A Holographic Study of Thermalization in

Strongly Coupled Plasmas

KEK, 21 June 2011

Masaki Shigemori(KMI Nagoya)

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CollaboratorsVijay Balasubramanian (U Penn),Berndt Müller (Duke)Alice Bernamonti, Ben Craps, Neil Copland, Wieland Staessens (VU Brussels)Jan de Boer (Amsterdam)Esko Keski-Vakkuri (Helsinki)Andreas Schäfer (Regensburg)

(Arxiv: 1012.4753, 1103.2683)

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An old & new question

What happens in nuclear matterat high temperature?

Low T

quarks, gluons: confined

High T

quarks, gluons: deconfined

~1012 Kelvin

Quark-Gluon Plasma (QGP)

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Heavy ion collision

QGP created at RHIC, LHC “Little Bang” Strongly coupled Difficult to study

QGP

heavynucleu

sheavynucleu

s

Perturbative QCD: not applicableLattice simulations: not always powerful enough

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Holographic approach

QCD SYM AdS string“~” =AdS/CFT

Equilibrated QGP HI collision Thermalization

AdS black hole Energy injection? BH formation

A toy model for QCD General insights into strongly coupled

physics

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Questions we ask & (try to) answer

What is the measure for thermalization in the bulk? Local operators are not sufficient

, etc. Non-local operators do better job

, etc.

What is the thermalization time?When observables become identical to the thermal ones

AdS boundary

local op

nonlocal op

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Outline0. Intro1. Review

1.1 HI collision1.2 Lattice1.3 Holographic approach

2. Holographic thermalization2.1 Models & probes for therm’n2.2 The model & results2.3 Summary

Mostly based on Casalderrey-Solana et al. 1101.0618

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1.1 QCD and heavy ion collision

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QCD at high T Conjectured phase diagram

Taken from FAIR website

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Heavy Ion Collision

Taken from BNL website

collision freeze-outnon-equilibriumstate

equilibratedQGP

Idea:

Actual:

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Scales in HI collision (1)

fm = Fermi = femtometer = m sec = 3 yoctoseconds

Aunucleus

10 fm

p n

1 fm

milimicr

onano

picofemt

oattozept

oyoct

o

hydrogen atom:

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Scales in HI collision (2)RHIC: Au LHC: Pb

0 .1 fm

Energy density on collision (RHIC, 5x crossover energy

density)8000 hadrons produced

(RHIC)

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Stages in HI collisionkinemati

c freezeout

chemicalfreezeout

(hadronization)

equilibration

hydrodynamic

evolution

(cf. )

equilibrated QGP

non-equilibrium

state

collision

𝑡

𝑇=150−180 MeV

𝑇=90MeV

hadronic gas

energy density~10 GeV/fm3 (RHIC, t=0.4 fm)~60 GeV/fm3 (LHC, t=0.25 fm)

thermalization time

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Bjorken flow Boost invariance (in central rapidity region)

(beam direction)

phenomenologicallyconfirmed

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Elliptic flow

beam direction

more hadron

s

less

𝑑𝑁𝑑2𝐩𝑇 𝑑𝑦

∝1+2𝑣2 cos (2𝜙 )+…

“elliptic flow” parametrizes asymmetryin hadron multiplicity

is important for determining hydro properties of QGP

𝜙

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Phenomenology of HI collision (1)

Ideal hydro after explains Very fast thermalization Larger spoils agreement Perturbative QCD fails:

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Phenomenology of HI collision (2) QGP has very small shear viscosity - entropy density

ratio:

The smallest observed in nature (water ~380, helium ~9)

(ultracold atoms at the unitarity point has similar ) Perturbative QCD fails:

Short mean-free-path no well-defined quasiparticle Strongly coupled; perturbative approach invalid LHC: comparably strongly coupled; comparably small

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Phenomenology of HI collision (3)

All hadron species emerge from a single common fluid

Baryon chemical potential is small: Jet quenching: 𝜇

𝑇 You are here!

QGP

jet

quark plowing through

Medium modifies jets Jets modifies medium

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Phenomenology of HI collision (4) Quarkonia: mesons made of

production suppressed

proton =

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A field theory model of HI collision Color glass condensate

multiplicity

rapidity

mesons

nucleons with large rapidity

Most particles originate from the “wake” of the

nuclei

color flux

collision

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1.2 Lattice approach

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Lattice QCD (1) Good at thermodynamics

Deconfinement is crossover

More conformal for larger (but non-conformal at )

Still, deconfined (Polyakov loop ≠ 0) Made of quarks & gluons But can’t be treated perturbatively

≲¿

CFT is not a crazy model

trace

ano

mal

y

sites in one dir

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Lattice QCD (2) At pioneering stage about transport properties

Real-time correlators difficult Shear viscosity

Not (yet) applicable to non-equilibrium properties

𝜂𝑠=

1.2−1.74𝜋 (𝑇=1.2−1.7𝑇 𝑐)

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1.3 Holographic approach

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Holographic approach Want to study strongly coupled phenomena in QCD Toy model: SYM

SYM String theoryin AdS5

holo. dualAdS/CFTStrong coupling

VacuumThermal state(equilibrated

plasma)Thermalization

Weak couplingEmpty AdS5

AdS BH

BH formation

𝑧 𝑧=0

AdS boundary

UVIR

horizon

𝑧=𝑧 𝐻

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Difference between QCD and Is SYM really “similar” to QCD???

