Thermal Transport and Energy Loss in Non-critical Holographic QCD

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1 Thermal Transport and Energy Loss in Non-critical Holographic QCD Umut Gürsoy University of Utrecht DAMTP - Cambridge - November 5, 2009 U.G., E. Kiritsis, F. Nitti, L. Mazzanti ongoing U.G., E. Kiritsis, F. Nitti, G. Michalogiorgakis arxiv: U.G., E. Kiritsis, F. Nitti, L. Mazzanti arxiv: U.G., E. Kiritsis, F.Nitti arxiv: U.G., E. Kiritsis arxiv: Thermal Transport and Energy Loss in Non-critical Holographic QCD p.1

2 QCD at high Temperature QCD at extreme conditions, vital to understand our universe. RHIC (Au + Au) probes QCD about MeV with s = 200 GeV per nucleon. A simulation of universe 10 6 secs after big-bang. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.2

3 QCD at high Temperature QCD at extreme conditions, vital to understand our universe. RHIC (Au + Au) probes QCD about MeV with s = 200 GeV per nucleon. A simulation of universe 10 6 secs after big-bang. RHIC data (e.g. elliptic flow, nuclear modification factors) prefer an almost but not fully ideal fluid model for QGP. Strongly coupled QGP with shear viscosity η/s pqcd fails! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.2

4 QCD at high Temperature QCD at extreme conditions, vital to understand our universe. RHIC (Au + Au) probes QCD about MeV with s = 200 GeV per nucleon. A simulation of universe 10 6 secs after big-bang. RHIC data (e.g. elliptic flow, nuclear modification factors) prefer an almost but not fully ideal fluid model for QGP. Strongly coupled QGP with shear viscosity η/s pqcd fails! Most observables (viscosity coefficients, energy loss of hard partons, drag force, momentum broadening ) involve dynamical phenomena. Lattice QCD not adequate, huge errors in analytic continuation from Euclidean to Lorentzian time. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.2

5 QCD at high Temperature QCD at extreme conditions, vital to understand our universe. RHIC (Au + Au) probes QCD about MeV with s = 200 GeV per nucleon. A simulation of universe 10 6 secs after big-bang. RHIC data (e.g. elliptic flow, nuclear modification factors) prefer an almost but not fully ideal fluid model for QGP. Strongly coupled QGP with shear viscosity η/s pqcd fails! Most observables (viscosity coefficients, energy loss of hard partons, drag force, momentum broadening ) involve dynamical phenomena. Lattice QCD not adequate, huge errors in analytic continuation from Euclidean to Lorentzian time. HOLOGRAPHIC APPROACH IS VERY PROMISING! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.2

6 Logic of this talk Thermal Transport and Energy Loss in Non-critical Holographic QCD p.3

7 Logic of this talk Consider SU(N) gluon plasma at finite temperature in equilibrium Thermal Transport and Energy Loss in Non-critical Holographic QCD p.3

8 Logic of this talk Consider SU(N) gluon plasma at finite temperature in equilibrium Write down the minimal (5D, 2 ) holographic dual theory. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.3

9 Logic of this talk Consider SU(N) gluon plasma at finite temperature in equilibrium Write down the minimal (5D, 2 ) holographic dual theory. Fix the background completely by qualitative requirements from gauge theory + fitting equilibrium lattice data Thermal Transport and Energy Loss in Non-critical Holographic QCD p.3

10 Logic of this talk Consider SU(N) gluon plasma at finite temperature in equilibrium Write down the minimal (5D, 2 ) holographic dual theory. Fix the background completely by qualitative requirements from gauge theory + fitting equilibrium lattice data Perturb away from equilibrium and compute transport coefficients Compare with RHIC data + simulations. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.3

11 Outline Minimal Model: improved holographic QCD" Thermodynamics in improved holographic QCD Review of RHIC physics Transport and energy loss - Viscosity: shear and bulk - Spectral densities at large ω - Energy loss of heavy quarks: Drag force - Langevin dynamics and diffusion constants Discussion and Outlook Thermal Transport and Energy Loss in Non-critical Holographic QCD p.4

12 Minimal holographic dual Thermal Transport and Energy Loss in Non-critical Holographic QCD p.5

13 Minimal holographic dual Only two dimension-4 operators: the stress-tensor T µν and the gluon condensate TrF 2. SVZ sum rules: sufficient to capture low-energy dynamics Thermal Transport and Energy Loss in Non-critical Holographic QCD p.5

14 Minimal holographic dual Only two dimension-4 operators: the stress-tensor T µν and the gluon condensate TrF 2. SVZ sum rules: sufficient to capture low-energy dynamics Most economic gravitational set-up: 5D: Minkowski + radial coordinate (energy scale) metric g µν T µν and dilaton φ TrF 2 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.5

