Dilatons, Thermodynamics and Transport Coefficients in High T QCD

Transcription

1 Issues Dilatons, and Transport Coefficients in High Temperature QCD, D. Kharzeev and E. Levin 1 1 Nuclear Theory Group, Physics Department, Brookhaven National Laboratory, Upton, New York USA. Also: E. Ruiz Arriola and L.L. Salcedo 2 2 Dep. Física Atómica, Molecular y Nuclear, Granada U., Spain. April 22nd 2009, Institute for Theoretical Physics, Heidelberg, Germany. Some References: E. Megías et al. PRD75 (2007), NPB186 (2008), arxiv: (2009), paper in preparation (2009).

2 Issues Issues 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

3 Issues Issues 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

4 Issues Issues 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

5 Issues Issues 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

6 Issues Issues 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

7 QCD and 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

8 QCD and 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

9 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

10 QCD and QCD Lagrangian: L QCD = 1 4 F a µν F a µν + f q a f (iγ µd µ m f )q a f ; In the limit of massless quarks (m f = 0), it is Invariant under Chiral Left Right transformations. Invariant under Scale (x λx) transformations. Partition function (gluodynamics m f ) [ Z = DĀµ,a exp 1 ] 4g 2 d 4 x( F µν a )2

11 QCD and The Classical scale invariance is broken by quantum effects. They introduce a mass scale Λ QCD. Under a scale transformation: g g + σβ(g), (µ e σ µ) where β(g) = µ µ g(µ) is the beta function. L QCD L QCD + σβ(g) g L QCD. The scale anomaly can be computed as the trace of the Energy-Momentum Tensor, so it is also known as : θ µ µ = β(g) g L QCD = β(g) 2g F a µν F a µν Note that β(g) = 0 = θ µ µ = 0.

12 QCD and At finite temperature, from the partition function of Gluodynamics Z : log Z (1/4g 2 0 ) = V T ( F a µν )2. After renormalization and using standard thermodynamics relations: ( 4 T ) T β(g) log Z = T V 2g (F µν a )2 = θ µ µ = ǫ 3P. The trace anomaly is related to the interaction measure (ǫ 3P)/T 4, which provides a measure of the interaction. In PT up to two loops (J.I.Kapusta (1979)): ǫ 3P T 4 = N c(n 2 c 1) 1152π 2 β 0 g(t) 4 + O(g 5 )

13 QCD and The free gluon gas has = 0. Perturbation Theory says that the trace of the Energy-Momentum Tensor takes a nonzero value as a result of quantum corrections. Lattice data predicts a violent behaviour in powers of T. (Many groups: G. Boyd et al, NPB (1996), Y. Aoki et al (2006),...). PT predicts a smooth dependence on T, because g(t) log(t), so it is unable to reproduce this behaviour, even if more and more orders are included (Andersen, Ann.Phys.317,2005).

14 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

15 QCD and from Lattice data 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.

16 QCD and from Lattice data 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.

17 QCD and Perturbation theory vs Lattice data ǫ 3P T 4 = a,p + b T 2 (ε-3p)/t HTL 2-loops a + b (T c /T) T/T c (ε-3p)/t HTL 2-loops a T/T c Perturbation Theory and Hard Thermal Loops only yield a!!.

18 QCD and The fuzzy bag of Pisarski Low temperature (confined) glueball gas P glueball (T) e M G/T, M G T c P glueball (T c ) 0 High temperature (deconfined) free gluon gas P gluons (T) = b 0 2 T 4, b 0 = (N2 c 1)2π2 45 Pisarski s (temperature dependent) fuzzy bag, PTP 168 (2007). P(T) = P gluons (T) B fuzzy (T), T > T c, P(T c ) = P glueballs (T c ) = 0 Then B fuzzy = b 0 2 T 2 c T 2 = P = b 0 2 (T 4 T 2 T 2 c ) ǫ 3P T 4 = b 0 ( ) 2 Tc b 0 = 3.45 (3.5Fit!!!!) T

19 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

20 QCD and 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) exp( TV 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.

