Strongly Coupled Plasmas and Gauge/String Duality
Laurence Yaffe
University of Washington
UVa/UKy Colloquia, April 2007 – p. 1/28
Outline
1. Heavy ion collisions and quark gluon plasma (QGP)2. Strongly coupled versus weakly coupled systems3. Theoretical techniques4. Gauge/string duality5. Applications to strongly coupled non-Abelian plasmas6. Outlook
UVa/UKy Colloquia, April 2007 – p. 2/28
Relativistic heavy ion collisions
RHIC: Au+Au, 200 GeV/nucleonrunning since 2000
LHC: Pb+Pb, 5.5 TeV/nucleonstarting 2008
−v
v
UVa/UKy Colloquia, April 2007 – p. 3/28
Relativistic heavy ion collisions: timeline
time
?
0 “splat” parton liberation (small x gluons)very high density gluon “soup”
1 fm/c “thermalization” isotropization of phase space distribution
quark-gluon plasma
expansion, cooling
hadronization10 fm/c expanding hadron gas
˜̃detection
UVa/UKy Colloquia, April 2007 – p. 4/28
Relativistic heavy ion collisions: spacetime view
beam direction
time
freeze−out
hadronization
"thermalization"
π pγµµ+ −
partons
QGP
hadrons
UVa/UKy Colloquia, April 2007 – p. 5/28
Relativistic heavy ion collisions: ideal hydrodynamics?
Single particle spectraAu+Au, 130 GeV/nucleon
0 1 2 3
10−2
100
102
pT (GeV)
1/ 2
π dN
/dyp
Tdp T
(GeV
−2) π
+ PHENIXp PHENIXp STARπ+ hydrop hydro
most central0 0.5 1 1.5 2 2.5
10−6
10−4
10−2
100
102
pT (GeV)1/
2π d
N/dy
p Tdp
T (G
eV−2
) PHENIXSTARhydro
π−
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %
60 − 92 %
centrality
0 1 2 3 410−8
10−6
10−4
10−2
100 PHENIXSTARhydro
p
pT (GeV)
1/2π
dN/
dyp T
dpT
(GeV
−2)
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %60 − 92 %
centrality
0 0.5 1 1.5 2
10−6
10−4
10−2
100
pT (GeV)
1/2π
dN/
dyp T
dpT
(GeV
−2)
PHENIXhydro
K+
0 − 5 %
5 − 15 %
15 − 30 %
30 − 60 %
60 − 92 %centrality
U. Heinz & P. Kolb, hep-ph/0204061
Elliptic flow: momentum ¢rality dependence
0 1 2 3 40
5
10
15
20
25
pT (GeV)
v 2 (%
) STARPHENIXEOS QEOS H
h+/−
0 0.25 0.5 0.75 10
2
4
6
8
10
nch/nmax
v 2 (%
)
eWNsWNeBCsBCSTAR
h+/−
U. Heinz, nucl-th/0512051
QGP at RHIC: nearly ideal fluid
UVa/UKy Colloquia, April 2007 – p. 6/28
QGP at RHIC:
Nearly ideal fluid =⇒ low viscosity, short mean free path=⇒ strongly coupled plasma
Theoretical challenge: predict dynamic properties ofstrongly coupled QGP
viscosity
energy loss (dE/dx)
emission spectra...