15 Minimal holographic dual Only two dimension-4 operators: the stress-tensor T µν and the gluon condensate TrF 2. SVZ sum rules: sufficient to capture low-energy dynamics Most economic gravitational set-up: 5D: Minkowski + radial coordinate (energy scale) metric g µν T µν and dilaton φ TrF 2 Large N c limit gravitational interactions can be ignored G N 0. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.5

16 Minimal holographic dual Only two dimension-4 operators: the stress-tensor T µν and the gluon condensate TrF 2. SVZ sum rules: sufficient to capture low-energy dynamics Most economic gravitational set-up: 5D: Minkowski + radial coordinate (energy scale) metric g µν T µν and dilaton φ TrF 2 Large N c limit gravitational interactions can be ignored G N 0. Running coupling: Extremely crucial for a dynamical generation of Λ QCD and realistic T-dependence in thermodynamics E(T), S(T), F(T). Radial dependence in the dilaton λ t = e φ(r) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.5

17 Improved HQCD U.G, Kiritsis; U.G. Kiritsis, Nitti 07 Gravitational dual of pure YM in 2 effective 5D non-critical string theory: S = M 3 pn 2 c d 5 x g {R + ( λ)2 λ 2 } V (λ) + G.H. Running t Hooft coupling λ t (E) λ(r) (dilaton), Tr F 2 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.6

18 Improved HQCD U.G, Kiritsis; U.G. Kiritsis, Nitti 07 Gravitational dual of pure YM in 2 effective 5D non-critical string theory: S = M 3 pn 2 c d 5 x g {R + ( λ)2 λ 2 } V (λ) + G.H. Running t Hooft coupling λ t (E) λ(r) (dilaton), Tr F 2 Asymptotics of V (λ) fixed by gauge theory input: In the UV: V (λ) = v 0 + v 1 λ + v 2 λ Gaussian f.p. as λ 0 (UV) AdS, non-zero V 0. - Log running of λ t (b 0 log E) 1 non-zero V 1 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.6

19 Improved HQCD U.G, Kiritsis; U.G. Kiritsis, Nitti 07 Gravitational dual of pure YM in 2 effective 5D non-critical string theory: S = M 3 pn 2 c d 5 x g {R + ( λ)2 λ 2 } V (λ) + G.H. Running t Hooft coupling λ t (E) λ(r) (dilaton), Tr F 2 Asymptotics of V (λ) fixed by gauge theory input: In the UV: V (λ) = v 0 + v 1 λ + v 2 λ Gaussian f.p. as λ 0 (UV) AdS, non-zero V 0. - Log running of λ t (b 0 log E) 1 non-zero V 1 In the IR: V (λ) λ 4 3(log λ) 1 2 as λ Linear quark potential V q q = σ s L + Gapped and discrete glueball spectrum m 2 n n First order deconfinement transition at non-zero T c. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.6

20 The logic/critic of the approach Thermal Transport and Energy Loss in Non-critical Holographic QCD p.7

21 The logic/critic of the approach We expect V (φ) l 2 s l 2 so l/l s 1 One shows a posteriori: R s 1 near UV but R s 0 in the IR. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.7

22 The logic/critic of the approach We expect V (φ) l 2 s l 2 so l/l s 1 One shows a posteriori: R s 1 near UV but R s 0 in the IR. In the IR, we trust the 2 gravitational picture. In the UV, we trust pqcd. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.7

23 The logic/critic of the approach We expect V (φ) l 2 s l 2 so l/l s 1 One shows a posteriori: R s 1 near UV but R s 0 in the IR. In the IR, we trust the 2 gravitational picture. In the UV, we trust pqcd. How to glue" the two pieces of information consistently? Replace the unknown gravitational dual near the UV with phenomenological requirements from QCD. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.7

24 The logic/critic of the approach We expect V (φ) l 2 s l 2 so l/l s 1 One shows a posteriori: R s 1 near UV but R s 0 in the IR. In the IR, we trust the 2 gravitational picture. In the UV, we trust pqcd. How to glue" the two pieces of information consistently? Replace the unknown gravitational dual near the UV with phenomenological requirements from QCD. We shall see that the present model achieves this in a neat way. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.7

25 ihqcd Thermodynamics U.G., Kiritsis, Mazzanti, Nitti 08 Fix the dilaton potential: V = 12 l 2 {1 + V 0 λ + V 1 λ 4/3 log } (1 + V 2 λ 4 )1 3 + V 3 λ 2 2 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.8

26 ihqcd Thermodynamics U.G., Kiritsis, Mazzanti, Nitti 08 Fix the dilaton potential: V = 12 l 2 {1 + V 0 λ + V 1 λ 4/3 log } (1 + V 2 λ 4 )1 3 + V 3 λ 2 2 Two sol ns with AdS asymptotics ds 2 = e A(r) dt 2 f(r) + dx dr2 f(r) : Thermal Gas No horizon, compact Euclidean time thermal gas of glueballs. Black-hole Horizon at λ h gluon plasma. Hawking-Page transition deconfienment transition at T c. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.8

