Issues Non-Perturbative Thermal QCD from AdS/QCD * Collaborators: B. Galow, M. Ilgenfritz, J. Nian, H.J. Pirner, K. Veshgini Research Fellow of the Alexander von Humboldt Foundation Institute for Theoretical Physics, University of Heidelberg, Germany. QCD 2010: 15 th International QCD Conference July 1st 2010, Montpellier, France. Pirner et al. PLB679(2009), NPA833(2010), (EM) NPB834(2010), (EM) 0911.1680(2009); E.Megías et al., in preparation (2010).
Issues Issues 1 Motivation 2 3 at finite T
Issues Issues 1 Motivation 2 3 at finite T
Issues Issues 1 Motivation 2 3 at finite T
Motivation Motivation Interaction Measure in Gluodynamics Weak Coupling Expansion and Resummed Perturbation Theory E. Braaten and A. Nieto (1996), J.O. Andersen et al (1999). 0.5 0.4 ideal ( - 3 )/ 0.3 0.2 0.1 0 2 5-0.1-0.2 3 4 1 2 3 4 5 T/T c
Motivation Motivation ǫ 3P T 4 = a,p + b T 2 (ε-3p)/t 4 4 3.5 3 2.5 2 1.5 1 0.5 0-0.5 16 3 4 32 3 8 HTL 2-loops a + b (T c /T) 2 (ε-3p)/t 4 0.4 0.2 0-0.2-0.4 HTL 2-loops a 1 2 3 4 5 1 2 3 4 5 T/T c T/T c Perturbation Theory and Hard Thermal Loops only yield a!!.
Issues Motivation 1 Motivation 2 3 at finite T
Motivation 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.
Issues Motivation 1 Motivation 2 3 at finite T
Motivation 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 invariance 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
Issues Motivation 1 Motivation 2 3 at finite T
Motivation 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 Ansatz: β(α) = b 2 α+ ( 1 ( ) ) 2 [ β(α) exp 8 α 3α 9 0 M ] β(a) a 2 da, α = e φ. d 4 x h K. [ ( ) ( b2 b 2 α+ ᾱ β 0 α 2 b2 + 2ᾱ 2 β ] 0 ᾱ β 1 )α 3 e α/ᾱ. α << ᾱ = Ultraviolet: β(α) β 0 α 2 β 1 α 3 α >> ᾱ = Infrared: β(α) b 2 α
Motivation Heavy QQ potential and running coupling W(C) exp( S NG ) exp( t V q q (R)) V qq Ρ GeV 1 1 2 3 1 2 3 4 5 6 Ρ GeV 1 Α E 1.50 1.00 0.70 0.50 0.30 0.20 E GeV 1.0 1.5 2.0 3.0 5.0 7.0 10.0 15.0 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
Motivation Dilaton potential and warp factor V Α GeV 2 2 4 6 8 10 12 14 50 100 150 Α h z GeV 2 l s 2 1.2 1.0 0.8 0.6 0.4 0.2 h z l s 2 h z 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 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 (Confinement) 1.5 < b 2 < 2.37 (IR singularity repulsive to physical modes)
Issues Motivation at finite temperature 1 Motivation 2 3 at finite T
Motivation Black hole thermodynamics 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 + 1 ( ( ) 2 dτ f(r) dr 2 +r 2 dω 2 2 4rh 2 dρ 2 +ρ 2 + 1 ) ρ 0 2r h 4 dω2 2 }{{} dθ 2 τ τ = Periodicity: 2r h 2r h + 2π = τ τ + 4πr h =: τ +β 1 T = 8πMG 4 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
Issues Motivation at finite temperature 1 Motivation 2 3 at finite T
Motivation at finite temperature 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.