UVa/UKy Colloquia, April 2007 – p. 7/28
Weak coupling versus strong coupling
interactions “small”
mean free path,screening length
� particle separation,deBroglie wavelength
high mobility & viscosity
Ex: dilute gasesdilute, hot astrophysical plasmasQGP at T ≫ 1 GeV
| {z }
experimentallyinaccessible
interactions dominant
mean free path,screening length
.particle separation,deBroglie wavelength
low mobility & viscosity
Ex: liquidscool, dense plasmasQGP at T ≈ few × 100 MeV
| {z }
RHIC/LHC
UVa/UKy Colloquia, April 2007 – p. 8/28
Theoretical physics
Fantasy
L = · · ·
masses = · · ·dσ
dΩ= · · ·
P (T ), ν, ζ = · · ·
ACME
SolverUniversal
UVa/UKy Colloquia, April 2007 – p. 9/28
Theoretical physics
Fantasy
L = · · ·
masses = · · ·dσ
dΩ= · · ·
P (T ), ν, ζ = · · ·
ACME
SolverUniversal
Reality
Perturbation Numerical
Smoke &
Theory toolbox
Exactlytheory simulations
mirrors
modelssoluble
UVa/UKy Colloquia, April 2007 – p. 9/28
Perturbation theory
QEDα =
e2
4π~c=
1
137. · · · ,ge−2
2=
α
2π− 0.328
“ α
π
”2+ 1.181
“ α
π
”3− 1.510
“ α
π
”4+ · · ·
high energy QCD
αs(E) =g2(E)
4π∝ 1
ln(E/ΛQCD)
very high temperature QGPλdeBroglie
λmfp∼ g4(T )
sometimes works great...
UVa/UKy Colloquia, April 2007 – p. 10/28
Numerical simulations
Very useful in: Classical dynamical systemsdeterministic dynamics, no quantum fluctuations
Equilibrium classical statistical systemsstochastic dynamics ⇒ averaging over probability distribution
Equilibrium quantum theoriesground state properties, particle energies, ...(QCD lattice gauge theory)
Not so good for: Real-time dynamics of quantum field theoriesnasty interference in Feynman path integral,
〈φ̂(x1)φ̂(x2) · · · 〉 =R
Dφ(x) eiS[φ]/~ φ(x1) φ(x2)···
R
Dφ(x) eiS[φ]/~
many degrees of freedom: no good algorithm
UVa/UKy Colloquia, April 2007 – p. 11/28
String theory
1968 Introduced as phenomenological model of strong interactions Veneziano, ...But: unphysical tachyon, massless spin-2 particle, ...
1974 Reinterpreted as possible theory of quantum gravity Scherk, SchwarzBut: needs 26 dimensions, no apparent connection to real world
1984 Realization that 10-d superstrings are anomaly free Green, Schwarz, ...‘Anomaly free’ ⇒ quantum-mechanically consistent
1995 Realization that theory describes “D-branes” too Polchinski, ...Dynamical surfaces of dimension > 1 on which strings can end
1997 Realization that string theory on AdS5 × S5 spacetime Maldacena, ...is equivalent to a 4-dimensional quantum field theory
UVa/UKy Colloquia, April 2007 – p. 12/28
String theory controversy:
The “Theory of Everything,” or ?
Which view is right?
UVa/UKy Colloquia, April 2007 – p. 13/28
Progress in physics:
Often comes from asking the right question.
Also comes from learning what are bad questions:
What is simultaneous position and momentum of an electron?
What is the bare mass of an electron?
UVa/UKy Colloquia, April 2007 – p. 14/28
Progress in physics:
Often comes from asking the right question.
Also comes from learning what are bad questions:
What is simultaneous position and momentum of an electron?
What is the bare mass of an electron?
What is the fundamental gauge group?
Are there extra spacetime dimensions?
Is string theory “right”?
