Non-Perturbative Thermal QCD: AdS/QCD and Gluon Condensates

Transcription

1 Issues Non-Perturbative Thermal QCD: AdS/QCD and Gluon Condensates * B. Galow, M. Ilgenfritz, J. Nian, H.J. Pirner, K. Veschgini (ITP, Heidelberg, Germany) E. Ruiz Arriola, L.L. Salcedo (U. Granada, Spain) Instituto de Física Teórica, IFT-CSIC/UAM, Universidad Autónoma de Madrid, Spain and Department de Física and IFAE, Universitat Autònoma de Barcelona, Spain. Supported by the Alexander von Humboldt Foundation, Spanish CPAN and Juan de la Cierva. INT Program 11-3 Frontiers in QCD November 8, 2011, INT, Washington U., Seattle, USA. E. Megías et al. NPB834(2010), (2009), NP P.S.207(2010), PRD83(2011), PLB696(2011), AIP C.P.1343(2011); PRD80(2009),PRD81(2010).

2 Issues Issues 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

3 Issues Issues 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

4 Issues Issues 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

5 Issues Issues 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

6 Motivation Pressure of Gluodynamics Weak Coupling Expansion and Resummed Perturbation Theory E. Braaten and A. Nieto (1996), J.O. Andersen et al (1999) ideal / T/T c

7 Motivation Interaction Measure in Gluodynamics Weak Coupling Expansion and Resummed Perturbation Theory E. Braaten and A. Nieto (1996), J.O. Andersen et al (1999) ideal ( - 3 )/ T/T c

8 Motivation Trace Anomaly N c = 3, N f = 0 G. Boyd et al., Nucl. Phys. B469, 419 (1996). ǫ 3P T 4 (ε-3p)/t HTL 2-loops a + b (T c /T) T/T c = a + b T 2, b = (3.46±0.13)Tc 2, 1.13T c < T < 4.5T c.

9 Motivation Trace Anomaly N c = 3, N f = 0 G. Boyd et al., Nucl. Phys. B469, 419 (1996). ǫ 3P T 4 (ε-3p)/t a + b (T c /T) (T c /T) 2 = a + b T 2, b = (3.46±0.13)Tc 2, 1.13T c < T < 4.5T c.

10 Motivation ǫ 3P T 4 = HTL (µ T ) + b }{{} T 2 1/ log T (ε-3p)/t HTL 2-loops HTL 3-loops a + b (T c /T) T/T c (ε-3p)/t HTL 2-loops HTL 3-loops a T/T c Perturbation Theory and Hard Thermal Loops only yield a!!.

11 Issues Motivation Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

12 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model Scale Invariance and Confinement Consider a rectangular Wilson loop: ( ) W(C) = exp ig A µ dx µ C x t It is related to the potential V q q (R) acting between charges q and q: W(C) t exp( t V q q (R)) Scale transformations: t λt, R λr. The only scale invariant solution is the Coulomb Potential: V q q 1 R Running coupling and string tension break scale invariance: V q q (R) = 4 α s (R) 3 R +σr.

13 Issues Motivation Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

14 Soft-wall model of AdS/QCD Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model dsqcd 2 = h(z) ds2 = h(z) l2 ( dt 2 z 2 + d x 2 + dz 2). }{{} AdS 5 h(z) = 1 = Conformal h(z) 1 = Non conformal Breaking of scaling invariance in QCD is given by the running coupling: ǫ 3p T 4 = β(α s) Fµν. 2 4α 2 s where β(α s ) = µ dαs dµ and α s(e) 1/ log(e/λ). = Assume an ansatz for conformal symmetry breaking similar to 1-loop running coupling (H.J. Pirner & B. Galow 09): h(z) = log(ǫ) log(ǫ+(λz) 2 ), z 1 E. Other ansatz: h(z) = e 1 2 cz2 Andreev & Zakharov 07

15 Soft-wall model of AdS/QCD H.J.Pirner & B.Galow 09 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model W(C) exp( S NG ) exp ( t V Q Q(R) ) V ren. Q Q (R)[GeV] R[GeV 1 ] 2 V q q a R +σr, a = 0.48, σ = (0.425 GeV)2 = ǫ = Λ 2 l 2 s = 0.48, Λ = 264 MeV.

