Introduction The model Thermodynamics Hydrodynamics Numerics

Holographic Hydrodynamics

from 5d Dilaton–Gravity

Liuba Mazzanti(University of Santiago de Compostela)

based on:U. Gursoy, E. Kiritsis, L. M., F. Nitti, [arXiv:1006.3261]U. Gursoy, E. Kiritsis, L. M., F. Nitti, [arXiv:0903.2859]U. Gursoy, E. Kiritsis, L. M., F. Nitti, [arXiv:0812.0792]U. Gursoy, E. Kiritsis, L. M., F. Nitti, [arXiv:0804.0899]

see also:U. Gursoy, E. Kiritsis, G. Michalogiorgakis, F. Nitti, [arXiv:0906.1890]

U. Gursoy, E. Kiritsis, F. Nitti, [arXiv:0707.1349]U. Gursoy, E. Kiritsis, [arXiv:0707.1324]

Kolymbari, September 12, 2010

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 1 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Motivations and Goals: Gluon Plasma and Holography

1 Motivations: phase diagram & hydrodynamics of the plasma

4D gauge theory: deconfinement phase transitionlattice results for thermodynamic functions (equation of state, . . . )heavy ion collision experiments for hydrodynamic and transport coefficients

2 Goals: 5d holography for the plasma

setup: confinement & running coupling for 4D YMcan it match the thermodynamics from lattice?if yes, can it give the hydrodynamic and transport coefficients ?

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 2 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Outline

5d dilaton–gravity setup: zero T and finite T (TG & BH)

Phase transition and thermodynamics

Drag force from worldsheet background

Diffusion constants and “jet quenching parameters”

Numerics

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 3 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Thermodynamics and Hydrodynamics

Finite temperature: gauge/gravity correspondence Hawking-Page’83,Witten’98

Gravity solutions with periodic time: thermal gas and black hole

m gauge/gravity

Gauge theory at finite temperature

Langevin diffusion: momentum broadeningCasalderrey-Teany’06, Gubser’06,deBoer-Hubeny-Rangamany-Shigemori’08, Son-Teaney,Giecold-Iancu-Muller’09

quark moving worldsheetdrag force ⇔ background

diffusion fluctuations

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 4 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Thermodynamics and Hydrodynamics

Finite temperature: gauge/gravity correspondence Hawking-Page’83,Witten’98

Gravity solutions with periodic time: thermal gas and black hole

m gauge/gravity

Gauge theory at finite temperature

Langevin diffusion: momentum broadeningCasalderrey-Teany’06, Gubser’06,deBoer-Hubeny-Rangamany-Shigemori’08, Son-Teaney,Giecold-Iancu-Muller’09

quark moving worldsheetdrag force ⇔ background

diffusion fluctuationsp0 Dp

¦

2

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 4 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

5d Dilaton–Gravity Holography: Action Gursoy-Kiritsis’07

Field/operator correspondence

gµν ⇔ T µν

φ ⇔ trF 2

Action in the Einstein frame

SE = −M3N2c

2

∫

d5x√−g

[

R − 4

3(∂φ)2 + V (φ)

]

Dilaton potential determined phenomenologically: bottom-up

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 5 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Black Hole Solution for 5d Dilaton–Gravity Gursoy-Kiritsis-Mazzanti-Nitti’08

Finite T : asymptotic AdS BH + dilaton

ds2 = b(r)2[

dr2

f(r)− f(r)dt2 + dx2

]

λ = λ(r), V = V (λ)

Uniqueness: integration constants

1 Λ ⇔ strong coupling scale

2 λh ⇔ T

3 “good singularity”

4 UV normalization ⇒ f(0) = 1

5 unphysical

t

r

t

r

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 6 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Zero T : Asymptotic Freedom and Confinement

UV geometry ⇔ YM β–function (V = 43W 2

o,φ − 6427W 2

o ) (r → 0)

β(λ) = −b0λ2 − b1λ

3 + . . . = −9

4λ2 ∂λ log Wo as λ → 0

b0 and b1 fixed from YM

λ = Nceφ ≃ − 1

b0 log rΛ, b ≃ ℓ

r

“

1 + 49

1log rΛ

”

and V ≃ V0 + V1λ

IR geometry ⇔ confinement via Wilson loop (r → ∞)

