QCD Matter above Tc. Atsushi NAKAMURA RIISE, Hiroshima University 5 th International Workshop on Extreme QCD Frascati, Aug.

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1 QCD Matter above Tc Atsushi NAKAMURA RIISE, Hiroshima University 5 th International Workshop on Extreme QCD Frascati, Aug.6-8, 2007 QCD

2 Collaboration with R. Gupta and S. Sakai M. Chernodub and V. Zakharov Y. Nakagawa, T. Saito, H. Toki and D. Zwanziger M.Hamada, M.Yahiro and H.Kono Trial to construct A Picture of QCD Matter above Tc

3 Contents 1. Motivation 2. Transport Coefficients 3. Confinement Picture 1. Gribov-Zwanziger 2. Quark Propagators 3. Magnetic Degrees of Freedom at T>Tc

4 T T c Temperature exceeds Tc at RHIC, and probably at SPS (and surely will exceed at LHC)

5 ε Bj ( τ) = m T τπ R 2 dn dy Bjorken( 83) τ: proper time y: rapidity R: effective transverse radius m T : transverse mass If we take τ=1fm/c ε c from lattice PHENIX( 05)

6 T = 1/ N a t t at 0 (continuum limit) N t ε Y.Aoki et al., hep-lat/ ε-3p p MILC Collaboration, Nf=2+1 hep-lat/ Red Nt=4 Black Nt=6

7 LHC RHIC T/ Tc

8 Like a Fluid? v 2 Lines: Hydrodynamics calc. with QGP type EoS. p T (GeV)

9 Collective Flow (which does no exist if the matter is a gas) y z z

10 Quarks flow! n=2 for mesons n=3 for baryons!

11 Matter interacts strongly with Partons. Figure stolen from Chujo s talk

12 Jet Quenching

13 Confinement/Deconfinement Simple Picture I can see only a colorless state from outside? Confinement Potential is screened at finite temperature. Deconfinement Free Quark Gluon Gas

14 Transport Coefficients A Step towards Gluon Dynamical Behavior. They can be (in principle) calculated by a well established formula (Linear Response Theory). They are important to understand QGP which is realized in RHIC (and CERN-SPS) and LHC. QCD Hydro-Model Experimental Data

15 RHIC-data Big Surprise! Hydro-dynamical Model describes RHIC data well! Oh, really? At SPS, the Hydro describes well one-particle distributions, HBT etc., but fails for the elliptic flow.

16 Hydro describes well v2 Hydrodynamical calculations are based on Ideal Fluid, i.e., zero shear viscosity.

17 Another Big Surprise! The Hydrodynamical model assumes zero viscosity, i.e., Perfect Fluid. Phenomenological Analyses suggest also small viscosity. Oh, really?

18 Liquid or Gas? Frequent Momentum Exchange Perfect fluid Opposite Situation Ideal Gas

19 Karsch and Wyld (1987) Masuda, Nakamura and Sakai (Lattice 95) Sakai, Nakamura, Saito(QM97,Lattice 98) (Improved Action) Aarts and Martinez-Resco (2002) Sakai, Nakamura (2004) Anisotropic Lattice caliburation for improved gauge actions Nakamura and Sakai (2005) Aarts, Allton, Foley, Hands, Kim (2007) Meyer (2007) Luescher-Weiz 2-level Shear Viscosity Smearing Good MEM Bulk Viscosity Clover Operator Two-Level Algorithm

20 Linear Response Theory Zubarev Non-Equilibrium Statistical Thermodynamics Kubo, Toda and Saito Statistical Mechanics

21 T = T + µν µν eq t 3 ε ( t' t) ρ β σ µν ρσ eq + dx' dte ' ( T ( xt, ), T ( xt ', ')) ( u) where ( Tµν ( x, t), Tρσ ( x', t')) eq 1 (, )( A ( ', ') ( ', ') ) τ Aτ dτ Tµν x t e Tρσ x t e Tρσ x t 0 eq eq ij i j j i T = η( u + u )/2 T 0i = χ β 1 x t i β + u α α ( (, ) ) p p = ς u α eq α 1 p T 3 i i