UVa/UKy Colloquia, April 2007 – p. 14/28
Strings on AdS5 × S5
AdS5︸ ︷︷ ︸const. negative
curvature
× S5︸︷︷︸const. positive
curvature
: 10-dimensional solution of General Relativityds2 = R2
du2
u2+ u2(−dt2 + d~x 2) + d~Σ25
ff
���
curvatureradius
AAK
AdS radialcoordinate
���
S5 metric
u → ∞ = AdS boundary = 4-d Minkowski space
IIB string theory on AdS5 × S5 : two adjustable (dimensionless) parametersR = curvature radius of AdS5 = radius of S5
`s = fundamental string length = 1/√
string tensionR/`s = curvature radius in string units
gs = string coupling = amplitude for string splitting/joining
“Easy” limit: gs � 1 and R/`s � 1 =⇒ classical (super)-gravity
UVa/UKy Colloquia, April 2007 – p. 15/28
Maldacena conjecture:
IIB string theory on AdS5 × S5 is exactly equivalent tomaximally supersymmetric 4-d Yang-Mills theory (“N = 4 SYM”)N =4 SYM : Non-Abelian gauge theory w. massless fermions & scalars
Two adjustable parameters:λ = interaction strength (’t Hooft coupling)
Nc = # “colors” = rank of SU(Nc) gauge group
Scale invariant (conformal) field theory (“CFT”)
AdS/CFT duality : Not (yet) rigorously proven, but supported by much evidenceMatching symmetries, operator dimensions, · · ·
AdS radius ∼ energy scale
Dictionary: (R/`s)4 = λ, gs = λ/(4πNc)
Weakly coupled string theory ⇐⇒strongly coupled, large Nc gauge theory
Many generalizations
UVa/UKy Colloquia, April 2007 – p. 16/28
N = 4 SYM vs. QCD: zero temperature
QCD N = 4 SYM
particles:mesons (π, K , ρ, · · · )baryons (n, p, ∆, · · · )
noneno S-matrix
RG flow:asymptotic freedomg2(E) ↘ as E ↗
nonefixed coupling λ
fundamental scale: ΛQCDnone
scale invariant CFT
Completely different properties!
UVa/UKy Colloquia, April 2007 – p. 17/28
N = 4 SYM vs. QCD: high temperature
QCD N = 4 SYMdeconfined non-Abelian plasma X X
Debye screening X X
finite spatial correlation length X X
neutral fluid hydrodynamics X X
Complete qualitative agreement
Try using N = 4 SYM as model for QGPWorks well at weak coupling
UVa/UKy Colloquia, April 2007 – p. 18/28
Gauge/string duality for hot plasma
N = 4 SYM at non-zero temperature dualto strings in AdS black hole geometry
uhhorizon
∞AdS boundary
u
x
black hole
ds2AdS−BH = R2
du2
f(u)+ f(u)(−dt2 + d~x 2)
ff
, f(u) = u2»
1 −“ uh
u
”4–
Hawking temperature = plasma temperature, T = π uh
Classical dynamics in 5-d AdS black hole geometry ⇒exact results in strongly coupled (λ → ∞, Nc → ∞) SYM plasma
UVa/UKy Colloquia, April 2007 – p. 19/28
Strongly coupled SYM plasma: selected results
Shear viscosity η = π8 N2c T 3From black hole absorption cross-section.Policastro, Son, Starinets
Viscosity/entropy density ηs
=~
4πkBUniversal lower bound?Kovtun, Son, Starinets
Heavy quark energy loss dEdx
=dp
dt= −π
√λT 2
2mp
From classical string dynamicsHerzog, Karch, Kovtun, Kozcaz, L.Y.
Photo-emission spectrum
2 4 6 8 10kT
0.005
0.01
0.015
0.02
0.025
0.03
0.035
kd
Gdk
SYM, Λ=0.75
SYM, Λ®¥
From Maxwell’s equationsHuot, Kovtun, Moore, Starinets, L.Y.
UVa/UKy Colloquia, April 2007 – p. 20/28
Heavy quark energy loss
Adding heavy quarks to N =4 SYM ⇔adding D7 brane to gravity dual
Quark moving through N =4 plasma ⇔open string moving in AdS-BH geometry
uh
u
horizon
AdS boundary
heavy mesonfree quark
∞
m
u
x
black hole
D7−brane
String dynamics:
world sheet coordinates: Xµ(σ, τ) ⇒ x(u, t) for planar world sheet, σ=u, τ=taction: −S = T0(world sheet area) = T0
Rdσdτ
√−g
induced metric: −g =“
∂X∂τ
· ∂X∂σ
”2−
“∂X∂τ
”2 “∂X∂σ
”2
= L4h
1 − u2f(u)
ẋ2 + u2f(u) x′2i
stationary solution: ẋ(u) = v =⇒ easy variational problem
first integral: x′(u) = ±v u2h
u2f(u)
UVa/UKy Colloquia, April 2007 – p. 21/28
Heavy quark energy loss: trailing string solution
x(u, t) = x0 + v(t + F (u))
F (u) = 12uh
hπ2− arctan u
uh− arccoth u
uh
i
Constant momentum flow down string= external force needed to maintain velocity= -(plasma drag force acting on moving quark)
v
quarku
x
D7−brane
black hole
A bit more work: −v vSeparating quark-antiquark pair (∼ cc̄ jets)HKKKY
World sheet fluctuations =⇒ ddt〈p2⊥〉, ddt 〈p2‖〉
Gubser; Casalderry-Solana, Teaney
Linearized gravitational corrections =⇒ 〈Tµν(x)〉P. Chesler, L.Y.