21 Issues QCD and Effective theory at finite Temperature 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

22 QCD and Effective theory at finite Temperature Scale invariance x µ λx µ is lost in real world. Several attempts to formulate an effetive theory of broken scale invariance in terms of Goldstone bosons (the dilaton): J. Schechter 1980, A.A. Migdal 1982, D. Kharzeev Formally couple the gluodynamics to the conformal background gravity g µν (x) = e h(x) δ µν, where h(x) single scalar field. S = d 4 x ( 1 g 8πG R 1 ) 4 gµν g λσ Fµλ a F νσ a e2h θ µ µ Upon substitution of θ µ µ = β(g) 2g F aµν Fµν a and Legendre transformation: L = ǫ v 1 ( m 2 2 eχ/2 ( µ χ) 2 + ǫ v + c 4 (F µν a )2) e χ (1 χ) 1 4 (F µν a )2

23 QCD and Effective theory at finite Temperature L = ǫ v m 2 1 Properties: ( 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) 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

24 Issues QCD and Effective theory at finite Temperature 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

25 QCD and Effective theory at finite Temperature Effective Theory at Finite Temperature Matsubara formalism: A µ (x 0 + β, x) = A µ (x 0, x), χ(x 0 + β, x) = χ(x 0, x) A µ (x 0, x) = + n= e iωnx 0 A µ (ω n, x), ω n = 2πTn Propagator: A µ A µ 1. At high T, non static modes ω p 2 +ωn 2 n 0 decouple = effective theory of static modes: A µ ( x), χ( x) Z = Dimensional Reduction DA µ (x 0, x)dχ(x 0, x) e R β 0 dx 0 R d 3 x L(A µ,χ) DA µ ( x)dχ( x) e β R d 3 x L 3D (A µ,χ)

26 QCD and Effective theory at finite Temperature Effective Theory at Finite Temperature Integration of ω n 0 at 1-loop and expansion in powers of A µ = 3 dim. red. theory of Gluodynamics. L 3YM = 1 4 (F ij a )2 1 ( ( i A 0,a ) 2 + m 2 ) D 2 A2 0,a + O(A 4 0 ) Integration of ω n 0 at 1-loop and expansion in powers of χ = 3 dim. red. theory with dilatons. L 3D = ( ǫ v cl 3YM ) (e χ (1 χ) m2 T 2 ) 24 ǫ v eχ/2 (1 + χ) }{{} V(χ,T) We assume µ χ = 0 = no kinetic term for dilatons. L 3YM

27 QCD and Classical equations of motion Effective theory at finite Temperature Classical equation for χ: 0 = (χe χ/2 + m2 T 2 ) (3 + χ) ( ǫ v cl 3YM ). 48 ǫ v }{{} ρ(t) Classical equation for A µ : where: F(χ, T) = c 4 Solution gives: i (F i A 0,a ) m 2 D FA 0,a = 1 4 gc aδ( x) [ e χ (1 χ) ρ(t)e χ/2 (1 + χ) ]. 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

28 QCD and Classical equations of motion Effective theory at finite Temperature How to fix E 0? At T = 0: χe χ/2 ( ǫ v + c 4 (F a µν) 2) min = 0 A valid solution is (F a µν) 2 min = 2( B a2 E a2 ) 2 E a2 min = 4 ǫ v /c Unlike in pqcd (F a µν) 2 min = 0, the minimun of the effective theory corresponds to a finite chromoelectric field E 0. χ r [GeV -1 ] T=0 T=T c T=2Tc V(χ) T=0.1T c T=0.3T c T=1.5T c χ

29 QCD and The Gluon Propagator Effective theory at finite Temperature From the Yang-Mills equations of motion: A 0,a ( x) = After Fourier transformation: E 0 e m Drˆn + gc a F(χ)m D F(χ)4πr e m Drˆr. A 0,a ( g k) = F(χ)( k 2 + md 2 ) 8πE 0 F(χ)( k 2 + m, D 2 )2 so the gluon propagator is D 00 ( 1 k) = k 2 + md 2 + mg 2 ( k 2 + m, D 2 m2 )2 G = 8πE 0 g