22 Energy Momentum Tensors 1 T = 2 Tr( F F δ F F ) µν µσ νσ µν ρσ ρσ 4 ( T = 0) µµ = 2 µν µν U ( x) exp( ia gf ( x)) 2 Fµν = log Uµν / ia g or ( ) 2 Fµν = Uµν Uµν /2ia g

23 η t t 1 3 ε ( t 1 1 t) = dx' dte dt' < T ( xt, ) T ( x', t') > t t η t t 3 ε ( t 1 1 t) χ = dx' dt1e dt ' < T01( x, t) T01( x ', t ') > T 3 ε ( t 1 1 t) + ς = dx' dte dt' < T ( xt, ) T ( x', t') > η : Shear Viscosity χ T : Heat Conductivity µν ( x', t') ς T µν : Bulk Viscosity ( x, t) ret we do not consider in Quench simulations. ret t 1 < t' < t < t 1 e ε ( t1 t) t

24 Ansatz for the Spectral Functions We measure Matsubara Green Function on Lattice (in coordinate space). < T (, t x) T (0) >= G (, t x) = FT.. G (, p) µυ µυ β β ωn ρ ( p, ) G ( p, i ω ρ ω β n) = d ω i ω ω We assume (Karsch-Wyld) A γ γ π ( m ω) + γ ( m + ω) + γ = 2 and determine three parameters, A, m, γ. We need large Nt! n

25 Lattice and Statistics Iwasaki Improved Action β=3.05 : 1.3M sweeps β=3.20 : 1.2M sweeps β=3.30 : 1.3M sweeps β=3.05 : 3.0M sweeps β=3.20 : 2.5M sweeps β=3.30 : 2.0M sweeps β=3.05 : 0.6Msweeps β=3.30 : 0.8M sweeps β=3.05 : 6.0M sweeps β=3.30 : 6.0M sweeps Quench

26 Nt=8

27 η s 1! 4π Kovtun, Son and Starinets, hepth/ η = s 1 4π for N=4 supersymmetric Yang- Mills theory in the large N. Policastro, Son and Starinets, Phys Rev. Lett. 87 (2001)

28 Nakamura and Sakai, 2005

29 Viscosity by Lattice, 2007 c

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31 Kharzeeva and Tuchinb ς s T T c 1.56

32 Fluctuations in MC sweeps Standard Action Improved Action

33 Improve Operator for Fµν F µν = F µν =

34 Anisotropic Lattice? Anisotropic lattice has matured and will help us to get more data points to determine the spectral function.

35 η determined on different anisotropic lattices

36 Status of Transport Coefficients, 2007 Old Method (We employ an Ansaz for the spectral function) Numerically, no discrepancy is observed agains the Ansaz, so far. High statistic reduces (2 ~ 6 M. 10 times more than before) error bars, and η ~ 0.1 Bulk viscosity has small positive values. Many techniques are tested (improved Kernel of MEM, improved operators, 2-level-Algorithm by Luescher-Weiz etc). All seem to work Then let s go

37 In order to understand Deconfinement, we need to understand the Confinement Mechanism Gribov-Zwanziger Picture Monopole Picture

38 1 1 = ( tr ( ( )) ( )) ( ( ) (, ) ( )) i i ρ ρ H d 3 x E x 2 B 2 x d 3 xd 3 y xv x y y M = D = + ga ( 2 ) Faddeev-Popov Operator 1 1 (, ) (, ) 3 2 V( x, y) = d z ( z ) M xz M zy 2 If M =, i.e. Abelian Case V x 1 y i.e, Coulomb Potential

39 Gribov picture 1 st Gribov Region λ > i 0 λ i : Eigen values of Faddeev -Popov Eigen Values locate near the boundary, and they make the confinement Potential.

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41 Time-time gluon propagator in Coulomb gauge D ( x, t) =< A ( x, t) A (0,0) > = V () r δ () t + P(,), x t r = x coul Color-Coulomb instantaneous part ( antiscreening ); this term may produce the color confinement. Non-instantaneous vacuum polarization part (screening) ; this term causes a screening effect ( quark-pair creation ). D 00 is invariant under the renormalization.