Wo ∼ λ23 (log λ)P/2

singularity at ∞ ⇒ P ≥ 0 (P = 1/2 for linear confinement)magnetic screening and discrete gapped glueballs for all confining backgrounds

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 7 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Zero T : Asymptotic Freedom and Confinement

UV geometry ⇔ YM β–function (V = 43W 2

o,φ − 6427W 2

o ) (r → 0)

β(λ) = −b0λ2 − b1λ

3 + . . . = −9

4λ2 ∂λ log Wo as λ → 0

b0 and b1 fixed from YM

λ = Nceφ ≃ − 1

b0 log rΛ, b ≃ ℓ

r

“

1 + 49

1log rΛ

”

and V ≃ V0 + V1λ

IR geometry ⇔ confinement via Wilson loop (r → ∞)

Wo ∼ λ23 (log λ)P/2

singularity at ∞ ⇒ P ≥ 0 (P = 1/2 for linear confinement)magnetic screening and discrete gapped glueballs for all confining backgrounds

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 7 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Black Hole: UV and IR Horizon

UV: same log expansion as the thermal–gas (rh → 0)

λ(r) = λo(r)[

1 + 458 G(T )r4 log Λr + . . .

]

f(r) = 1 − 14B(T )r4 + . . .

Holography

G(T ): vev of the ∆ = 4 operator ⇒ gluon condensate G(T ) ∼ T4

log2 T

B(T ): thermodynamic quantity B(T ) = TS

IR: good singularity singled out (rh → ∞)

b(r) ∼ bo(r), λ(r) ∼ λo(r)

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 8 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Black Hole: UV and IR Horizon

UV: same log expansion as the thermal–gas (rh → 0)

λ(r) = λo(r)[

1 + 458 G(T )r4 log Λr + . . .

]

f(r) = 1 − 14B(T )r4 + . . .

Holography

G(T ): vev of the ∆ = 4 operator ⇒ gluon condensate G(T ) ∼ T4

log2 T

B(T ): thermodynamic quantity B(T ) = TS

IR: good singularity singled out (rh → ∞)

b(r) ∼ bo(r), λ(r) ∼ λo(r)

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 8 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Phase Transition

FM3

P V3=

S − So

βM3P V3

= 15G(T ) − B(T )

4

⇓

First order phase transition ⇔ confining backgrounds

minimum temperature

0.01 0.02 0.03 0.04 0.05Λh

0.16

0.18

0.20

0.22

0.24

THΛhLm0

1.00 1.05 1.10 1.15

T

Tc0

FHTLm0

4 Nc2 V3

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 9 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Holographic Thermodynamics

Trace Anomaly vs. Gluon Condensate: the Trace Anomaly Equation

From dilaton fluctuation and field/operator correspondence

1

4

β(λ)

λ2〈trF 2〉 = e − 3p = 60G(T )M3N2

c

Trace: e − 3p ∝ condensate ⇒ latent heat ∼ N2c

Pressure, Energy, Entropy: p, e, s ∼ N2c for T > Tc ⇒ deconfinement

Sound speed: c2s → 1/3 at high–T , small at Tc

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 10 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Thermodynamic Result Boyd-et al.’05 (SU(3)), Lucini-Teper-Wenger’05 (large–Nc),

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free gas

e

T4 Nc2

3 s

4 T3 Nc2

3 p

T4 Nc2

0 1 2 3 4 5

T

Tc0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ì

ììììììììì ì ì ì ì ì ì ì

LhNc2T4

1 2 3 4 5

T

Tc

0.1

0.2

0.3

0.4

e- 3 p

T4 Nc2

ì

ì

ì

ì

ììììììììììì

ìì ìì ì ì ì ì ì

ì ìfree gas

0 1 2 3 4 5

T

Tc

0.1

0.2

0.3

0.4cs

2

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 11 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Holographic Langevin Diffusion

Heavy quark dynamics:

d~p

dt+ ηD~p = ~ξ, 〈ξiξj〉 = κijδ(t − t′)

↑viscous force

↑diffusion constants

⇔ momentum broadening

ηD = − 1

γMωImGR(ω)|ω=0, κ = Gsym(ω)|ω=0

Gsym and GR are correlators of F(t), the instantaneous force on the quark

Field/operator correspondence

F ⇔ XM

correlators are computed holographically from the string fluctuations δXM

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 12 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Holographic Langevin Diffusion

