Termodynamics and Transport in Improved Holographic QCD

Transcription

1 Termodynamics and Transport in Improved Holographic QCD p. 1 Termodynamics and Transport in Improved Holographic QCD Francesco Nitti APC, U. Paris VII Large Swansea July Work with E. Kiritsis, U. Gursoy, L. Mazzanti, G. Michalogiorgakis , , , ,

2 Termodynamics and Transport in Improved Holographic QCD p. 2 Phenomenological Holography and QCD Bottom-up Holography: Construct a gravity background such that, using holography rules, we can model the properties of the desired gauge theory (e.g. AdS/QCD, Pomarol and Da Rold 05; Erlich et al 05).

3 Termodynamics and Transport in Improved Holographic QCD p. 2 Phenomenological Holography and QCD Bottom-up Holography: Construct a gravity background such that, using holography rules, we can model the properties of the desired gauge theory (e.g. AdS/QCD, Pomarol and Da Rold 05; Erlich et al 05). Minimal setup: 5 dimensions (4 space-time + radial (RG) direction) restrict to a small number of fields use hints from (what is known about) 5D non-critical string theory

4 Termodynamics and Transport in Improved Holographic QCD p. 2 Phenomenological Holography and QCD Bottom-up Holography: Construct a gravity background such that, using holography rules, we can model the properties of the desired gauge theory (e.g. AdS/QCD, Pomarol and Da Rold 05; Erlich et al 05). Minimal setup: 5 dimensions (4 space-time + radial (RG) direction) restrict to a small number of fields use hints from (what is known about) 5D non-critical string theory Aim: a realistic holographic model of the deconfined QGP (to compare with lattice and RHIC)

5 Outline Termodynamics and Transport in Improved Holographic QCD p. 3

6 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography

7 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography Setup and Vacuum solutions

8 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography Setup and Vacuum solutions Finite Temperature Solutions Black hole solutions Phase transition

9 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography Setup and Vacuum solutions Finite Temperature Solutions Black hole solutions Phase transition Thermodynamics

10 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography Setup and Vacuum solutions Finite Temperature Solutions Black hole solutions Phase transition Thermodynamics Transport: Shear and Bulk Viscosity

11 Termodynamics and Transport in Improved Holographic QCD p. 3 Outline Bottom-up holography Setup and Vacuum solutions Finite Temperature Solutions Black hole solutions Phase transition Thermodynamics Transport: Shear and Bulk Viscosity Heavy quark diffusion

12 Termodynamics and Transport in Improved Holographic QCD p. 4 Precursors AdS/QCD (Pomarol and Da Rold 05; Erlich et al 05) the metric is the only bulk field in the gauge sector The 5D spacetime is AdS 5 truncated by a IR cut-off with no explicit dynamics confinement is imposed via boundary conditions e A(r) r

13 Termodynamics and Transport in Improved Holographic QCD p. 4 Precursors AdS/QCD (Pomarol and Da Rold 05; Erlich et al 05) the metric is the only bulk field in the gauge sector The 5D spacetime is AdS 5 truncated by a IR cut-off with no explicit dynamics confinement is imposed via boundary conditions One step further: Soft-Wall AdS/QCD (Karch et al 06) hard cut-off replaced by a dilaton that grows in the UV the metric is still exactly AdS 5 (No mass gap; gluons are not confined) 2 bulk fields now, but still no dynamics (no action, no field equations): cannot compute free energies consistently.

14 Termodynamics and Transport in Improved Holographic QCD p. 5 Improved Holographic QCD Gursoy, Kiritsis, FN , Make the soft-wall dynamical and interpret it as the holographic dual of the glue sector. Treat metric-dilaton as a dynamical system (including full backreaction). Breaking of conformal symmetry, mass gap, confinement, and all non-perturbative dynamics driven by the dilaton dynamics (aka the Yang-Mills coupling).