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44 Eigen-Value Distribution of Faddeev- Popov Operator does not change drastically below and above Tc. Consequently, the Instantanous Potential remains Linear-Rising behavior even above Tc. The Polarization Part P makes the Potential screened. D ( x, t) =< A ( x, t) A (0,0) > = V () r δ () t + P(,), x t r = x coul Always Linear Rising

45 Quark Propagators above Tc Quarks become free from the Confinement, but we suspect they are not quasi-free. Then how they behave?

46 Quark Propagators G( p, p ) = 4 i ia p4 pi γ4p4 + ib p4 pi γipi + C p4 pi = 1 (, ) (, ) (, ) 1 1 A( p, p ) iγ p + ib ( p, p ) γ p + C ( p, p ) i i i i 4 i = 1 iγ p ib ( p, p ) γ p + C ( p, p ) i i i 4 i i i i + 4 i A( p, p ) p B ( p, p ) p C ( p, p ) B( p, p ) i 4 i 2 2 A p4 pi B ( p, p ) = (, ) C( p, p ) i 4 i 2 2 A p4 pi C ( p, p ) = (, ) C. D. Roberts and S. M. Schmidt, nucl-th/

47 Quark Propagators Wave Function: A ( q, q ) 2 2 free 4 i 4 qi = 2 2 Aq ( 4, qi ) Zq (, ) J-I. Skullerud and A. G. Williams Phys. Rev. D63, C. D. Roberts and A. G. Williams, hep-ph/ P. O. Bowman et. al., Lattice Hadron Physics, (Springer) Mass Function: M ( q, q ) = C ( q, q ) C ( q, q ) i 4 i free 4 i P. O. Bowman et. al., Lattice Hadron Physics, (Springer)

48 Momentum Dependence (Wilson fermion)

49 Pole Mass C ( p ) p C p p p C p ( 4, i ) = 4 + (0) + p p = C ( p4 ) 2 C (0) 1+ 2 C (0) p p 4 2 C ( p ) p = 1+ 2 C (0) 2 p 4 2 p4 = 0 2 M 2 pole = C 2 (0) C ( p ) 1+ 2 C (0) p 4 p = 0 2 4

50 Pole Mass in Quark Propagators (Quench) (Very) k Preliminary ( ) k > 2?

51 Magnetic Degrees of Freedom? M. Chernodub and V. Zakharov hep-lat/ J. Liao and E. Shuryak hep-ph/

52 Fitting to extrapolate mass Gz ( ) = C cosh( mz ( N/2)) z 5 at z > 1/ T m/t 4 3 Magnetic Electric Nakamura, Sakai and Saito, Phys. Rev. D69, (2004) hep-lat/ T/T c

53 Spatial Wilson Loops T / T = 4 C T / T = 2.67 C T / T = 2 C Karsch, Laermann, and Luetgemeier, Phys.Lett. B346 (1995) 94 It is well known that Spatial Wilson Loops give a Confinement Potential. even above Tc.

54 Confinement is due to monopole condensation Center vortex mechanism Del Debbio, Faber, Greensite, Olejnik, 97 a realization of spaghetti (Copenhagen) vacuum Center strings are classified with respect to the center Z N of the SU(N) gauge group Confinement is due to vortex percolation

55 Observation of monopoles in the vortex chains: monopole is a defect, at which the flux of the vortex alternates.

56 SU (2) T C (Very) k Preliminary ( ) k > 2 ε 3p

57 Vino Vianco Sono ritornato, e molto contento. Grazie!! I was a Pos-Doc from at Frascati. I started Lattice QCD Study here and wrote the first paper about SU(2) Color Lattice at Finite Density (Phys.Letters 149B, 1984), i.e., I was born here as an XQCD citizen.