Heavy quark dynamics:

d~p

dt+ ηD~p = ~ξ, 〈ξiξj〉 = κijδ(t − t′)

↑viscous force

↑diffusion constants

⇔ momentum broadening

ηD = − 1

γMωImGR(ω)|ω=0, κ = Gsym(ω)|ω=0

Gsym and GR are correlators of F(t), the instantaneous force on the quark

Field/operator correspondence

F ⇔ XM

correlators are computed holographically from the string fluctuations δXM

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 12 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Trailing String Gursoy-Kiritsis-Mazzanti-Nitti’10

Worldsheet background:

X1 = vt + x(r), X2 = X3 = 0

SNG = − 12πℓ2s

∫

drdt b2√

1 − v2

f + fx2

v

worldsheet horizon

BH horizon

boundary

Worldsheet Horizon

The induced metric has a horizon at rs with temperature: f(rs) = v2

4πTs =

√

f f

√

4b

b+

f

f

∣

∣

∣

∣

∣

rs

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 13 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Worldsheet Background: the Drag Force

Drag force ⇒ classical momentum lost by the moving string to the horizon

ηD = − πx

γvM=

1

γM

b2(rs)

2πℓ2s

Diffusion time ⇒ attenuation time for the momentum

τD ≡ 1

ηD= γM

2πℓ2s

b2(rs)

Comparison to AdS5 (λAdS fixed)

Ts,AdS = T/√

γ

τD,AdS = 2M

π√

λAdST2momentum–independent

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 14 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Worldsheet Fluctuations

Worldsheet onshell action to second order

X1 = vt + x(r) + δX1, X2 = δX2, X3 = δX3

S(2)NG = − 1

2πℓ2s

∫

drdtGαβ

2∂αδX ∂βδX

Gαβ

⊥ = Z2Gαβ

‖= b2

Z3

−Z2f+v2

f2 vx

vx f − v2

!

, Z ≡ b2√

f−v2q

b4f−b4sv2

Leading Asymptotics from eom I =⊥, ‖boundary: δXI ∼ cI

sour + cIvevr3

horizon: δXI ∼ cIin (rs − r)

− iω4πTs + cI

out (rs − r)iω

4πTs

Retarded wave functions ΨR: cout = 0 and csour = 1

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 15 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Diffusion Constants and Jet Quenching Gursoy-Kiritsis-Mazzanti-Nitti’10

GR = − 12πℓs

GrαΨ∗R ∂αΨR

∣

∣

bound., Gsym = − coth

(

ω2Ts

)

ImGR

⇓κ = lim

ω→0Gsym = lim

ω→0coth

(

ω

2Ts

)

Jr

Jr is a conserved current (number current)q ≡ 〈∆p2〉/L at strong coupling

⇓

κ⊥ =1

2vq⊥ =

1

πℓ2s

b2sTs, κ‖ = vq‖ =

16π

ℓ2s

b2s

f2s

T 3s

Generalized Einstein relation to diffusion time:

τDκ⊥ = 2γMTs

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 16 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Diffusion Constants and Jet Quenching Gursoy-Kiritsis-Mazzanti-Nitti’10

GR = − 12πℓs

GrαΨ∗R ∂αΨR

∣

∣

bound., Gsym = − coth

(

ω2Ts

)

ImGR

⇓κ = lim

ω→0Gsym = lim

ω→0coth

(

ω

2Ts

)

Jr

Jr is a conserved current (number current)q ≡ 〈∆p2〉/L at strong coupling

⇓

κ⊥ =1

2vq⊥ =

1

πℓ2s

b2sTs, κ‖ = vq‖ =

16π

ℓ2s

b2s

f2s

T 3s

Generalized Einstein relation to diffusion time:

τDκ⊥ = 2γMTs

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 16 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

WKB Approximation for Large Frequencies

1 WKB for ωrs ≫ 1: ΨR ∼ C1 cos[

∫

ωZf−v2 + θ1

]

+ C2 sin[

∫

ωZf−v2 + θ2

]

2 Coefficients determined by boundary and horizon behavior

Spectral Densities ρ ≡ − 1π ImGR

Infinite mass: cubic in ω

ρ⊥ ≃ γ−2ρ‖ ≃ ℓ2γ2

π2ℓ2sω3λ

43tp

Finite mass, high velocities (γωrQ ≫ 1): linear in ω

ρ⊥ = γ−2ρ‖ ≃ ℓ2γ2

π2ℓ2sω3r2

Qb2Qλ

43Q

»