QCD confines at low T, but doesn’t confineAt QCD deconfines and is similar to

is CFT but QCD isn’tQCD is near conformal at high T ( achieved initially)

is supersymmetric but QCD isn’t at T>0, susy broken

QCD is asymptotically freeExperiments suggest it’s strongly coupled at high T

…

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Holo approach: equilibrium Shear viscosity

𝜂𝑠=

14𝜋 (1+ 15𝜁 (3 )

𝜆3 /2+…)

Holography:

Cf. weak coupling:𝜂

𝑠=𝐴

𝜆2 log (𝐵/√𝜆)

𝜆

𝜂 / 𝑠

1/ 4𝜋 universal

theory (A,B) dependent

No quasiparticles No propagating light DoFs

Only quasi-normal modes (same as QGP)

[Policastro+Son+Starinets]…

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Holo approach: near-equilibrium Parton energy loss [Herzog et al.][Gubser]

Brownian motion / transverse momentum broadening Brownian

motion

[de Boer+Hubeny+Rangamani+MS][Son+Teaney]…

Can derive Langevin eq.

force

Quark = string ending on boundary Extract drag coeff / diffusion const.

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2. Probing thermalizationby holography

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Non-equilibrium phenomena (Near-)equilibrium physics (dual: BH)

Well-studied as we saw Thermalization process (dual: BH formation)

Poorly understood Occurs fast: (cf. ) pQCD not applicable Lattice QCD insufficient

Non-equil. physics of strongly coupled systems:terra incognita

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Basics of holography AdS spacetime

𝑡

UVIR

AdSboundary

bulk

𝑧

𝒙

𝑧=0

𝑧=∞

AdS spacetime(“bulk”)

𝑑𝑠2=𝑅𝐴𝑑𝑆2 𝑑 𝑧2−𝑑𝑡 2+𝑑𝒙❑

2

𝑧2

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Holographic probes – local op 1-point function

𝜙 (𝑧 , 𝑥 )=𝛼(𝑥 )𝑧 4−Δ+…+𝛽 (𝑥 )𝑧Δ+…

𝜙𝒪 bulk field

boundary

operator

perturbation vevΔ𝑆= ∫ 𝑑4 𝑥𝛼 (𝑥 ) 𝒪(𝑥) ⟨𝒪 (𝑥 ) ⟩=𝛽 (𝑥 )

𝑧 𝑧=0

boundary

1-point function

knows only about UV

UVIR

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Holographic probes – non-local op 2-point function

Non-local ops. know also about IR

𝑧 𝑧=

0

boundary

⟨𝒪 (𝑥 )𝒪 (𝑥 ′ ) ⟩ ∼𝑒−𝑚ℒ

𝑥𝑥 ′geodesic

ℒ

UVIR

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2.1 Models & probes of thermalization

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Models for holographic thermalization

Initial cond? Not clear how to implement

initial cond in bulk Various scenarios

• Collision of shock waves [Chesler+Yaffe]…

• Falling string [Lin+Shuryak]…

• Boundary perturbation [Bhattacharyya+Minwalla]…

Thermalization

BH formation

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The bulk spacetime

shock wave

AdSBH

empty AdS

turn off pert.turn on pert.

AdS boundary

𝑡𝑧

event

horizo

n

singularity

UVIR

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Probes of thermalization (1)

Instantaneous thermalization? Inside = empty AdS

Outside = AdS BH 1-point function

instantaneously thermalizes

1-pt func:same as that in thermal state

shock wave

AdSBH

empty AdS

turn off pert.turn on pert.AdS boundary

𝑡𝑧

event

horizo

n

singularity

[Bhattacharyya+Minwalla]

UVIR

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Probes of thermalization (2)

Local operators are not sufficient They know only about near-bndy

region They probe UV only

Non-local operators do better job They reach deeper into bulk

They probe IR also They know more detail

about thermalization process

AdS boundary

local opnon-localop

𝑡𝑟

UVIR

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Non-local operators 2-point function

Bulk: geodesic (1D) Wilson line

Bulk: minimal surface (2D) Entanglement entropy

Bulk: codim-2 hypersurface

AdS boundary

bulk

geodesic

AdS boundary

bulk

minimal surface

Let’s study these during BH formation process!