30 QCD and Strong Coupling in the Infrared Effective theory at finite Temperature Define the electric susceptibility of the QCD vacuum: ǫ(χ, T) = 4F(χ, T) The effective Lagrangian can be written (Āa µ = ga a µ): L 3D = ǫ v f(χ, T) 1 ǫ(χ, T) ( ( F a 4 g 2 ij ) 2 + 2( i Ā 0,a ) 2 + 2m DĀ2 2 0,a + ) This implies the behaviour of the renormalized strong coupling: α s (χ, T) = α s(m 0 ) ǫ(χ, T) = α s (M 0 ) ǫ(χ, 0) + f(χ)t 2

31 Issues QCD and Effective theory at finite Temperature 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

32 QCD and Effective theory at finite Temperature From the gluon propagator, we get: (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 3 β(g) )2 = (Pert.) }{{} π g m2 G m DT. }{{} T 4 T 2 It s straightforward to compute all the thermodynamics quantities: P(T) T dt Pressure : T 4 = T (T 3 ) = (Pert.) + }{{} 2π β(g)m2 G }{{ T 2 } T 0 1/T 2 Entropy : s T 3 = + 4P T 4 = (Pert.) + 3 β(g) mg 2 m D π g T 3.

33 QCD and Effective theory at finite Temperature Lattice data: G. Boyd et al. NPB469 (1996). (ε-3p)/t pert HTL 1-loops HTL 2-loops pert + b (T c /T) 2 P/P ideal P pert P HTL + P NP P pert +P NP T/T c T/T c m G = 1.18 ± 0.10 GeV

34 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

35 QCD and Fluid slightly disturbed from equilibrium = compute θ µν as an expansion in gradients θ ij = δ ij P η [ }{{} i u j + j u i 23 ] δ ij k u k ζ δ }{{} ij k u k Shear Bulk where u i θ0 i ǫ+p (local flow velocity). Mathematical tool: Effective kinetic theory = Boltzmann equation f( x, p, t) t + v p x f( x, p, t) = C[f], Perturbative results (P.Arnold et al 2000, 2006): η = c η T 3 g 4 log g 1, g 2 ζ = c ζ log g 1 T 3 v p p E p.

36 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

37 QCD and f t + v p x f = C[f], It can be linearized (P.Arnold et al 2000): v p p E p. f(x, p) = f eq (x, p) + f 1 (x, p) = f eq (x, p) + f eq (1 ± f eq )β 2 X i j (x)i i j χ( p ) Boltzmann equation can be written: S( p) = [Cχ]( p), where S = Tf eq (1 ± f eq )q( p), q( p) = p vp 3 vs 2 βe p β. It follows from a maximization of the functional: Q[χ] = χ, S 1 χ, Cχ, 2 f, g := f( p)g( p) p Finally: η = 2 15 Q[χ max]

38 QCD and Collision integrals (A) (B) (C) (D) (E) (F) (G) (H) (I) (J ) χ, Cχ = β3 8 abcd p i M ab cd 2 δ( i p i )f a eq f b eq (1±f c eq )(1±f d eq )(χa +χ b χ c χ d ) 2 M diagram C 2 s2 + u 2 t 2 ( s 2 + u 2) ( ) 2 1 t + m2 G t 2.

39 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

40 QCD and shear entropy Lattice data Meyer PRD76 (2007) T Tc m G = 0.9(2) GeV η 1 s 4π T 3 ( η = c η g 4 log g 1 1 mg 2 + 2g 2 (1 + g 2 ) 2 T g 2 m 4 ) 1 G 12g 4 (1 + g 2 ) 3 T 4, T m G = Perturbative result is recovered. T m G = Power corrections become dominant.