15 Improved Holographic QCD Gursoy, Kiritsis, FN , Make the soft-wall dynamical and interpret it as the holographic dual of the glue sector. Treat metric-dilaton as a dynamical system (including full backreaction). Breaking of conformal symmetry, mass gap, confinement, and all non-perturbative dynamics driven by the dilaton dynamics (aka the Yang-Mills coupling). Bulk Field Content Dimension 4 operators in Pure YM 4D Operator Bulk field Coupling TrF 2 Φ κn c e Φ TrF 2 T µν g µν gµν T µν λ = κn c g 2 Y M = eφ (finite in the large N limit). Termodynamics and Transport in Improved Holographic QCD p. 5

16 Termodynamics and Transport in Improved Holographic QCD p. 6 The Setup only lowest dimension operators are considered action limited to 2-derivatives ( S E = MpN 3 c 2 d 5 x g [ R 4 ]) 3 ( Φ)2 V (Φ)

17 Termodynamics and Transport in Improved Holographic QCD p. 6 The Setup only lowest dimension operators are considered action limited to 2-derivatives ( S E = MpN 3 c 2 d 5 x g [ R 4 ]) 3 ( Φ)2 V (Φ) V (Φ) fixed phenomenologically. N c appears only as an overall factor. Effective Planck scale N 2 c is large. the parameter M p can be fixed via thermodynamics in terms of V ( )

18 Termodynamics and Transport in Improved Holographic QCD p. 7 Vacuum Solution Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) )

19 Termodynamics and Transport in Improved Holographic QCD p. 7 Vacuum Solution Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) ) UV: e A, λ 0, V (λ) 12 l 2 ( 1 + v0 λ + v 1 λ 2... ) Asymptotically AdS solution; l 2 ads = 12/V (0) finite

20 Termodynamics and Transport in Improved Holographic QCD p. 7 Vacuum Solution Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) ) UV: e A, λ 0, V (λ) 12 l 2 ( 1 + v0 λ + v 1 λ 2... ) Asymptotically AdS solution; l 2 ads = 12/V (0) finite IR: λ large, e A 0; V λ 4/3 (log λ) 1/2

21 Termodynamics and Transport in Improved Holographic QCD p. 7 Vacuum Solution Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) ) UV: e A, λ 0, V (λ) 12 l 2 ( 1 + v0 λ + v 1 λ 2... ) Asymptotically AdS solution; l 2 ads = 12/V (0) finite IR: λ large, e A 0; V λ 4/3 (log λ) 1/2 Get asymptotic freedom, confinement, mass gap, linear glueball spectrum.

22 Termodynamics and Transport in Improved Holographic QCD p. 7 Vacuum Solution Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) ) UV: e A, λ 0, V (λ) 12 l 2 ( 1 + v0 λ + v 1 λ 2... ) Asymptotically AdS solution; l 2 ads = 12/V (0) finite IR: λ large, e A 0; V λ 4/3 (log λ) 1/2 Get asymptotic freedom, confinement, mass gap, linear glueball spectrum. β(λ) dλ d log E = (dλ/dr) (da/dr)

23 Termodynamics and Transport in Improved Holographic QCD p. 8 Vacuum Solution e A(r) e Cr2, λ e 3 2 Cr2 r exp A Λ r r

24 Termodynamics and Transport in Improved Holographic QCD p. 8 Vacuum Solution e A(r) e Cr2, λ e 3 2 Cr2 r exp A Λ r r In the string frame: S = gs λ 2 (R +...), e A s λ 2/3 e A

25 Termodynamics and Transport in Improved Holographic QCD p. 8 Vacuum Solution e A(r) e Cr2, λ e 3 2 Cr2 r exp A Λ r r exp A s In the string frame: S = gs λ 2 (R +...), e A s λ 2/3 e A r r

26 Termodynamics and Transport in Improved Holographic QCD p. 8 Vacuum Solution e A(r) e Cr2, λ e 3 2 Cr2 r exp A Λ r r exp A s In the string frame: S = gs λ 2 (R +...), e A s λ 2/3 e A r r Wilson Loop area law with tension: σ c = 1 2πl 2 s e 2A(r ) λ 4/3 (r )

27 Termodynamics and Transport in Improved Holographic QCD p. 9 Finite Temperature 4D Pure YM theory exhibits a first order deconfining phase transition around T c 260MeV.

28 Karsch, hep-lat/ Termodynamics and Transport in Improved Holographic QCD p. 9 Finite Temperature 4D Pure YM theory exhibits a first order deconfining phase transition around T c 260MeV. Can the holographic setup reproduce the phase transition and the YM equation of state found on the lattice?