Motivation δ 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 2 + 3 b b + 3ḃ b ḟ f = b2 V(φ) f z 0 zh 0 Conformal solution: V(φ) = 12 l 2, φ = 0 = b(z) = l ( z, f(z) = 1 z du b(u) 3 u b(u) 3 z h ) 4, T = 1 πz h
Motivation [ β(α) = b 2 α+ b 2 α+( b2 ᾱ β 0 = V(α) = 12 l 2 T GeV 1.0 0.8 0.6 0.4 ( T c 272MeV 1 0 5 10 15 20 Α h Input: ) ( ) ) 2 [ β(α) exp 8 α 3α 9 0 1.2 f 1.0 0.8 0.6 0.4 0.2 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 10. 0.0 0.0 0.2 0.4 0.6 0.8 1.0 z GeV 1 T = 368 MeV
Motivation at finite temperature [ β(α) = b 2 α+ b 2 α+( b2 ᾱ β 0 Input: ) α 2 +( β2 2ᾱ 2 β 0 ᾱ β 1 ) α 3 ] e α/ᾱ z l b z 1.00 0.95 0.90 0.85 0.80 0.0 0.2 0.4 0.6 0.8 Α z 0.5 0.4 0.3 0.2 0.1 z GeV 1 0.2 0.4 0.6 0.8 T = 368MeV z GeV 1
Issues Motivation at finite temperature 1 Motivation 2 3 at finite T
Motivation 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 = 32 45 π2 T 3 =: s ideal (T) One can compute all the thermodynamics 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
Motivation at finite temperature Input β(α). e 3p T 4 3.0 P T 4 2.0 2.5 2.0 1.5 1 T 2 1.5 1.0 1.0 0.5 0.5 0.0 T Tc 0.0 T Tc 0 1 2 3 4 5 0 1 2 3 4 5 N c = 3, N f = 0, b 2 = 2.3, ᾱ = 0.46 p(t) T 4 = π3 l 3 (1 43 16G β 0α h + 29 ) (4β20 3β 1 )α 2h +... 1, α h 5 β 0 log(πt/λ)
Motivation at finite temperature Free Energy from: Bekenstein-Hawking entropy Classically Free Energy from: Gibbons-Hawking action S = A 4G 5 βf = S reg = G = πg 5 15 = F = V 3 16πG 5 ( 15G C ) f 4 β(α) α 2 TrF 2 µν, C f = 4πTb 3 (z h ) T s b T (z) = b 0 (z) [1+ Gl ( 3 z4 1+ 19 ) ] 12 β 0α 0 (z)+c2 b α2 0(z)+ +, z 0
Motivation at finite temperature Α z Α 0 z z 4 GeV 4 0.011 0.012 0.013 0.014 0.015 0.02 0.04 0.06 0.08 0.10 z GeV 1 α T (z) α 0 (z) z 4 = 45 8 G l 3 β 0 GeV 0.01 0.01 0.02 0.03 0.04 0.2 0.3 0.4 0.5 0.6 T GeV ( ( 11 1+ 6 β 0 β ) ) 1 α 0 (z)+c α β 2α0(z)+ 2 0 = Corrections in α 0 are very important for agreement.
Motivation at finite temperature Flavor dependence of the phase transition 300 250 200 TcAgree 150 very well with 100 50 J.Braun & H.Gies JHEP 1005 (2010). 0 0 2 4 6 8 10 N f ( T c = T Nf =0 1 κnf +O(Nf 2 )) κ = 0.1145, T Nf =0 = 273.3 MeV. Ex : T Nf =4 = 150 MeV.
Conclusions: Motivation at finite temperature AdS/QCD serves as a powerful tool to study the non-perturbative regime of QCD at zero and finite temperature. We consider a model of conformal symmetry breaking based on dilatons. Numerical results of equation of state of QCD from Bekenstein-Hawking entropy formula and Gibbons-Hawking action agree each other. Good and unified description for: Equation of State (trace anomaly, pressure, entropy,...) Polyakov loop. Spatial Wilson loops. Good description of flavor dependence for the phase transition temperature. We are working in other observables: Heavy Q Q free energy. Glueball spectrum. Mesons spectrum....
Motivation at finite temperature Backup slides
Polyakov Loop Motivation L(T) := P = 1.2 DXe Sw = semiclassically P = i at finite temperature S reg 0 = c R T + l αh 2πls 2 da e D(a) b(a)a 4 3 T 0 w i e S i w 0 e S 0 1.0 L 2 T 0.8 0.6 0.4 0.2 Lattice 2 + 1 flavors: Petreczky, PRD70 (2004), PRD77(2008). 0.0 0 1 2 3 4 5 6 L(T) = exp [c 0 + l2 2ls 2 T Tc( α 4 3 h 1+ 4 9 β 0α h + 1 )] ( ) 161β 2 81 0 + 72β 1 α 2 h +O(αh 3 ), ( ) 3 In contradiction with PT: L PT (T) = exp πα 2 +O(α 2 ). 4 3
Motivation Spatial Wilson Loops 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 [ ] σ s (T) = 1 2πls 2 α 4 3 h b 2 (z h ) = l2 2ls 2 πt 2 α 4 3 h 1 8 9 β 0α h + 2 81 (25β2 0 2β 1 )αh 2 +O(α3 h ). This is in contradiction with pqcd: σ pqcd T 2 α 2 s(t). See also Kajantie et al. PRD80 (2009). l x