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AdS/CFT Calculations of Parton Energy Loss

Jorge Casalderrey-Solana

Lawrence Berkeley National Lab.

In collaboration with D. Teaney

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Why N = 4 Yang Mills?The QGP at T=(1-2)Tc may be strongly coupled:

Strong flow observed at RHIC consistent with ideal hydro.

Transport requires cross sections 10 times larger than pQCD.

Estimates and calculations on QGP shear viscosity yield small values.

N=4 Yang Mills can be solved at strong coupling via AdS/CFT Pressure at strong coupling is ¾ PStephan-Boltzmann .

Lattice QCD similar deviations are observed at T=(1-2)Tc .

Shear viscosity conjectured minimal bound. 41s

But N=4 is not QCD (scale invariant, supersymmetry…)However AdS/CFT is the only method available to address strongly coupled gauge theories

Strong jet quenching (opaque medium).

s(T) is large.

Non perturbative access to real time dynamics of gauge theories.

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Density Matrix of a Heavy QuarkEikonalized E=M >> T

mediumQ

momentum change

medium correlations

observation

Fixed gauge field propagation =color rotation

duAi

e

Evolution of density matrix:

Un-ordered Wilson Loop!

Introduce type 1 and type 2 fields as in no-equilibrium and thermal field theory (Schwinger-Keldish)

0

00 ,2

, zr

xt

tvzr

xt 0,2

,

tvzr

xt 0,2

,

00 ,2

, zr

xt

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Momentum BroadeningTransverse momentum transferred

Transverse gradient Fluctuation of the Wilson line

Four different correlators:

tie 1y

2y0

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Wilson Line From Classical Strings

If the branes are not extremal they are “black branes” horizon

Heavy Quark Move one brane to ∞. The dynamics are described by a classical string between black and boundary barnes

Nambu-Goto action minimal surface with boundary the quark world-line.

Which String Configuration corresponds to the 1-2 Wilson Line?

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Kruskal Map

Black hole two copies of the (boundary) field theory (Maldacena)

r

t

r0

Herzog & Son: Fluctuations on (R, L) can be matched so that field correlators have the correct analytic properties (KMS relations)

Each (boundary) fields are identified with type 1 and 2.

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Fluctuations (static) of the Quark World LineSmall fluctuation problem (linearized)Near the horizon (u1/r2

0)

4/1);( iti ueuY

4/* 1);( iti ueuY

infalling

outgoing

),(),(

1lim

'*

2/10

22

uYuYu

TRG uuR

Son-Herzog prescription:

30 Im

2lim TG

TRu

KMS relation (static quarks in equilibrium)

Ng 2

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Consequences for Heavy QuarksHeavy probe on plasma => Brownian Motion (Langevin dynamics)

HQ Diffusion coefficient (Einstein):

Putting numbers:

It is not QCD but… QCD at weak coupling:

Different number of degrees of freedom:

Liu, Rajagopal, Wiedemann

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3

2

1T

MTD

Drag Force (Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser)

Direct computation: HQ forced to move with velocity v:

Wilson line x=vt at the boundary

Energy and momentum flux through the string:

same Fluctuation-dissipation theorem

iDii pt

dt

dp Langevin: '' tttt Tyy

2

2

12 v

vT

dt

dp

MTT

D 2

Einstein relation:

Drag force valid for ultra relativistic particle.

Add an external electric field to valance the dragv

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Broadening of a Fast ProbeFluctuations of the “bending” string:

World sheet horizon at 1z

Complications:Discontinuity in the past horizon

Similar to KMS relation but: GR is infalling in the world sheet horizon The temperature of the correlator is that of the world sheet black hole (blue shift?)TTws

New Scale! TTws

3T Diverges in ultra relativistic limit!

But the brane does not support arbitrary large electric fields (pair production) 2TM

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Computation of (Radiative Energy Loss)q̂(Liu, Rajagopal, Wiedemann)

t

L

r0

Dipole amplitude: two parallel Wilson lines in the light cone:

2TM

LLqS eeW

ˆ4

1For small transverse distance:

Order of limits:

11 v M2

String action becomes imaginary for

323.5ˆ TNgqSYM entropy scaling

fmGeVqQCD2126ˆ

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Conclusions

AdS/CFT provides a rigorous way to address the physics at strong coupling.

The computed transport coefficient have remarkable features

Large valuesUnusual coupling dependencesEnergy dependence of the momentum broadening

Many other application of AdS/CFT to Heavy Ion phenomenology (fields associated to probe, hydrodynamics, production of fireball)

The applicability of these results demand phenomenological work to explain them in a way which can be translated to QCD.

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Boundary Conditions for Fluctuations

RL

F

P

V=0

U=0

LR

RLuYuY RL ,0

,);();(.

Son, Herzong (Unruh):

Negative frequency modes LR YeYY ***

Positive frequency modes LR YeYY iV

iU

near horizon

LR

RLuYuY RL

,0

,);();(

*

.*

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Fluctuations of moving string

String solution at finite v

discontinous across the “past horizon” (artifact)

Small transverse fluctuations in (t,u) coordinates

;1);( 4/ uFueuY iitii

;1

1);( 0

2/4/ uFuueuY

iitio

Both solutions are infalling at the AdS horizon

0z

1z

1z

Which solution should we pick?

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World Sheet HorizonWe introduce

The induced metric is diagonal

Same as v=0 when zz ˆ

World sheet horizon at 1z

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Fluctuation MatchingThe two modes are infalling and outgoing in the world sheet horizon

)ˆ;ˆ(ˆ);( uYuY i )ˆ;ˆ(ˆ);( * uYuY o

Close to V=0 both behave as v=0 case same analyticity continuation

The fluctuations are smooth along the future (AdS) horizon.

Along the future world sheet horizon we impose the same analyticity condition as for in the v=0 case.

0ˆ U0U

(prescription to go around the pole) 1z ˆ