29 Termodynamics and Transport in Improved Holographic QCD p. 10 Finite Temperature Solutions Gursoy, Kiritsis, Mazzanti, FN , Two kinds of solutions with τ τ + 1/T and R 3 spatial sections

30 Termodynamics and Transport in Improved Holographic QCD p. 10 Finite Temperature Solutions Gursoy, Kiritsis, Mazzanti, FN , Two kinds of solutions with τ τ + 1/T and R 3 spatial sections Thermal gas (Confined phase) : ds 2 = e 2A 0(r) ( dr 2 + dτ 2 + dx 2 i ), Φ = Φ0 (r), 0 < r < +.

31 Termodynamics and Transport in Improved Holographic QCD p. 10 Finite Temperature Solutions Gursoy, Kiritsis, Mazzanti, FN , Two kinds of solutions with τ τ + 1/T and R 3 spatial sections Thermal gas (Confined phase) : ds 2 = e 2A 0(r) ( dr 2 + dτ 2 + dx 2 i ), Φ = Φ0 (r), 0 < r < +. Black hole (Deconfined phase - Gluon Plasma): ( ) dr ds 2 = e 2A(r) 2 f(r) + f(r)dτ2 + dx 2 i, Φ = Φ(r), 0 < r < r h A(r) r 0 A 0 (r), f(r) r 0 1, f(r h ) = 0, T = f(r h ) 4π, S = (4πM3 pn 2 c V 3 ) e 3A(r h)

32 Termodynamics and Transport in Improved Holographic QCD p. 11 Deconfinement Transition The gravity dual indeed displays a rich phase structure with a deconfininement phase transition: a minimum BH temperature T min ; Two BH branches (big and small) for T > T min ; Big BHs dominates above the critical temperature T c > T min.

33 Termodynamics and Transport in Improved Holographic QCD p. 12 Thermodynamics of the deconfined phase Thermodynamic observables are computed by geometric quantities: pressure p = F/V 3 M 3 pn 2 c gr +... entropy density s M 3 pn 2 c Area/V 3 energy density ǫ = Mass/V 3 = p + Ts

34 Termodynamics and Transport in Improved Holographic QCD p. 12 Thermodynamics of the deconfined phase Thermodynamic observables are computed by geometric quantities: pressure p = F/V 3 M 3 pn 2 c gr +... entropy density s M 3 pn 2 c Area/V 3 energy density ǫ = Mass/V 3 = p + Ts At high T we recover conformal behavior: ǫ T s T 3 3 p T 4 3π4 (M p l) 3 Nc 2 Requiring a free gas with N 2 c d.o.f. fixes M p (M p l) = 1/(45π 2 ) 1/3

35 Termodynamics and Transport in Improved Holographic QCD p. 13 An Explicit Model Gursoy, Kiritsis, Mazzanti, FN Phenomenological potential, fixed small-λ and large-λ behavior.

36 Termodynamics and Transport in Improved Holographic QCD p. 13 An Explicit Model Gursoy, Kiritsis, Mazzanti, FN Phenomenological potential, fixed small-λ and large-λ behavior. V (λ) = 12 l 2 l: Overall energy unit (e.g. fixed by lowest glueball state)

37 Termodynamics and Transport in Improved Holographic QCD p. 13 An Explicit Model Gursoy, Kiritsis, Mazzanti, FN Phenomenological potential, fixed small-λ and large-λ behavior. V (λ) = 12 l 2 ( β 0λ l: Overall energy unit (e.g. fixed by lowest glueball state) β 0 fixed by the one-loop β-function

38 Termodynamics and Transport in Improved Holographic QCD p. 13 An Explicit Model Gursoy, Kiritsis, Mazzanti, FN Phenomenological potential, fixed small-λ and large-λ behavior. V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2 l: Overall energy unit (e.g. fixed by lowest glueball state) β 0 fixed by the one-loop β-function V 1, V 3 used for the fit to lattice data

39 Termodynamics and Transport in Improved Holographic QCD p. 13 An Explicit Model Gursoy, Kiritsis, Mazzanti, FN Phenomenological potential, fixed small-λ and large-λ behavior. V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2 l: Overall energy unit (e.g. fixed by lowest glueball state) β 0 fixed by the one-loop β-function V 1, V 3 used for the fit to lattice data Other model parameters: M p fixed by high-t regime l s /l fixed at zero-t by matching σ c (440 MeV ) 2.