1 + (γωrQ)2–−1

M ∼ 1/rQ,λtp = λ at turning point

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 17 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Spectral Densities — Infinitely Massive QuarksSymmetric Correlator

v=0.1

v=0.9

v=0.99

5 10 15 20 25 30Ω rs

0.1

10

1000

105

Gsym¦BGeV2

fmF

T 1. Tc

v=0.1

v=0.9

v=0.99

5 10 15 20 25 30Ω rs

1

100

104

106

108

GsymþBGeV2

fmF

T 1. Tc

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 18 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Spectral Densities — Finite Massive QuarksRetarded and Symmetric Correlator — Charm

v=0.1

v=0.9

v=0.99

5 10 15 20Ω rs

500

1000

1500

ReGR¦BGeV2

fmF

T 3. Tc

v=0.1

v=0.9

v=0.99

5 10 15 20Ω rs

2000400060008000

10 00012 00014 000

Gsym¦BGeV2

fmF

T 3. Tc

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 19 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Diffusion ConstantsRatio to AdS

Tc

1.5 Tc

3 Tc

1 10-1 10-2 10-3 10-4 10-51-v

0.1

0.2

0.3

0.4

0.5

0.6

Κ¦Κ¦conf

Tc

1.5 Tc

3 Tc

1 10-1 10-2 10-3 10-4 10-5 10-61-v0.0

0.1

0.2

0.3

0.4

0.5

0.6ΚþΚþconf

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 20 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Jet Quenching ParameterBottom and Charm Quarks: q vs. momentum

bottom

T=250 MeV

T=500 MeV

T=800 MeV

0 20 40 60 80pT HGeVL0

5

10

15

20

25

30q`

¦HGeV2fmL

charm

T=250 MeV

T=500 MeV

T=800 MeV

0 5 10 15 20pT HGeVL0

5

10

15

20

25

30q`

¦HGeV2fmL

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 21 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Summary

5D dilaton–gravity ⇔ 4D holographic large-Nc gauge theory

⇓

Confinement with discrete spectrum in agreement with lattice at low–T

1 Phase transition first order Hawking–Page confinement/deconfinementonly for confining backgrounds

2 Thermodynamics in good agreement with lattice3 Hydrodynamics compatible with phenomenological models and experiments

Further issues

1 Flavors

2 Chemical potentials

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 22 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Ansatz for the Potential Gursoy-Kiritsis-Mazzanti-Nitti’09

V (λ) =12

ℓ2

1 + V0λ + V1λ4/3

[

log(

1 + V2λ4/3 + V3λ

2)]1/2

1 Monotonic

2 Asymptotic freedom and confinement

3 Linear Regge trajectories

4 YM for V0, V2

+

V1 = 14 p/T 4 at T = 2Tc

V3 = 170 e/T 4 at T = Tc (latent heat)

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 23 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Thermodynamic ResultsSummary of Parameters

HQCD Nc = 3 Nc → ∞ Parameter

m0++/√

σ 3.37 3.56 * 3.37 ** ℓs/ℓ = 0.15

[

p(N2

c T 4)

]

T→∞π2/45 π2/45 π2/45 Mℓ = [45(2π)2]−1/3

[

p(N2

c T 4)

]

T=2Tc

1.2 1.2 •• - V 1 = 14

Lh

(N2c T 4

c ) 0.31 0.28 • 0.31 •• V 3 = 170

Table: *=Chen-et al.’05,**=Lucini-Teper’01,•=Boyd-et al.’96,••=Lucini-Teper-Wenger’05

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 24 / 25

Introduction The model Thermodynamics Hydrodynamics Numerics

Thermodynamic ResultsSummary of Results

HQCD Nc = 3 Nc → ∞

m0∗++/m0++ 1.61 1.56(11) * 1.90(17) **

m2∗++/m2++ 1.36 1.40(4) * 1.46(11) **

Tc/m0++ 0.167 - 0.177(7) ••

Table: *=Chen-et al.’05,**=Lucini-Teper’01,•=Boyd-et al.’96,••=Lucini-Teper-Wenger’05

Liuba Mazzanti (USC) Holographic Hydrodynamics from 5d Dilaton–Gravity Kolymbari, September 12, 2010 25 / 25