41 Issues QCD and 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

42 QCD and Two approximations: Diagramatic computation (Megías, Kharzeev and Levin, 2009). Kubo formula (D. Kharzeev et al. 2008): ζ = 1 9 lim 1 ω 0 d 4 x e iωt [θ µ µ ω (x), θµ µ (0)]. From low energy theorems of Energy-Momentum Tensor: ζ 1 [ T 5 ] 9ω 0 T (T) + 16 ǫ v, ω 0 ( ) GeV From previous results for the trace anomaly: g 2 ζ c ζ log(g 1 ) T ( mg 2 πω β 1 + β ) T. 0 g

43 QCD and Lattice data: H. Meyer PRL 100 (2008) bulk entropy 0.8 bulk entropy T Tc T Tc T > 1.2T c : m G = 0.8(1)GeV

44 QCD and Shear vs Lattice data: H. Meyer (2007), (2008) shear entropy, bulk entropy T Tc : m G = 0.9(2)GeV : m G = 0.8(1)GeV

45 Issues QCD and Polyakov loop 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

46 QCD and Polyakov loop A 2 0,a = (N2 c 1)T d 3 p (2π) 3 D 00( p) = 2 π m DT }{{} T 2 + m2 G T πm D }{{} T 0 1 Polyakov loop: L = N c Tr c e iga 0,a/T e g2 A 2 0,a /(4NcT 2 ) L(T) LO NLO T/T c

47 QCD and Gluon Asymmetry: Lattice data Polyakov loop 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 A 2 0,a NP 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 = g2 A 2 0,a }{{ P + g }} {{ = g 2 A 2 0,a } = c p + c NP /T 2. T 2 cte i

48 Issues QCD and Polyakov loop 1 2 QCD and 3 Effective theory at finite temperature 4 5 Polyakov loop

49 QCD and Polyakov loop Correlation functions of Polyakov loops define the free energy of a heavy qq pair (O.Kaczmarek et al, PLB543(2002)): e Fq q( x,t)/t+c(t) = 1 Nc 2 tr c Ω( x) tr c Ω ( 0). Pert. evaluation of Free Energy Expand Ω in powers of ga 0 : d 3 k A 0,a ( x)a 0,b ( y) = δ ab T (2π) 3 ei k ( x y) D 00 ( k). At leading order (O(g 2 )) and next to leading order (O(g 3 )): F 1 (r, T) = N2 c 1 ( g 2 2N c 4πr + g2 mg 2 T ) e m Dr 8πmD N2 c 1 g 2 m D 2N c 4π + g2 mg 2. 6πm D

50 QCD and Polyakov loop Singlet Free Energy N c = 3, N f = 0 Lattice data (O. Kaczmarek PRD70 (2004)) F 1 (r,t) [GeV] rt T/T c

51 QCD and Assymptotic limits Polyakov loop Taking the assymptotic limits: T 0: F 1 (r, T) T 0 N2 c 1 g 2 1 2N c 4π r + g3 mg 2 T r V q q (r). 6πm }{{ D } σ V q q (r) well known from lattice: S.Necco NPB622(2002). r : F (T) = F 1 (r, T) = N2 c 1 g 2 m D 2N c 4π + g2 m 2 G 6πm D. L(T) = e F (T)/2T also known from lattice: O.Kaczmarek. From a fit of V q q at T = 0 (F 1 (r, T = 0)): σ = (0.42(1) GeV) 2 = m G = 1.26(5) GeV. From a fit of the Polyakov loop (F 1 (r =, T)): c NP = (0.49(7) GeV) 2 = m G = 1.29(19) GeV.

52 QCD and Polyakov loop The chromoelectric field in the vacuum induces a dimension two gluon condensate: g 2 A 2 0,a NP = N2 c 1 g 2 T E 0. 8π m D Observable m G [GeV] g 2 A 2 0,a NP Polyakov loop 1.29(19) (0.84(6) GeV) 2 Heavy qq free energy 1.38(9) (0.90(5) GeV) (10) (0.77(6) GeV) 2 0.9(2) (0.59(13) GeV) 2 0.8(1) (0.52(7) GeV) 2 Energy Loss (in preparation) 0.95(15) (0.62(10) GeV) 2

53 Conclusions: QCD and Polyakov loop We describe strong-qgp regime of QCD within an effective theory of broken scale invariance. Dilatons induce a finite chromoelectric field in the vacuum which leads to confinement. Non perturbative behaviour near and above T c is characterized by power corrections in T. These power corrections are high energy trace of non-perturbative low energy effects. Lattice data of pressure, shear viscosity and bulk viscosity are well reproduced by just choosing a single parameter E 0. This result seems to imply an unified and coherent picture for the description of sqgp observables.