27 ihqcd Thermodynamics U.G., Kiritsis, Mazzanti, Nitti 08 Fix the dilaton potential: V = 12 l 2 {1 + V 0 λ + V 1 λ 4/3 log } (1 + V 2 λ 4 )1 3 + V 3 λ 2 2 Two sol ns with AdS asymptotics ds 2 = e A(r) dt 2 f(r) + dx dr2 f(r) : Thermal Gas No horizon, compact Euclidean time thermal gas of glueballs. Black-hole Horizon at λ h gluon plasma. Hawking-Page transition deconfienment transition at T c. Free energy from S BH S TG. Parameter fixing: V 0, V 2 fixed by scheme independent β-function coefficients (b 0 and b 1 ), V 1, V 3 fixed by the latent heat L h and S(2T c ) (lattice). Thermal Transport and Energy Loss in Non-critical Holographic QCD p.8

28 ihqcd Thermodynamics U.G., Kiritsis, Mazzanti, Nitti 08 Fix the dilaton potential: V = 12 l 2 {1 + V 0 λ + V 1 λ 4/3 log } (1 + V 2 λ 4 )1 3 + V 3 λ 2 2 Two sol ns with AdS asymptotics ds 2 = e A(r) dt 2 f(r) + dx dr2 f(r) : Thermal Gas No horizon, compact Euclidean time thermal gas of glueballs. Black-hole Horizon at λ h gluon plasma. Hawking-Page transition deconfienment transition at T c. Free energy from S BH S TG. Parameter fixing: V 0, V 2 fixed by scheme independent β-function coefficients (b 0 and b 1 ), V 1, V 3 fixed by the latent heat L h and S(2T c ) (lattice). M p fixed by high T asymptotics of the free energy in SU(N) YM: f π2 45 N2 T 4 : M p l = (45π) 1 3. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.8

29 ihqcd Thermodynamics contiued Deconfiniment transition at T c = 247 MeV (lattice: T c = 260 MeV.) Comparison to Boyd et al. 96 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.9

30 ihqcd Thermodynamics contiued Deconfiniment transition at T c = 247 MeV (lattice: T c = 260 MeV.) Comparison to Boyd et al. 96 The free energy: 0.01 F N c 2 T c 4 V T T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.9

31 ihqcd Thermodynamics contiued Deconfiniment transition at T c = 247 MeV (lattice: T c = 260 MeV.) Comparison to Boyd et al. 96 The free energy: F N 2 c T 4 c V 3 T T min T T c T c 0.03 Λ c Λ h Thermal Transport and Energy Loss in Non-critical Holographic QCD p.9

32 ihqcd Thermodynamics contiued Deconfiniment transition at T c = 247 MeV (lattice: T c = 260 MeV.) Comparison to Boyd et al. 96 The free energy: F N 2 c T 4 c V 3 T T min T T c T c 0.03 Λ c Λ h Big and Small black-hole solutions, like N = 4 on S 3 Existence of T min phase transition at T c > T min Thermal Transport and Energy Loss in Non-critical Holographic QCD p.9

33 Survey of thermodynamical quantities I Comparison to Panero Entropy density s / T 3, normalized to the SB limit SU(3) SU(4) SU(5) SU(6) SU(8) improved holographic QCD model T / T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.10

34 Survey of thermodynamical quantities III Comparison to Panero Pressure p / T 4, normalized to the SB limit SU(3) SU(4) SU(5) SU(6) SU(8) improved holographic QCD model T / T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.11

35 Survey of thermodynamical quantities IV Comparison to Panero Trace of the energy-momentum tensor / T 4, normalized to the SB limit of p / T SU(3) SU(4) SU(5) SU(6) SU(8) improved holographic QCD model T / T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.12

36 Survey of thermodynamic quantities V Comparison to Boyd et al. 96 Thermodynamic functions and the speed of sound: free gas e T 4 2 N c 3 s 4 T 3 2 N c 3 p T 4 N c T T c 2 c s free gas T T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.13

37 Heavy ion collisions (Review by Romatschke 09) SPS(CERN), RHIC(Brookhaven), upcoming LHC(CERN) Au + Au collisions at s/n GeV Thermal Transport and Energy Loss in Non-critical Holographic QCD p.14

38 Heavy ion collisions (Review by Romatschke 09) SPS(CERN), RHIC(Brookhaven), upcoming LHC(CERN) Au + Au collisions at s/n GeV Longitudinal evolution of the QG plasma after collision: 1.5 time [a.u] IV τ= const III τ= const II τ= const I collision -0.5 pre-collision longitudinal direction [a.u.] Thermal Transport and Energy Loss in Non-critical Holographic QCD p.14