16 Issues Motivation Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

17 The 5D Einstein-dilaton model Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 5D Einstein-dilaton model (Gürsoy et al. 08): 1 S 5D = d 5 x ( g R 4 ) 16πG 5 3 µφ µ φ V(φ) 1 8πG 5 One to one relation between β-function and dilaton potential V(φ): V(φ) = 12 l 2 ( 1 ( ) ) 2 [ β(α) exp 8 α 3α 9 0 Ansatz (E.Megías et al., NPB834, 2010): [ ( ) b2 β(α) = b 2 α+ b 2 α+ ᾱ β 0 α 2 + M ] β(a) a 2 da, α = e φ. ( b2 2ᾱ 2 β 0 ᾱ β 1 α << ᾱ = Ultraviolet: β(α) β 0 α 2 β 1 α 3 α >> ᾱ = Infrared: β(α) b 2 α d 4 x h K. )α 3 ] e α/ᾱ.

18 The 5D Einstein-dilaton model Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model W(C) exp( S NG ) exp ( t V Q Q(R) ) ds 2 = e 4 3 φ e 2A( dt 2 + d x 2) +e 4 3 φ l 2 e 2D dα 2 ρ(α 0 ) = ρ 2 ρ 2 α0 e D 3Ã α 4 dσ = 2le A0 3 dα, 0 1 α 8 3 e 4Ã V reg. (α Q Q 0) = 1 t Sreg. NG = V V s [ = 2lα ea 0 α0 α 4 3 e D+Ã dα π l s α 8 3 e 4Ã 0 dα α 4 3 e D+Ã ]

19 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model Heavy QQ potential at Short Distances Analytical result at short distances: V Q Q(ρ) = 2 l 2 α 4/3 0 (ρ) [ ] π l2 s ρ }{{} β 0 α }{{} 0 +(1.347β β 1 )α0 2 }{{} where the running coupling is [ ( 1.32l α 0 (ρ) = β 0 log ρ Ρ V QQ Ρ LO ) + β 1 β 0 log NNLO NLO LO Ρ GeV NLO ( β 0 log V PT NNLO ( ))] l +O(log 3) ρ Q Q (ρ) = N2 c 1 2N c Comparison with PT = ls l = 0.33 α PT(ρ) (1+O(α PT )) ρ

20 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model Heavy QQ potential and running coupling W(C) exp( S NG ) exp ( t V Q Q(R) ) V qq Ρ GeV Ρ GeV 1 Α E E GeV σ = (0.425 GeV) 2 = b 2 ᾱ = 3.51GeV l s Fit of running coupling = b 2 ᾱ = 5.09 ls = 1.45 GeV 1

21 Dilaton potential and warp factor Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model V Α GeV Α h z GeV 2 l s h z l s 2 h z z GeV 1 l s 2 V(α) (b 2 2 9)α 8 9 b 2, α ; 1 α(z) b 2, z z IR 4 (z IR z) 9 b2 2 1

22 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model Confinement and good IR singularity Effective Schrödinger potentials for glueballs 0 ++ and 2 ++ : V Schr z GeV Α z Tensor 2 Scalar z GeV 1 ] [ 2 z 2 + V Schr.(z) ψ n (z) = mnψ 2 n (z) Confining theory = 1.5 < b 2 IR singularity repulsive to physical modes = b 2 < 2.37 Best choice of parameters: b 2 = 2.3, ᾱ = 0.45, l = 4.39 GeV 1, ls = 1.45 GeV 1.

23 Issues Motivation The 5D Einstein-dilaton model at finite temperature 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

24 Schwarzschild black hole The 5D Einstein-dilaton model at finite temperature General Relativity with no source = Einstein-Hilbert action 1 S EH = d D x g R, R = g µν R µν 16πG D Classical solution δ δg µν = Einstein equations E µν R µν 1 2 g µνr = 0 = spherical R µν = 0 Schwarzschild solution in spherical coordinates (1915): ds 2 = g µν dx µ dx ν = f(r)dt f(r) dr 2 + r 2 dω 2 2, f(r) = 1 r h r r h is the horizon. Not physical singularity: R µνρσ R µνρσ = 12 r 2 h r 6. Large distance limit: g tt (r) r (1+2V Newton (r)) = r h = 2G 4 M.