40 Termodynamics and Transport in Improved Holographic QCD p. 14 Matching Pure YM Thermodynamics V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2

41 Termodynamics and Transport in Improved Holographic QCD p. 14 Matching Pure YM Thermodynamics V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2 V 1 14, V lattice data: Karsch, hep-lat/

42 Termodynamics and Transport in Improved Holographic QCD p. 14 Matching Pure YM Thermodynamics V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2 V 1 14, V (L h ) HQCD (L h ) lat 0.31N 2 c T 4 c lattice data: Karsch, hep-lat/

43 Termodynamics and Transport in Improved Holographic QCD p. 14 Matching Pure YM Thermodynamics V (λ) = 12 l 2 ( β 0λ + V 1 λ 4/3 [ log ( 1 + V 3 λ 2)] ) 1/2 V 1 14, V (L h ) HQCD (L h ) lat 0.31N 2 c T 4 c (T c ) HQCD 250MeV (T c ) lat 260MeV lattice data: Karsch, hep-lat/

44 Termodynamics and Transport in Improved Holographic QCD p. 15 Trace Anomaly and speed of sound ǫ 3p = T µ µ T c 2 s = s c V

45 Termodynamics and Transport in Improved Holographic QCD p. 15 Trace Anomaly and speed of sound ǫ 3p = T µ µ T c 2 s = s c V

46 Termodynamics and Transport in Improved Holographic QCD p. 16 Hydrodynamic Transport In the long-wavelength limit: T µν = (ǫ + p) u µ u ν + p g µν ( P µi P [η νj i u j + j u i 2 ) 3 g ij u ] + ζ g ij u

47 Termodynamics and Transport in Improved Holographic QCD p. 16 Hydrodynamic Transport In the long-wavelength limit: T µν = (ǫ + p) u µ u ν + p g µν ( P µi P [η νj i u j + j u i 2 ) 3 g ij u ] + ζ g ij u η : Shear viscosity. RHIC data consistent with very small η/s, Closest match: Strong coupling holographic computation in N = 4 SYM: η/s = (4π) IHQCD setup: same result as in N = 4 SYM (universal in 2-derivative models)

48 Termodynamics and Transport in Improved Holographic QCD p. 16 Hydrodynamic Transport In the long-wavelength limit: T µν = (ǫ + p) u µ u ν + p g µν ( P µi P [η νj i u j + j u i 2 ) 3 g ij u ] + ζ g ij u ζ : Bulk viscosity Vanishes in a conformal fluid (like N = 4 plasma) How signigicant is ζ close to T c? Computed holographically by the scalar fluctuation of the metric-dilaton system (Gubser et al, 08)

49 Termodynamics and Transport in Improved Holographic QCD p. 17 Bulk Viscosity Indication from lattice: raise of ζ/s close to T c (Meyer 08).

50 Termodynamics and Transport in Improved Holographic QCD p. 17 Bulk Viscosity Indication from lattice: raise of ζ/s close to T c (Meyer 08).

51 Termodynamics and Transport in Improved Holographic QCD p. 17 Bulk Viscosity Indication from lattice: raise of ζ/s close to T c (Meyer 08). Buchel s bound: ζ/η 2(1/3 c 2 s)

52 Termodynamics and Transport in Improved Holographic QCD p. 18 Drag Force on a Heavy Quark Trailing String picture:

53 Termodynamics and Transport in Improved Holographic QCD p. 18 Drag Force on a Heavy Quark Trailing String picture: Drag force: F = 1 v de dt = 1 2πl 2 s v e 2A(r s) λ(r s ) 4 3, f(rs ) = v 2

54 Termodynamics and Transport in Improved Holographic QCD p. 19 Drag Force on a Heavy Quark Define diffusion time: F = p/τ

55 Termodynamics and Transport in Improved Holographic QCD p. 19 Drag Force on a Heavy Quark Define diffusion time: F = p/τ in N = 4, τ conf 1/ λt 2

56 Termodynamics and Transport in Improved Holographic QCD p. 19 Drag Force on a Heavy Quark Define diffusion time: F = p/τ in N = 4, τ conf 1/ λt 2 in IHQCD τ = τ(p); no need to fix λ;

57 Termodynamics and Transport in Improved Holographic QCD p. 19 Drag Force on a Heavy Quark Define diffusion time: F = p/τ in N = 4, τ conf 1/ λt 2 in IHQCD τ = τ(p); no need to fix λ; τ(p) const. for p m q. We obtain τ charm 4.5fm/c for p 10 GeV, consistent with RHIC analysis.