39 Heavy ion collisions (Review by Romatschke 09) SPS(CERN), RHIC(Brookhaven), upcoming LHC(CERN) Au + Au collisions at s/n GeV Longitudinal evolution of the QG plasma after collision: 1.5 time [a.u] IV τ= const III τ= const II τ= const I collision -0.5 pre-collision longitudinal direction [a.u.] Equillibration time τ QGP 1 fm. Hydrodynamic phase (phase II) τ 5 10 fm. This phase relativistic fluid. Idealization: Infinite, static fluid. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.14

40 Bulk viscosity Thermal Transport and Energy Loss in Non-critical Holographic QCD p.15

41 Bulk viscosity The only other characteristic parameter of the fluid at O( u) the bulk viscosity, ζ: T µν = (ǫ + p)u µ u ν pg µν ( + P µα P [η νβ α u β + β u α 2 ) 3 g αβ u + O( u) 2 ] + ζg αβ u Thermal Transport and Energy Loss in Non-critical Holographic QCD p.15

42 Bulk viscosity The only other characteristic parameter of the fluid at O( u) the bulk viscosity, ζ: T µν = (ǫ + p)u µ u ν pg µν ( + P µα P [η νβ α u β + β u α 2 ) 3 g αβ u + O( u) 2 ] + ζg αβ u Shear viscosity η: For all 2 theories η s = 1 4π 0.08 Buchel and Liu 03 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.15

43 Bulk viscosity The only other characteristic parameter of the fluid at O( u) the bulk viscosity, ζ: T µν = (ǫ + p)u µ u ν pg µν ( + P µα P [η νβ α u β + β u α 2 ) 3 g αβ u + O( u) 2 ] + ζg αβ u Shear viscosity η: For all 2 theories η s = 1 4π 0.08 Buchel and Liu 03 Bulk viscosty ζ: What is already known from field theory and lattice? Thermal Transport and Energy Loss in Non-critical Holographic QCD p.15

44 Field theory computation Kharzeev, Tuchin,Karsch 07 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.16

45 Field theory computation Kharzeev, Tuchin,Karsch 07 Kubo s linear response theory: ζ = 1 9 lim ω 0 1 ω ImG R(ω, 0) = π 9 lim ω 0 1 ω ρ(ω) where G R (ω, p) = i R d 3 xdte iωt i p x θ(t) P 3 i,j=1 [T ii(t, x), T jj (0, 0)]. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.16

46 Field theory computation Kharzeev, Tuchin,Karsch 07 Kubo s linear response theory: ζ = 1 9 lim ω 0 1 ω ImG R(ω, 0) = π 9 lim ω 0 1 ω ρ(ω) where G R (ω, p) = i R d 3 xdte iωt i p x θ(t) P 3 i,j=1 [T ii(t, x), T jj (0, 0)]. Low-energy theorems (SVZ) at finite T: lim ω 0 G E (ω, 0) = T d dt T µ µ LHS by analytic continuation G E (ω, 0) = 2 0 dω ρ(ω) ω. RHS by thermodynamics T µ µ = e(t) 3p(T). Thermal Transport and Energy Loss in Non-critical Holographic QCD p.16

47 Field theory computation Kharzeev, Tuchin,Karsch 07 Kubo s linear response theory: ζ = 1 9 lim ω 0 1 ω ImG R(ω, 0) = π 9 lim ω 0 1 ω ρ(ω) where G R (ω, p) = i R d 3 xdte iωt i p x θ(t) P 3 i,j=1 [T ii(t, x), T jj (0, 0)]. Low-energy theorems (SVZ) at finite T: lim ω 0 G E (ω, 0) = T d dt T µ µ LHS by analytic continuation G E (ω, 0) = 2 0 dω ρ(ω) ω. RHS by thermodynamics T µ µ = e(t) 3p(T). One needs to make an ansatz for density to extract ζ: ρ(ω) = 9ζ π ω 2 0 (T)ω ω 2 0 (T)+ω2. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.16

48 Field theory computation Kharzeev, Tuchin,Karsch 07 Kubo s linear response theory: ζ = 1 9 lim ω 0 1 ω ImG R(ω, 0) = π 9 lim ω 0 1 ω ρ(ω) where G R (ω, p) = i R d 3 xdte iωt i p x θ(t) P 3 i,j=1 [T ii(t, x), T jj (0, 0)]. Low-energy theorems (SVZ) at finite T: lim ω 0 G E (ω, 0) = T d dt T µ µ LHS by analytic continuation G E (ω, 0) = 2 0 dω ρ(ω) ω. RHS by thermodynamics T µ µ = e(t) 3p(T). One needs to make an ansatz for density to extract ζ: ρ(ω) = 9ζ π ω 2 0 (T)ω ω 2 0 (T)+ω2. One finds that ζ/s increases as T T c from above, but the over-all size of ζ is ambigious... need to know ρ(ω) precisely. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.16

49 A lattice computation Meyer 08 Ζ s T Tc Huge statistical errors Huge systematic errors due to analytic continuation, assumption on the spectrum. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.17