25 Black hole thermodynamics The 5D Einstein-dilaton model at finite temperature ( Z = Tr e βh), β = 1 T Periodicity in euclidean time (τ = it): Φ(τ +β) = Φ(τ) *Regularity: Expansion around the horizon f(r h ) = 0; r = r h (1+ρ 2 ): dsbh 2 = f(r)dt ( ( ) 2 dτ f(r) dr 2 +r 2 dω 2 2 4rh 2 dρ 2 +ρ ) ρ 0 2r h 4 dω2 2 }{{} dθ 2 τ = Periodicity: 2r h τ 2r h + 2π = τ τ + 4πr h =: τ +β 1 T = 8πMG 4 *Thermodynamical interpretation of black holes: dm dm = TdS = S = T = 4πG 4M 2 A = 4πr 2 h = 16π(G 4M) 2 = S Black Hole (T) = A(r horizon) 4G D Bek-Hawking

26 Issues Motivation The 5D Einstein-dilaton model at finite temperature 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

27 The 5D Einstein-dilaton model at finite temperature The 5D Einstein-dilaton model at finite temperature 1 S 5D = 16πG 5 d 5 x ( g R 4 ) 3 µφ µ φ V(φ) 1 d 4 x h K. 8πG 5 M Finite temperature solutions (E. Kiritsis et al. JHEP (2009) 033): Thermal gas solution (confined phase): ds 2 th = b 2 0(z)( dt 2 + d x 2 + dz 2 ), t t + iβ Black hole solution (deconfined phase): ] dsbh 2 = b 2 (z) [ f(z)dt 2 + d x 2 + dz2 f(z) In the UV (z 0): flat metric b(z) l/z and f(0) = 1. There exists an horizon f(z h ) = 0. Regularity at the horizon = T = ḟ(z h) 4π.

28 The 5D Einstein-dilaton model δ The 5D Einstein-dilaton model at finite temperature Einstein equations δg µν : ( R µν 1 ) ( 4 2 g µνr 3 µφ ν φ 1 ( )) 4 2 g µν 3 ( φ)2 + V(φ) = 0 }{{}}{{} E µν T µν (a) (b) (c) f ḟ + 3ḃ = 0,= f(z) = 1 b 6ḃ2 b 2 3 b b = 4 3 φ 2, 6ḃ2 b b b + 3ḃ b ḟ f = b2 V(φ) f z 0 zh 0 Conformal solution: V(φ) = 12 ( l 2, l z φ = 0 = b(z) =, f(z) = 1 z du b(u) 3 du b(u) 3 z h ) 4, T = 1 πz h

29 The 5D Einstein-dilaton model [ β(α) = b 2 α+ b 2 α+( b2 ᾱ β 0 = V(α) = 12 l 2 T GeV ( T c 272MeV Α h Input: ) ( ) ) 2 [ β(α) exp 8 α 3α f The 5D Einstein-dilaton model at finite temperature α 2 +( b2 2ᾱ 2 β 0 ᾱ β 1 ) α 3 ] e α/ᾱ ] β(a) a 2 da, E.Megías NPB z GeV 1 T = 368 MeV

30 The 5D Einstein-dilaton model [ β(α) = b 2 α+ b 2 α+( b2 ᾱ β 0 Input: ) The 5D Einstein-dilaton model at finite temperature α 2 +( β2 2ᾱ 2 β 0 ᾱ β 1 ) α 3 ] e α/ᾱ z l b z Α z z GeV T = 368MeV z GeV 1

31 Issues Motivation The 5D Einstein-dilaton model at finite temperature 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

32 Motivation The 5D Einstein-dilaton model at finite temperature Postulate: Entropy of gauge theories is equal to the Bekenstein-Hawking entropy of their string duals. S(T) = A(z h) 4G 5 = V 3b 3 (z h ) 4G 5, z h 1 πt π High temperature limit: s(t) 3 l 3 T 4G 5 T 3 = π2 T 3 =: s ideal (T) One can compute all the thermodynamic quantities: s(t) = d dt p(t), ǫ 3p (T) T 4 = s T 3 4p T 4. In the free energy = contributions from big and small black holes: T αh ( ) dt p(t) = p(t 0 )+ d T s( T) = d α h s( α h ). T 0 + d α h

33 Motivation The 5D Einstein-dilaton model at finite temperature Input β(α). e 3P T P T Numeric Αh 2 Αh 3 SU 3 Panero SU 3 Boyd et al Numeric Αh Αh 2 Αh 3 SU 3 Panero SU 3 Boyd et al T Tc T Tc N c = 3, N f = 0, b 2 = 2.3, ᾱ = 0.45 p(t) T 4 = π3 l 3 ( G β 0α h + 29 ) (4β20 3β 1 )α 2h , α h 5 β 0 log(πt/λ)