58 Backup Slides

59 Very high Temperature

60 Entropy Density F f = = fv p U TS = Tlog Z = F S s = = V β ε + p β d p = d T d T β ' 4 β β ' 4 0 β0 T p We reconstruct p from Raw-Data by CP-PACS (Okamoto et al., Phys.Rev.D (1999) )

61 Spectral Function by Aarts and Resco lo ( ) w high ρ ω = ρ ( ω) + ρ ( ω) ρ ( ω ) b + b x lo w 2 4 T 1 2 = x c1x c2x x ω T /2 high ( Nc 1)( ω 4 mth ) ρ ( ω) = θ( ω 2 mth) [ n( ω) + 0.5] 2 80π ω Fitting with three parameters, c < 0 1? b c 1 1 m

62 Effect of High-Frequency part BW ρ = ρ + ρ β=3.3 high ρ ( ω ) b + b x ρ lo w 2 4 T 1 2 = x c1x c2x BW 3 ηa m th (201) A γ γ = π + ( m ω) + γ ( m+ ω) + γ (191) (194) (204) m th = x 1.8 ω T high ρ contribution is larger than BW ρ at t=1.

63 Why they are so noisy? RG improved action helps lot. Noise from Lattice Artifact? (Finite a correction?) Once we checked that there is not so much difference between ( ) F = U U /2i µν µν µν and for SU(2). But we should check it again. a F = log U / i µν µν

64 The situation reminds us Glue-Ball Case. (I thank Ph.deForcrand for discussions on this point.) Glue-Ball Correlators = ( τ ) (0) Large (extended) Operators work better, e.g., ν µ where = + +

65 Mmmm not works Fat Link

66 Anisotropic Lattice? Anisotropic lattice has matured and will help us to get more data points to determine the spectral function.

67 QCD G( n,n) = G( n n) = 1 N i ψ ( n )ψ (n) i = 1 N i W 1 ( n n) i G(p 4, p i ) = N µ n µ =0 p 4 = 2π N 4 (n ) exp(i(p n))g(n) p i = 2π N i n i

68 Quark Propagators 1 G(p 4, p i ) = ia(p 2 4, p 2 i )γ 4 p 4 + ib(p 2 4, p 2 i )γ i p i + C(p 2 4, p 2 i ) 1 1 = A(p 2 4, p 2 i ) iγ 4 p 4 + i B (p 2 4, p 2 i )γ i p i + C (p 2 4, p 2 i ) = 1 A(p 4 2, p i 2 ) iγ 4 p 4 i B (p 2 4, p 2 i )γ i p i + C (p 2 4, p 2 i ) p B 2 (p 2 4, p 2 i )p 2 i + C 2 (p 2 4, p 2 i ) B (p 2 4, p 2 i ) = B(p 2 4, p 2 i ) A(p 2 4, p 2 i ) C (p 2 4, p 2 i ) = C(p 2 4, p 2 i ) A(p 2 4, p 2 i ) p 4 p 4 + iµ C. D. Roberts and S. M. Schmidt, nucl-th/

69 : Z(q 4,q i ) = A (q 2 free 4,q 2 i ) A(q 2 4,q 2 i ) J-I. Skullerud and A. G. Williams Phys. Rev. D63, C. D. Roberts and A. G. Williams, hep-ph/ P. O. Bowman et. al., Lattice Hadron Physics, (Springer) : M (q 4,q i ) = C (q 2 4,q 2 i ) C free (q 2 4,q 2 i ) P. O. Bowman et. al., Lattice Hadron Physics, (Springer)

70 Momentum Dependence (Wilson fermion)

71 Momentum Dependence (Wilson fermion vs. Clover fermion) T = 1.37T C

72 Momentum Dependence (Wilson fermion)

73 3 (Wilson fermion vs. Clover fermion) T = 1.37T C

74 1 p 0 2 M (p i 2 ) = 0 p B 2 (p 2 4, p 2 i )p 2 i + C 2 (p 2 4, p 2 i ) (p 0 2 p 4 2 ) p i = 0 C (p 2 4,0) p 2 4 C (p 2 4 ) = C (0) + C (p 4 p ) p 4 2 =0 p 4 2

75 Pole Mass C ( p ) p C p p p C p ( 4, i ) = 4 + (0) p 4 p = 0 2 M pole = 1+ 2 C (0) C (p 4 2 p 4 C 2 (0) 1+ 2 C (0) C (p 4 p ) p 4 2 =0 2 ) p 4 2 =0 p C 2 (0) 1+ 2 C (0) C (p 4 p ) p 4 2 =0