58 Termodynamics and Transport in Improved Holographic QCD p. 20 To Summarize... Positive features: Strict relationships between confinement, mass gap and phase transitions. Smooth connection to the correct UV of the theory An explicit model that can reproduce at the quantitative level glueball spectrum thermodynamics several transport properties of large-n pure Yang-Mills theory

59 The UV needs some rethinking (hints that the AdS asymptotics must be in the string frame) Termodynamics and Transport in Improved Holographic QCD p. 20 To Summarize... Positive features: Strict relationships between confinement, mass gap and phase transitions. Smooth connection to the correct UV of the theory An explicit model that can reproduce at the quantitative level glueball spectrum thermodynamics several transport properties of large-n pure Yang-Mills theory Some drawbacks: No explicit construction of gauge theory as decoupling limit. No derivation of the gravity side from a fundamental theory

60 Termodynamics and Transport in Improved Holographic QCD p. 21 Perspectives Toward QCD Include chiral sector Meson spectra and chiral condensate More thermodynamics ( χsb/restoration ) Finite baryon density

61 Termodynamics and Transport in Improved Holographic QCD p. 21 Perspectives Toward QCD Include chiral sector Meson spectra and chiral condensate More thermodynamics ( χsb/restoration ) Finite baryon density... stay tuned.

62 Termodynamics and Transport in Improved Holographic QCD p. 22 Spectrum at T = 0 Glueball spectrum: spectrum of normalizable scalar and tensor fluctuations around the T = 0 background.

63 Termodynamics and Transport in Improved Holographic QCD p. 22 Spectrum at T = 0 Glueball spectrum: spectrum of normalizable scalar and tensor fluctuations around the T = 0 background. Numerical computation in the explicit HQCD model shows good agreement with lattice data m 0 ++ m 0 ++ m 2 ++ m 0 ++ T c m 0 ++ HQCD Lat. N c = 3 (Chen et al.) Lat. N c = (Lucini, Teper) (11) 1.90(17) (4) 1.46(11)

64 Termodynamics and Transport in Improved Holographic QCD p. 23 Vacuum Solution - UV Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) )

65 Vacuum Solution - UV Non-constant V (Φ): solutions have running coupling. ds 2 = e 2A(r) ( dr 2 + η µν dx µ dx ν), λ(r) = e Φ(r), (E 4D e A(r) ) UV: e A, λ 0; r 0 e A(r) = l r V (λ) 12 l 2 ( 1 + v0 λ + v 1 λ 2... ) ( ) , λ 1 b 0 log rλ +..., b 0 = 9 8 v 0 β(λ) dλ d log E = b 0λ l = V (0)/12 is the asymptotic AdS radius; Λ is an integration constant: the dynamically generated strong coupling scale. Termodynamics and Transport in Improved Holographic QCD p. 23

66 Termodynamics and Transport in Improved Holographic QCD p. 24 Vacuum Solution - IR V (λ) Monotonic in the range λ (0, ) (no fixed points)

67 Termodynamics and Transport in Improved Holographic QCD p. 24 Vacuum Solution - IR V (λ) Monotonic in the range λ (0, ) (no fixed points) λ(r) grows and A(r) shrinks as r increases

68 Termodynamics and Transport in Improved Holographic QCD p. 24 Vacuum Solution - IR V (λ) Monotonic in the range λ (0, ) (no fixed points) λ(r) grows and A(r) shrinks as r increases For large λ take: V λ 4/3 (log λ) 1/2

69 Termodynamics and Transport in Improved Holographic QCD p. 24 Vacuum Solution - IR V (λ) Monotonic in the range λ (0, ) (no fixed points) λ(r) grows and A(r) shrinks as r increases For large λ take: V λ 4/3 (log λ) 1/2 Confinement (Wilson Loop area law) Spectrum with mass gap and linear glueball trajectories m 2 n n