34 Motivation The 5D Einstein-dilaton model at finite temperature Free Energy from: Bekenstein-Hawking entropy Classically Free Energy from: Gibbons-Hawking action S = A 4G 5 βf = S reg := S BH S th = = F = V ( 3 15G C ) f 16πG 5 4 E. Kiritsis et al. JHEP (2009) 033. G = πg 5 15 β(α) α 2 TrF 2 µν, C f = 4πTb 3 (z h ) T s b T (z) = b 0 (z) [1+ Gl ( 3 z ) ] 12 β 0α 0 (z)+c2 b α2 0(z)+ +, z 0

35 Motivation The 5D Einstein-dilaton model at finite temperature Α z Α 0 z z 4 GeV GeV T GeV z GeV α T (z) α 0 (z) z 4 = 45 8 G l 3 β 0 ( ( β 0 β ) ) 1 α 0 (z)+c α β 2α0(z)+ 2 0 = Corrections in α 0 are very important for agreement.

36 The 5D Einstein-dilaton model at finite temperature Flavor dependence of the phase transition Tc Agrees quite well with J. Braun & H. Gies JHEP 1005 (2010), κ = N f ( T c = T Nf =0 1 κnf +O(Nf 2 )) κ = , T Nf =0 = MeV, T Nf =4 = MeV, T Lattice N f =0 Lattice = 270±2 MeV, TN f =4 = MeV.

37 Polyakov Loop Motivation L(T) := P = DX e Sw = semiclassically P = i The 5D Einstein-dilaton model at finite temperature w i e S i e S 0 S reg 0 = 1 [ zh zc ] 2πls 2 dz α 4/3 (z)b 2 (z) dz α 4/3 0 (z)b T 0(z) LR Numeric Αh 4 3 Αh 7 3 Αh 10 3 ΑT 2 Lattice Gluodynamics SU(3): S.Gupta et al., PRD77(2008). l s = 2.36 GeV 1, z c = 0.43 GeV 1 T Tc 0 [ 2 4( l 2 L(T) = exp 2ls 2 α 4 3 h β 0α h + 1 )] ( ) 161β β 1 α 2 h +O(αh 3 ), ( ) 3 In contradiction with PT: L PT (T) = exp πα 2 +O(α 2 ). 4 3

38 Spatial Wilson Loops The 5D Einstein-dilaton model at finite temperature Rectangular Wilson loop in (x,y) plane: W(C) = ( ) ly exp ig A µ dx µ e l y V(d), C V(d) d σ s d l y S NG = l2 2πls 2 l y l x/2 l x/2 dx h(z) z 2 1+ ( xz) 2, X(x, y) = (x, y, 0, 0, z(x)) f(z) σ s (T) = 1 2πl 2 s α 4/3 h bh 2 l x

39 Spatial Wilson Loops The 5D Einstein-dilaton model at finite temperature Spatial string tension: Lattice data: G. Boyd et al. NPB469 (1996) 1.4 T Σs SU 3 Α 4 3 h Numeric Α 7 3 h Alanen 09 Α 10 3 h Α 13 3 h σ s (T) = 1 2πls 2 α 4 3 h b 2 (z h ) = l2 2ls 2 πt 2 α 4 3 h [ This is in contradiction with pqcd: σ pqcd T 2 α 2 s(t). See also Kajantie et al. PRD80 (2009). l s = 1.94 GeV 1, χ 2 /d.o.f. = T Tc ] β 0α h (25β2 0 2β 1 )αh 2 +O(α3 h )

40 Issues Motivation The 5D Einstein-dilaton model at finite temperature 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

41 Pressure and Running Coupling Pressure from AdS/QCD (UV expansion): The 5D Einstein-dilaton model at finite temperature P AdS/QCD = P SB (1 2.33α h α 2 h +O(α3 h )) Pressure from perturbation theory: P pqcd = P SB (1 1.19α T + 5.4α 3/2 T +O(α 2 T )) A comparison at high T demands α AdS/QCD α pqcd / P PSB SU 3 Α h Α h 2 Α h 3 PT Α T T c