50 Questions to be answered by holography How significant is ζ near T c? If so, what is the holographic reason for the rise near T c? More generally: Do we see similar profile for ζ/s(t) in holographic models? Thermal Transport and Energy Loss in Non-critical Holographic QCD p.18

51 Holographic computation Kubo s linear response theory: ζ = 1 9 lim ω 0 1 ω ImG R(w, 0) Holography: Im G R Flux of isotropic gravitons absorbed at the horizon h ii h ii h iih ii. More complicated than shear because h ii mix with dilaton fluctuations δφ. Derive the fluctuation equations for h ii, pick up the gauge δφ = 0, Fluctuations decouple in the smart gauge! Gubser et al 08: Define X = φ /A ( ) ( ) h ii + 3A + 2 X X + f f h ii + ω 2 f X f 2 fx h ii = 0 Boundary conditions: - h ii (φ = ) = 1 and, - In-falling wave at horizon h ii c b (r h r) iω 4πT Read off c b (ω, T) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.19

52 Results I: Comparison to Meyer 08 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.20

53 Results I: Comparison to Meyer 08 Ζ s T Tc - Near UV, vanishes as expected: ideal gluon gas at high T - Near T c Peak, much smaller than lattice expectations! - Agreement with another holographic model Gubser et al. 08 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.20

54 Effects of bulk viscosity on observables Thermal Transport and Energy Loss in Non-critical Holographic QCD p.21

55 Effects of bulk viscosity on observables Hydrodynamic simulations: H.Song, U.Heinz 09 ζ/s ζ/s= { C*2η/s(1/3 - c s2 ) (QGP) 0 (HRG) C=2 C= T (GeV) The function C(T) = v 2 ζ(t) 2η(1/3 c 2 s(t)) Au+Au, b=7 fm, EOS L e 0 =30 GeV/fm 3, τ 0 =0.6 fm/c T dec =130 MeV Buchel 07 A dynamical bound on ζ C(T) > 1. η/s ζ/s 0 C=0 1/4π C=0 1/4π C=1 1/4π C= p T (GeV) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.21

56 Effects of bulk viscosity on observables Hydrodynamic simulations: H.Song, U.Heinz 09 ζ/s ζ/s= { C*2η/s(1/3 - c s2 ) (QGP) 0 (HRG) C=2 C= T (GeV) The function C(T) = v 2 ζ(t) 2η(1/3 c 2 s(t)) Au+Au, b=7 fm, EOS L e 0 =30 GeV/fm 3, τ 0 =0.6 fm/c T dec =130 MeV Buchel 07 A dynamical bound on ζ C(T) > 1. η/s ζ/s 0 C=0 1/4π C=0 1/4π C=1 1/4π C= p T (GeV) However direct lattice Meyer 08 ζ/s 0.8 near T c! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.21

57 Results II: The function C(T) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.22

58 Results II: The function C(T) Ζ Η c 2 s T T c Thermal Transport and Energy Loss in Non-critical Holographic QCD p.22

59 Results II: The function C(T) Ζ Η c 2 s T T c - Buchel s bound is satisfied: C(T) > 1. - Indeed C 1.5 as the hydro simulations assumes - Effects of ζ on the elliptic flow are small: At p T = 0.5 GeV: η/s = % suppression, ζ/s(t c ) = % suppression in v 2. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.22

60 Holographic reason for the peak Ζ s T Tc Thermal Transport and Energy Loss in Non-critical Holographic QCD p.23

61 Holographic reason for the peak Ζ s T Tc - Holographic explanation of the rise: due to small BH branch! - Existence of T min rise near T c. - Color confinement in zero-t theory peak near T c at finite T! U.G, Kiritsis, Nitti, Mazzanti 08 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.23

62 Spectral densities Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

63 Spectral densities The fluctuation equations for shear and bulk fluctuations can also be solved for finite ω. to obtain ρ(ω) = 1 π Im G R(ω). One has ρ(ω) = 1 π Im G R(ω, p) = F(ω) 16π 2 G 5, Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

64 Spectral densities The fluctuation equations for shear and bulk fluctuations can also be solved for finite ω. to obtain ρ(ω) = 1 π Im G R(ω). One has ρ(ω) = 1 π Im G R(ω, p) = F(ω) 16π 2 G 5, Numerical computation for arbitrary ω is ongoing work. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

65 Spectral densities The fluctuation equations for shear and bulk fluctuations can also be solved for finite ω. to obtain ρ(ω) = 1 π Im G R(ω). One has ρ(ω) = 1 π Im G R(ω, p) = F(ω) 16π 2 G 5, Numerical computation for arbitrary ω is ongoing work. One can obtain exact results by WKB approximation: Transform the fluctuation equations into Schrodinger form and apply WKB for ω/t 1: U.G., Kiritsis, to appear ρ s (ω) = N2 c 360π 2ω4, ρ b (ω) = N2 c 540π 2ω4 b 2 0λ 2 t. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

