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)
4
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
5
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
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“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.
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