42 Pressure and Running Coupling The 5D Einstein-dilaton model at finite temperature Simple model: β(α) = { β pert (α) ˆβ 2 α 4 if α ˆα β pert (ˆα) ˆβ 2ˆα 4 3(α ˆα) if α > ˆα

43 Debye Screening Motivation The 5D Einstein-dilaton model at finite temperature S AdS/QCD = 1 4G5 2 dz d 4 x g(f V MN F V MN + F A MN F AMN ) Debye mass: md 2 = g2 Π 00 (ω = 0, k 2 = md 2) Screening mass in the absence of an external field (Gorsky et al. 10): S AdS/QCD = V 4D 2G5 2 dz b(z)( z V 0 (z)) 2 e.o.m.: z (b(z) z V 0 (z)) = 0, V 0 (z h ) = 0 & V 0 (z = 0) = µ. z ) V 0 (z) = µ (1 C h du b 1 (u) 0 = Π 00 = 1 V 4D 2 S µ 2 = m2 D = 4π G 2 5 [ zh 1 α h du b (u)] 1 0 T g2 h T 2

44 Debye Screening Motivation The 5D Einstein-dilaton model at finite temperature Debye Mass (E. Megías et al. PLB696(2011)) md T Α 0.5 Α 0.02 m D T, m D T, m D T 4ΠΑ h 1 2 m D T (4πα h) 1/2 T ˆα = 0.02 = Good P and bad m D. ˆα = 0.5 = Bad P and good m D T T c (1 29 β 0α h +O(α 2h ) )

45 Dimension Two Condensates? DIMENSION TWO GLUON CONDENSATE In collaboration with: E. Ruiz Arriola and L.L. Salcedo (U. Granada, Spain).

46 Issues Motivation Dimension Two Condensates? 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

47 Dimension 2 condensates? Dimension Two Condensates? Standard wisdom: There are no local gauge invariant operators of dimension 2. But, are there operators of dimension 2 relevant for phenomenology? There exists a nonlocal gauge invariant condensate A 2 min = 1 VT min g d 4 x ( ga µ g + g µ g ) 2, which reduces to the A 2 µ condensate in the Landau gauge. Example: Can f 2 π be written as the expectation value of a gauge invariant local operator?

48 Dimension 2 condensates? Dimension Two Condensates? In perturbation theory one must fix the gauge. There is no complete gauge fixing. Thus, there is always a dimension 2 local operator which is invariant under the residual gauge invariance. Dimension-2 objects have been seen in fixed gauge lattices: (Skullerad, Rodriguez-Quintero, Oliviera 2006).

49 Dimension Two Condensates? Quark propagator on the Lattice in Landau Gauge P. Bowman, W. Broniowski and E. Ruiz Arriola, PRD70: (2004). S 1 (p) = p/a(p) B(p) A(Q) = 1+ πα s(µ 2 ) A 2 µ N c Q 2 πα s(µ 2 ) G 2 µ 3N c Q 4 +,

50 Issues Motivation Dimension Two Condensates? 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

51 Dimension Two Condensates? and Polyakov loop ( ) Dµν(p ab 2 ) = δab δ µν k 2 δab δ µν k 2 1+ m2 G k 2 At finite temperature: D 00 ( k) = 1 k 2 +m 2 D A 2 0,a = (N 2 c 1)T Polyakov loop: 1 L = N c Tr c e iga 0,a/T ( 1+ m2 G k 2 +m 2 D d 3 k (2π) 3 D 00( k) = 2 π m DT e g2 A 2 0,a /(4NcT 2) +O(g 6 ) 0.4 g 2 A 2 0,a NP = (0.84(6) GeV) L(T) }{{} T Narison & Zakharov 99 ) Megías 06 + m2 G T A 2 πm 0,a P + A 2 0,a NP }{{ D } T 0 LO NLO T/T c

52 Gluon Asymmetry: Lattice data Dimension Two Condensates? Gluon asymmetry N c = 2, N f = 0 Lattice data (M. Chernodub, E.M. Ilgenfritz, PRD78 (2008)) A 2 /T x x x8 fit (T/T c = ) high-t limit of the fit A 2 /T T/T c T/T c A 2 = g 2 A 2 0,a 1 g 2 A 2 i,a 3 A 2/T 2 1/T 2 g 2 A 2 0,a = g 2 A 2 0,a P + g 2 A 2 0,a NP = g 2 A 2 0,a /T 2 = c P + c NP /T 2. }{{}}{{} T 2 cte i