66 Spectral densities The fluctuation equations for shear and bulk fluctuations can also be solved for finite ω. to obtain ρ(ω) = 1 π Im G R(ω). One has ρ(ω) = 1 π Im G R(ω, p) = F(ω) 16π 2 G 5, Numerical computation for arbitrary ω is ongoing work. One can obtain exact results by WKB approximation: Transform the fluctuation equations into Schrodinger form and apply WKB for ω/t 1: U.G., Kiritsis, to appear ρ s (ω) = N2 c 360π 2ω4, ρ b (ω) = N2 c 540π 2ω4 b 2 0λ 2 t. Contrast them with the perturbative Yang-Mills computations: H. Meyer 07 ρ s (ω) = N2 c 160π 2ω4, ρ b (ω) = N2 c 64π 2ω4 b 2 0λ 2 t. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

67 Spectral densities The fluctuation equations for shear and bulk fluctuations can also be solved for finite ω. to obtain ρ(ω) = 1 π Im G R(ω). One has ρ(ω) = 1 π Im G R(ω, p) = F(ω) 16π 2 G 5, Numerical computation for arbitrary ω is ongoing work. One can obtain exact results by WKB approximation: Transform the fluctuation equations into Schrodinger form and apply WKB for ω/t 1: U.G., Kiritsis, to appear ρ s (ω) = N2 c 360π 2ω4, ρ b (ω) = N2 c 540π 2ω4 b 2 0λ 2 t. Contrast them with the perturbative Yang-Mills computations: H. Meyer 07 ρ s (ω) = N2 c 160π 2ω4, ρ b (ω) = N2 c 64π 2ω4 b 2 0λ 2 t. Dependence on ω, N c and λ t match perfectly! Overall coefficients do not we expect α corrections in UV. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.24

68 Energy loss of a heavy quark Highly energetic partons produced in head-on nuclei collisions are very important probes Example: When m λt a heavy quark moving the plasma, e.g. charm with m = 1.4 GeV equilibration time τ e τ QGP Thermal Transport and Energy Loss in Non-critical Holographic QCD p.25

69 Energy loss of a heavy quark Highly energetic partons produced in head-on nuclei collisions are very important probes Example: When m λt a heavy quark moving the plasma, e.g. charm with m = 1.4 GeV equilibration time τ e τ QGP In weakly coupled QGP: main source of energy loss is collisions with thermal gluons and quarks. D. Teaney 03 What happens in a strongly coupled plasma? Thermal Transport and Energy Loss in Non-critical Holographic QCD p.25

70 Holographic computation Herzog et al; Gubser 06 Holography: Represent the (infinitely) heavy quark with a trailing string moving with constant v: horizon momentum loss r heavy quark boundary v x1 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.26

71 Holographic computation Herzog et al; Gubser 06 Holography: Represent the (infinitely) heavy quark with a trailing string moving with constant v: horizon momentum loss r heavy quark boundary v x1 Energy loss of a heavy quark in a hot wind: F = dp dt = 1 de v dt = µ(p)p + ξ(t) Stochastic force ξ(t) yields momentum broadening: ξ(t)ξ(t ) = κ(p)δ(t t ) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.26

72 Holographic computation Herzog et al; Gubser 06 Holography: Represent the (infinitely) heavy quark with a trailing string moving with constant v: horizon momentum loss r heavy quark boundary v x1 Energy loss of a heavy quark in a hot wind: F = dp dt = 1 de v dt = µ(p)p + ξ(t) Stochastic force ξ(t) yields momentum broadening: ξ(t)ξ(t ) = κ(p)δ(t t ) What is the drag coefficient µ, equilibration time τ e, diffusion costants κ at strong coupling? Thermal Transport and Energy Loss in Non-critical Holographic QCD p.26

73 Results: Energy loss Standard calculation: - Pick up the stattic gauge: σ 0 = t, σ 1 = r. - String ansatz x 1 = vt + δ(r) - Minimize the area (in the string frame!) - Compute the WS momentum flowing into the BH horizon Thermal Transport and Energy Loss in Non-critical Holographic QCD p.27

74 Results: Energy loss Standard calculation: - Pick up the stattic gauge: σ 0 = t, σ 1 = r. - String ansatz x 1 = vt + δ(r) - Minimize the area (in the string frame!) - Compute the WS momentum flowing into the BH horizon F = 1 v de dt = 1 2πl 2 s v e 2A(r s) λ(r s ) 4 3, r s defined by f(r s ) = v 2. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.27

75 Results: Energy loss Standard calculation: - Pick up the stattic gauge: σ 0 = t, σ 1 = r. - String ansatz x 1 = vt + δ(r) - Minimize the area (in the string frame!) - Compute the WS momentum flowing into the BH horizon F = 1 v de dt = 1 2πl 2 s Relativistic limit, v 1: F = l2 l 2 s v e 2A(r s) λ(r s ) 4 3, r s defined by f(r s ) = v 2. r 45 Ts(T) 4N 2 c v 1 v 2 β 0 4 log[1 v 2 ] Non-relativistic limit v 0: F = l2 l 2 s 45π s(t) N 2 c 2 3 λ(rh ) 4 3 2π v + Thermal Transport and Energy Loss in Non-critical Holographic QCD p.27