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2.2 The model and results

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The model: Vaidya AdS Null infalling shock wave in AdS (Vaidya AdS

spacetime)𝑑𝑠2= 1𝑧 2

[− (1−𝑚 (𝑣 )𝑧𝑑)𝑑𝑣2−2𝑑𝑧 𝑑𝑣+𝑑 �⃗�2]

Thin shell limit𝑚 (𝑣 )=𝑀𝜃 (𝑣 )

We study AdSD with D=3,4,5 field theory in d=2,3,4

𝑧=0𝑧=∞

UV (AdS boundary)IR

BH forms at late time

𝑣𝑧

�⃗�

Simple setup amenable to detailed study

sudden, spatially homogen. injection of energy

Sudden & spatially homog. injection of energy

⨂

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Nonlocal probes we consider

Bulk spacetim

e

Dim of bulk

probe Operator

AdS3 1 Geodesic = EE

AdS41 Geodesic

2 Wilson line = EE

AdS51 Geodesic2 Wilson line3 EE

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What we computed AdS3

Can be solved analytically in thin shell limitCf. Numerically done in [Abajo-Arastia+Aparicio+Lopez 1006.4090]

AdS4 Numerical

Cf. Partly done in [Albash+Johnson 1008.3027]

AdS5 Numerical

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Geodesics in AdS3 (1) Equal-time geodesics

Geodesic connecting two boundary points at distance ,at time after energy was deposited into system

Refraction cond across shock wave

bulk

shock wave

geodesic

𝑥𝑧

𝑣=𝑡0

ℓ

𝑣=0

AdS boundary

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Geodesics in AdS3 (2) Analytic expression

ℒ−ℒ therm=2 log [ sinh (𝑟𝐻𝑡 0 )𝑟𝐻 √1−𝑐2 ] ,

ℓ= 1𝑟𝐻 [ 2𝑐𝑠 𝜌 + ln ( 2 (1+𝑐 )𝜌 2+2𝑠 𝜌−𝑐  

2 (1+𝑐 )𝜌2−2 𝑠 𝜌−𝑐   )] , 𝜌=12 coth (𝑟𝐻 𝑡0 )+ 12 √coth2 (𝑟𝐻 𝑡0 )− 2𝑐

𝑐+1

𝑟𝐻≡√𝑀 ,

Equal-time geodesics for fixed and

Equal-time geodesics for fixed and

𝑣=𝑡0

ℓ

𝑣=0ℒ

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Result: geodesic lengthℒ−ℒtherm

ℓ=1

ℓ=2

ℓ=3

ℓ=4

AdS3 AdS4 AdS5

Given fixed compute as function of Renormalize by subtracting thermal value If we wait long enough, (thermalize) Larger It takes longer to thermalize ― “Top-down”

𝑣=𝑡0

ℓ

𝑣=0ℒ

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“Thermalization time” from geodesicsAdS3 AdS4 AdS5

𝜏1/2

𝜏𝑐𝑟𝑖𝑡

𝜏𝑚𝑎𝑥𝜏𝑐𝑟𝑖𝑡

𝜏𝑐𝑟𝑖𝑡𝜏𝑚𝑎𝑥 𝜏𝑚𝑎𝑥

𝜏1/2 𝜏1/2

has steepest slope becomes equal to the thermal value

: becomes half the thermal value

is as expected from causality bound Others would indicate : superluminous propagation??

Should look at observable that gives largest

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“Causality bound”

𝑥

𝑡

ℓ2

𝑂ℓ

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Other non-local observables

Similar behavior for all probes – “Top-down” thermalization

AdS4

AdS5

CircularWilson loop

RectangularWilson loop

Entanglement Entropy

𝒜−𝒜therm

𝒜−𝒜therm

N/A

ℓ

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Thermalization time

AdS3

AdS4

AdS5

geodesic circularWilson loop

Rectangular Wilson loop

EE fora sphere

EE for disk/sphere saturates the causality bound Infinite rectangle doesn’t have one scales – reason for larger ?

𝜏1/2

𝜏𝑐𝑟𝑖𝑡𝜏𝑚𝑎𝑥

N/A N/A N/A

N/A

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Infinite rectangleℓ

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2.3 Summary

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Summary Studied various nonlocal probes during

thermalization after sudden injection of energy

Different probes show different thermalization time

Largest for codim-2 probes Equilibration propagates at speed of light

“Top-down” thermalization Thermalization proceeds from UV to IR “Built-in” in the bulk,

but not in weakly-coupled field theory

shock wave

AdSBH

empty AdS

event

horizo

n

singularity

UVIR

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An “estimate” of therm’n time in HI collision

𝜏crit=ℓ /2

ℓ 𝑇=300−400MeV

𝜏crit 0.3 fm/𝑐Reasonably

short!Cf.

𝜏 pQCD≳2.5 fm/𝑐

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Future directions More realistic backgrounds Toward emergent boost invariance

Falling string [Lin-Shuryak] Approach to Janik-Peschanski solution

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Thanks!

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Extra material

58

“Swallowtail” phenomenon [Albash+Johnson]𝒜−

𝒜 therm

For given , there are three possible minimal area surfaces

Rectangular Wilson loop for AdS4

ℓ(𝒜−𝒜

therm)

59

“Swallowtail” in quasi-static shellℒren

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EE as “coarse-grained” entropy

(Left) Maximal growth rate of entanglement entropy density vs. diameter of entangled region for d = 2; 3; 4 (top to bottom). (Middle) Same plot for d = 2, larger range of .(Right) Maximal entropy growth rate for d = 2.