Gluon propagator at high T : A. Maas, J. Wambach, RA, EPJ C37 (2004) 335; C42 (2005) 93. A. Cucchieri, T. Mendes and A.R. Taurines, PR D67 (2003) Chromomagnetic gluon propagator Z(0,q 3 )/q 4 g Lattice data Continuum extrapolated 3 120, β= , β= , β= q/g 3 Gribov-Zwanziger / Kugo-Ojima scenario / positivity violation applies in confined and deconfined phase! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

49 Partial gluon confinement at any T Gribov-Zwanziger / Kugo-Ojima scenario / positivity violation at any T : No infrared singularities, c.f. Linde (1980), because no chromomagnetic mass! D. Zwanziger, hep-ph/ No surprise: three-dimensional YM theory confining area law for spatial Wilson loop Coulomb string tension 0 at any T Nature of deconfinement? Gluon plasma? R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

50 Quark confinement in the Coulomb gauge D. Zwanziger, Phys. Rev. Lett. 90 (2003) One-gluon-exchange in Coulomb gauge QCD is confining. D 00 ( x, t) V C ( x )δ(t) + non inst. terms lim R V Wilson (R) lim R V C (R) As V Wilson (R) = σr +... one obtains: V C (R) = σ c R with σ c σ. OVERCONFINING lattice calculations: σ c 3 σ, i.e. σ c MeV. J. Greensite, S. Olejnik and D. Zwanziger, Phys. Rev. D 69 (2004) [arxiv:hep-lat/ ] A. Nakamura and T. Saito, Prog. Theor. Phys. 115 (2006) 189 [arxiv:hep-lat/ ]. R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

51 Quark confinement in the Coulomb gauge R.A. and P.A. Amundsen, Nucl. Phys. B306 (1988) 305 Infrared singular instantaneous interaction: Quark propagator vanishes, i.e. quarks confined, nevertheless: quark mass function and condensate well-defined! Quark DS-equation i S 1 (p) = /p m Σ(p) -1 = -1 Σ(p), S 1 (p) but dp 0 S(p) qq σ 3/2 c R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

52 Quark confinement in the Coulomb gauge R.A. and P.A. Amundsen, Nucl. Phys. B306 (1988) 305 Infrared singular instantaneous interaction: Quark propagator vanishes, i.e. quarks confined, nevertheless: quark mass function and condensate well-defined! Quark DS-equation i S 1 (p) = /p m Σ(p) -1 = -1 Σ(p), S 1 (p) but dp 0 S(p) qq σ 3/2 c R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

53 Quark confinement in the Coulomb gauge R. A., M. Kloker, A. Krassnigg, and R. Wagenbrunn, Phys. Rev. Lett. 96 (2006) (color singlet) meson properties well-defined! colored diquarks confined! nevertheless well-defined size! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

54 Quark confinement in the Coulomb gauge R. A., M. Kloker, A. Krassnigg, and R. Wagenbrunn, Phys. Rev. Lett. 96 (2006) (color singlet) meson properties well-defined! colored diquarks confined! nevertheless well-defined size! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

55 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

56 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! (Kugo Ojima Confinement, Oehme Zimmermann superconvergence, Gribov Zwanziger horizon condition,... ) Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

57 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

58 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

59 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

60 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

61 Summary Summary on QCD Green functions in covariant gauge Gluons confined by ghosts: Positivity violated! Gluons removed from S matrix! Infrared-finite strong running coupling in Yang-Mills theory! Conformal Nature of Infrared Yang-Mills theory! Analytic structure of gluon propagator: effectively one parameter! Chiral symmetry dynamically broken! In 2- and 3-point function! Quark confinement: In IR dominantly scalar! Positivity violation at any temperature! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

62 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

63 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

64 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

65 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

66 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32

67 Summary Summary on QCD Green functions in Coulomb gauge Ghosts and Coulomb gluons IR enhanced! One-gluon-exchange (D 00 ) overconfines! Transverse gluons: IR vanishing = Positivity violated! c.f. talk by Hugo Reinhardt Infrared-finite strong running coupling in Yang-Mills theory! Quark confinement: 4D quark propagator vanishes! Dynamical chiral symmetry breaking; color singlet quark condensate well-defined! R. Alkofer (Graz) QCD Green Functions Oct. 17, / 32