76 Comparison to conformal case The AdS result: F conf = π 2 λt 2 v 1 v 2 Fix l s in our model by the lattice string tension Fix λ = 5.5 in N = 4 SYM: Thermal Transport and Energy Loss in Non-critical Holographic QCD p.28

77 Comparison to conformal case The AdS result: F conf = π 2 λt 2 v 1 v 2 Fix l s in our model by the lattice string tension Fix λ = 5.5 in N = 4 SYM: F Fc F Fc v 1 10 v 4 10 v 7 10 v T Tc 1.01 T Tc 1.48 T Tc 1.99 T Tc T Tc v We clearly see the effects of asymptotic freedom! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.28

78 Comparison schemes Thermal Transport and Energy Loss in Non-critical Holographic QCD p.29

79 Comparison schemes An important detail:: How to compare to QCD? Direct scheme: T QGP = T our Thermal Transport and Energy Loss in Non-critical Holographic QCD p.29

80 Comparison schemes An important detail:: How to compare to QCD? Direct scheme: T QGP = T our In the range 1.5T c < T < 3T c E QGP E GP T 4 Alternative schemes: E QGP = E our or s QGP = s our We try all possible schemes. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.29

81 Predictions for experiments Thermal Transport and Energy Loss in Non-critical Holographic QCD p.30

82 Predictions for experiments Equilibration times for charm and bottom: Τ fm bottom charm p GeV Solid: direct, dashed: energy, dot-dashed entropy schemes. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.30

83 Predictions for experiments Equilibration times for charm and bottom: Τ fm bottom charm p GeV Solid: direct, dashed: energy, dot-dashed entropy schemes. Some experimental studies + models PHENIX col. 06, van Hees et al 05: For p = 10 GeV, τ e 4.5 fm (charm) We have 3 < τ e < 5.5 fm Thermal Transport and Energy Loss in Non-critical Holographic QCD p.30

84 Predictions for experiments Equilibration times for charm and bottom: Τ fm bottom charm p GeV Solid: direct, dashed: energy, dot-dashed entropy schemes. Some experimental studies + models PHENIX col. 06, van Hees et al 05: For p = 10 GeV, τ e 4.5 fm (charm) We have 3 < τ e < 5.5 fm BETTER THAN AdS CASE! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.30

85 Langevin dynamics of heavy quarks Thermal Transport and Energy Loss in Non-critical Holographic QCD p.31

86 Langevin dynamics of heavy quarks Diffusion constants κ describe the momentum broadening of heavy quarks floating through the plasma. ξ(t)ξ(t ) = κ(p)δ(t t ). Thermal Transport and Energy Loss in Non-critical Holographic QCD p.31

87 Langevin dynamics of heavy quarks Diffusion constants κ describe the momentum broadening of heavy quarks floating through the plasma. ξ(t)ξ(t ) = κ(p)δ(t t ). String theory computation: In Fourier space κ = lim ω 0 G sym (ω) = lim ω 0 coth( ω 4T s )Im G R (ω) where T s is the world-sheet temperature. G R extracted from fluctuations on the trailing string solution: X 1 = vt + ζ(r) + δx 1, X T = δx T. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.31

88 Langevin dynamics of heavy quarks Diffusion constants κ describe the momentum broadening of heavy quarks floating through the plasma. ξ(t)ξ(t ) = κ(p)δ(t t ). String theory computation: In Fourier space κ = lim ω 0 G sym (ω) = lim ω 0 coth( ω 4T s )Im G R (ω) where T s is the world-sheet temperature. G R extracted from fluctuations on the trailing string solution: X 1 = vt + ζ(r) + δx 1, X T = δx T. There is a [ horizon" on the world-sheet: ] ds 2 = b 2 (f(r) v 2 )dτ 2 dr + 2 f v 2 b 4 (r s )/b 4 (r) WS horizon at f(r s ) = v 2. Compute G R from the flux of fluctuations absorbed at r s. κ = 2 b 2 (r πl 2 s )T s, s κ = 32π l 2 s b 2 (r s ) f (r s ) 2 T 3 s Thermal Transport and Energy Loss in Non-critical Holographic QCD p.31

89 Physical picture t rs: WS horizon x rh: BH horizon r Thermal Transport and Energy Loss in Non-critical Holographic QCD p.32

90 Physical picture t rs: WS horizon x r rh: BH horizon A black-hole horizon on the WS at r s : Fluctuations on the string fall into the horizon energy loss Thermal Transport and Energy Loss in Non-critical Holographic QCD p.32

91 Physical picture t rs: WS horizon x r rh: BH horizon A black-hole horizon on the WS at r s : Fluctuations on the string fall into the horizon energy loss However, there is Hawking radiation at r s towards the boundary momentum broadening. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.32