53 Heavy qq Free Energy Dimension Two Condensates? Singlet Free Energy N c = 3, N f = 0 Lattice data (O. Kaczmarek PRD70 (2004)) F 1 (r,t) [GeV] e Fq q( x,t)/t+c(t) = 1 Nc 2 tr c Ω( x) tr c Ω ( 0), g 2 A 2 0,a NP = (0.90(5) GeV) 2 rt T/T c

54 Motivation Dimension Two Condensates? From the gluon propagator, one gets (Megías 09): (F a µν) 2 = 2 (B a i ) 2 (E a i ) 2 2 i A 0,a i A 0,a = 6 π m2 G m DT T 2. It reproduces the thermal behaviour of the trace anomaly: T 4 ǫ 3P = β(g) 2g (F µν) a 2 3 β(g) = (Pert.) }{{} π g m2 G m DT. }{{} T 4 T 2 It is straightforward to compute all the thermodynamic quantities: Pressure : Entropy : See also O.Andreev s talk. P(T) T 4 = T dt T (T ) = (Pert.) }{{} s 4P = + T 3 T 4 = } (Pert.) {{} T 0 + T π β(g)m2 G }{{ T 2 } 1/T 2 3 β(g) mg 2 m D π g T }{{ 3. } 1/T 2

55 (ε-3p)/t Motivation Dimension Two Condensates? Lattice data: G. Boyd et al. NPB469 (1996). pert HTL 1-loops HTL 2-loops pert + b (T c /T) T/T c P/P ideal P pert P HTL + P NP P pert +P NP T/T c Values of the dimension two gluon condensate from a fit of: Observable g 2 A 2 0,a NP Polyakov loop (3.22±0.07T c ) 2 Heavy qq free energy (3.33±0.19T c ) 2 Trace Anomaly (2.86±0.24T c ) 2

56 Dimension Two Condensates? from ET of scale invariance Scale invariance x µ λx µ is broken by quantum effects. Effective theory of scale invariance D.Kharzeev 08. L = ǫ v m 2 1 ( 2 eχ/2 ( µ χ) 2 + ǫ v + c 4 (F µν) a 2) e χ (1 χ) 1 }{{} 4 (F µν) a 2 V(χ) Properties: χ = 1: No coupling between dilatons and gluons = dynamics of color fields is pert. It happens at some scale M 0 5 6T c. No color fields = minimum of dilaton pot. at V(χ = 0) = ǫ v. V(χ) T>Tc T=Tc T= χ ǫ v + c 4 (F a µν) 2 { > 0 T < T c = 0 T = T c < 0 T > T c

57 Dimension Two Condensates? from ET of scale invariance D. Kharzeev, E. Levin & E. Megías 09 Classical equation for A µ gives: Ei a = i A 0,a = E 0 gc a F(χ) e m Drˆn+ F(χ)4πr 2(1+m Dr)e m Drˆr Unlike in pqcd (F a µν) 2 min = 0, the minimun of the effective theory corresponds to a finite chromoelectric field E 0. (F a µν) 2 min = 2( B a2 E a2 ) B 0 2E 2 0 = 4 ǫ v /c In momentum space: A 0,a ( k) = so the gluon propagator is g F(χ)( 8πE 0 k 2 +md 2) F(χ)(, k 2 +md 2)2 D 00 ( 1 k) = k 2 + md 2 + mg 2 ( k 2 + m, D 2 m2 )2 G = 8πE 0. g

58 Issues Motivation Dimension Two Condensates? 1 Motivation 2 Scale Invariance and Confinement Soft-wall model of AdS/QCD The 5D Einstein-dilaton model 3 The 5D Einstein-dilaton model at finite T 4 Dimension two Condensates?

59 Dimension Two Condensates? Duality between perturbative series and power corrections Perturbative-Non perturbative duality (Narison & Zakharov, PLB679 (2009)): Large order perturbative series are dual to non-perturbative contributions. V qq (r) 4 α s +σ r 3 r V qq (r) 1 N a n α n r s(r)+σ N r n=1 Both formulas fit lattice data very well. = Perturbative resummation could be written in terms of condensates, starting from dimension two.