UVa/UKy Colloquia, April 2007 – p. 22/28
Heavy quark energy loss: wakes
〈Tµν(x)〉 ⇐ asymptotic near-boundary behavior of gµν
(x’(u’),u’)
x
δ〈Tµν(x)〉 ⇐ δgµν(x, u)˛˛˛u=∞
moving string
=R
du′ Gµν,αβ| {z }
(x,∞; x′(u′), u′) παβ(u′)| {z }
stress-energydensity ofmoving string
“bulk-to-boundary”graviton Green’s functionin AdS-BH background
Energy density, v = 0.75 c
(velocity of sound vs = c/√
3 = 0.577 c)
Supersonic shock in viscous plasma
Results valid on all length scales
P. Chesler
UVa/UKy Colloquia, April 2007 – p. 23/28
Heavy quark wake: v = 0.75 c
P. Chesler
UVa/UKy Colloquia, April 2007 – p. 24/28
Heavy quark wake: v = vs = 0.577 c
P. Chesler
UVa/UKy Colloquia, April 2007 – p. 25/28
Heavy quark wake: v = 0.25 c
P. Chesler
UVa/UKy Colloquia, April 2007 – p. 26/28
Outlook
Remarkable recent progress using gauge/string duality to map non-equilibrium, stronglycoupled plasma dynamics into “easy” classical dynamics (in curved space).
Comparisons of RHIC data and N =4 SYM predictions currently underway.
Much more to do: light quark dynamics,1/λ corrections,
important!���9 ���1/Nc corrections,
results for more QCD-like theories (with known gravity duals), ...
Goal: controlled extrapolation from λ=∞, N =4 SYM to real QGP near T = Tc.
interaction strength λ-.01 .1 1 10 100PSfrag replacements
QCDT ≫ Tc
QCDT & Tc
N =4 SYMλ � 1
N =4 SYMλ � 1
UVa/UKy Colloquia, April 2007 – p. 27/28
Photon yield: comparison to RHIC data
0.5 1.5 2.5 3.5 4.5pT (GeV)
10−610−510−410−310−210−1
100101
dN/d
2 pTd
y (G
eV−2
)
PHENIX Data (Preliminary) Thermal (Ti=236 MeV)Thermal + pQCD
Preliminary results, Jan-e Alam
N = 4 SYM spectrum plus perturbative QCD for high p⊥ tail=⇒ good fit with reasonable value of temperature T .
UVa/UKy Colloquia, April 2007 – p. 28/28
OutlineRelativistic heavy ion collisionsRelativistic heavy ion collisions: timelineRelativistic heavy ion collisions: spacetime viewRelativistic heavy ion collisions: ideal hydrodynamics?QGP at RHIC:Weak coupling kern 27pt versus kern 27pt strong couplingTheoretical physicsTheoretical physics
Perturbation theoryNumerical simulationsString theoryString theory controversy:Progress in physics:Progress in physics:
Strings on $AdS_5 imes S^5$Maldacena conjecture:$Nfour $ SYM vs. QCD:~~ darkred zero temperature$Nfour $ SYM vs. QCD:~~ darkred high temperatureGauge/string duality for hot plasmaStrongly coupled SYM plasma: selected resultsHeavy quark energy lossHeavy quark energy loss: trailing string solutionHeavy quark energy loss: wakesHeavy quark wake: $v = 0.75 , c$Heavy quark wake: $v = vs = 0.577, c$Heavy quark wake: $v = 0.25, c$OutlookPhoton yield: comparison to RHIC data