92 Numerical results for jet quenching Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

93 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

94 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

95 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. If Langevin dynamics satisfied: ˆq = 2κ v. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

96 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. If Langevin dynamics satisfied: ˆq = 2κ v. In the extreme relativistic limit v 1, one derives: (st) 3 ( ) 4 4 b 0 4 log(1 v 2 3 ) κ (45π2 ) 3 4 2π 2 l 2 l 2 s (1 v 2 ) 1 4 Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

97 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. If Langevin dynamics satisfied: ˆq = 2κ v. In the extreme relativistic limit v 1, one derives: (st) 3 ( ) 4 4 b 0 4 log(1 v 2 3 ) κ (45π2 ) 3 4 2π 2 l 2 l 2 s (1 v 2 ) 1 4 ˆq = 5.2 (direct), 12 (energy), (entropy) GeV 2 /fm, for a charm quark traveling at p = 10GeV at T = 250 MeV. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

98 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. If Langevin dynamics satisfied: ˆq = 2κ v. In the extreme relativistic limit v 1, one derives: (st) 3 ( ) 4 4 b 0 4 log(1 v 2 3 ) κ (45π2 ) 3 4 2π 2 l 2 l 2 s (1 v 2 ) 1 4 ˆq = 5.2 (direct), 12 (energy), (entropy) GeV 2 /fm, for a charm quark traveling at p = 10GeV at T = 250 MeV. Much better than pure AdS! ( 2 6 GeV 2 /fm) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

99 Numerical results for jet quenching An extremely important observable in heavy ion-collisions. One can define ˆq = p2 T L. Average momentum acquired by a parton, traveling distance L in the medium. From the nuclear modification factors R AA at RHIC and comparison with hydro simulations: ˆq 5 15 GeV 2 /fm. If Langevin dynamics satisfied: ˆq = 2κ v. In the extreme relativistic limit v 1, one derives: (st) 3 ( ) 4 4 b 0 4 log(1 v 2 3 ) κ (45π2 ) 3 4 2π 2 l 2 l 2 s (1 v 2 ) 1 4 ˆq = 5.2 (direct), 12 (energy), (entropy) GeV 2 /fm, for a charm quark traveling at p = 10GeV at T = 250 MeV. Much better than pure AdS! ( 2 6 GeV 2 /fm) ASYMPTOTIC FREEDOM very important for the enhancement. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.33

100 Summary and Outlook Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

101 Summary and Outlook Bulk viscosity and energy loss for hard probes and ultra-relativistic quarks in improved holographic QCD. Results comparable to expectations from lattice or data. ζ/s peak near T c lower than lattice expectations. Drag force, jet-quenching well within expectations, better than AdS. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

102 Summary and Outlook Bulk viscosity and energy loss for hard probes and ultra-relativistic quarks in improved holographic QCD. Results comparable to expectations from lattice or data. ζ/s peak near T c lower than lattice expectations. Drag force, jet-quenching well within expectations, better than AdS. Higher order hydrodynamic coefficients Expanding plasma (non-static configurations) Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

103 Summary and Outlook Bulk viscosity and energy loss for hard probes and ultra-relativistic quarks in improved holographic QCD. Results comparable to expectations from lattice or data. ζ/s peak near T c lower than lattice expectations. Drag force, jet-quenching well within expectations, better than AdS. Higher order hydrodynamic coefficients Expanding plasma (non-static configurations) Meson sector. Melting of heavy mesons, etc. Finite chemical potential and phase diagram in µ c T. Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

104 Summary and Outlook Bulk viscosity and energy loss for hard probes and ultra-relativistic quarks in improved holographic QCD. Results comparable to expectations from lattice or data. ζ/s peak near T c lower than lattice expectations. Drag force, jet-quenching well within expectations, better than AdS. Higher order hydrodynamic coefficients Expanding plasma (non-static configurations) Meson sector. Melting of heavy mesons, etc. Finite chemical potential and phase diagram in µ c T. Embedding in 5D non-critical string theory? Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

105 Summary and Outlook Bulk viscosity and energy loss for hard probes and ultra-relativistic quarks in improved holographic QCD. Results comparable to expectations from lattice or data. ζ/s peak near T c lower than lattice expectations. Drag force, jet-quenching well within expectations, better than AdS. Higher order hydrodynamic coefficients Expanding plasma (non-static configurations) Meson sector. Melting of heavy mesons, etc. Finite chemical potential and phase diagram in µ c T. Embedding in 5D non-critical string theory? THANK YOU! Thermal Transport and Energy Loss in Non-critical Holographic QCD p.34

Termodynamics and Transport in Improved Holographic QCD

Termodynamics and Transport in Improved Holographic QCD p. 1 Termodynamics and Transport in Improved Holographic QCD Francesco Nitti APC, U. Paris VII Large N @ Swansea July 07 2009 Work with E. Kiritsis,