60 Dimension Two Condensates? Vqq r fm r fm N c = 3, N f = 0 : Tree Level α s = 0.19, σ = 4.99 fm 2 (dim-2). (E.Megías, E.Ruiz Arriola & L.L.Salcedo, PRD81(2010)): In a fit including Pert. and Non Pert. contributions, one should see: Smaller contributions from condensates at increasing perturbative orders. σ N 0 (duality). Rather strong statistical correlation between Λ QCD and condensates (Λ QCD is the only scale in QCD). r(σ N,Λ QCD ) ±1.

61 Dimension Two Condensates? Correlations: Perturbative - Non Perturbative Fit using PT at N 3 LL, N. Brambilla et al. PRD80 (2009). Σ fm Σ fm fm < r < 0.17fm 0.085fm < r < 0.83fm r(δ,σ) = r(δ,σ) = α s (µ) : δ = log(µ/λ QCD )

62 Dimension Two Condensates? Correlations: Perturbative - Non Perturbative Order δ σ[fm 2 ] r(σ,δ) χ 2 /dof Tree Level 4.99(11) loop 3.66(7) 4.25(6) loop 4.54(10) 4.12(6) N 3 LO 4.31(13) 4.07(6) N 3 LL 4.19(14) 3.85(7) σ N decreases with N. Very high correlation r(σ,λ QCD ) 1.

63 Dimension Two Condensates? More examples of non-perturbative corrections: Polyakov loop (Megías et al. JHEP 0601 (2006)): ( L(T) = L PT (T) exp C ) 2 4N c T 2. Heavy qq free energy (Megías et al. PRD75 (2007)): F qq (r, T) = 4 α s 3 r e m Dr 2 3 α sm D +... }{{} PT Trace anomaly (Megías et al. PRD80 (2009)): παs ( C 2 1 2e m D r ) 6m D } {{ } NP C 2 ǫ 3p T 4 = 11 3 α2 s +O(α 5/2 s ) + 33 }{{} 4π α s T 2. }{{} PT NP Dimension two gluon condensate: C 2 g 2 A 2 0,a..

64 Polyakov loop Motivation Dimension Two Condensates? L(T) LO NLO T/T c PT : δ = log(µ/2πt) NP : b L = C 2 6T 2 c bl < T/T c < 6 r(δ, b L ) = Order L P (6T c ) δ b L r(b L,δ) χ 2 /dof L P = const 1.121(8) 1.72(5) O(α 3/2 ) 1.125(11) 0.72(20) 2.23(16) O(α 2 ) 1.123(9) 0.06(18) 2.15(11)

65 Trace Anomaly Motivation Dimension Two Condensates? (ε-3p)/t pert (ǫ 3P) T HTL 1-loops HTL 2-loops pert + b (T c /T) T/T c PT : δ = log(µ/2πt) NP : b = 3β g = (ǫ 3P) pert T 4 C 2 T 2 c b b ( ) 2 Tc T < T/T c < 4.54 r(δ, b ) = 0.983

66 Trace Anomaly Motivation Dimension Two Condensates? Fit using HTLpt at 1-loop Andersen 00, 2-loops Andersen 02, 3-loops Andersen 09 Order HTL /T 4 T=4.5Tc δ b r(b,δ) 1-loop ± (40) loops ± (28) loops 0.08(5) 2.3 ± ± Order δ b c r(b,δ) r(b, c ) 1-loop 0.7 ± ± ± loops 0.47(80) 3.58(40) 0.1 ± loops 2.3 ± ± ± (ǫ 3P) T 4 = (ǫ 3P) pert T 4 + b ( Tc T ) 2 ( ) 4 Tc + c. T

67 Conclusions: Motivation Dimension Two Condensates? AdS/QCD serves as a powerful tool to study the non-perturbative (NP) regime of QCD at zero and finite temperature. Model of conformal symmetry breaking in 5D based on dilatons. Numerical results of e.o.s. of QCD from Bekenstein-Hawking entropy formula and Gibbons-Hawking action agree. Good and unified description for: Equation of State (trace anomaly, pressure, entropy,...) Flavor dependence for T c. Polyakov loop and Spatial Wilson loops. Dilemma: Equation of State? Running coupling & Debye screening. The NP behaviour of QCD near and above T c is characterized by power corrections in T. These power corrections are signals of the contribution of a dimension two gluon condensate. Study of correlations gives us a tool to understand the duality between perturbative series and power corrections. Lattice data of observables at T = 0 and T 0 seem to confirm this duality.