Heavy Quark Propagation in an AdS/CFT plasma

Jorge Casalderrey-SolanaLBNL

Work in collaboration with Derek Teaney

Outline

Langevin dynamics for heavy quarks

Calibration of the noise

Broadening from Wilson Lines

AdS/CFT computationBroadening at finite velocity

Langevin Dynamics

TMT11~

Heavy Quark M>>T moves slowly MTvth ~

de Broglie w. l. HQ is classical

)(tpdtdp

D

Random (white) noise )'()()'(),( tttt

Einstein relations

MTD 2

22TD

TMp ~

T1~

MT

Medium properties

The Langevin model to is used to describe charm and bottom quarks

Fits to data allow to extract the diffusion coefficient

TD

263

(Moore & Teaney, van Hees & Rapp)

Heavy Quarks at RHIC

How to Calibrate the NoiseIn perturbation theory colTp 2

But we cannot use diagrams!

M>>T long deflection timeDT

M

D

11

tpdtdp

D )(tFtdtdp

Dt /1

medt

xtExtQTxtQxdtF aa ,,,)( 3

)0()( FtFdt

Eq

Thermal average

t

t

duAi

ettU 00,

Broadening from Wilson Lines

00 ),( tQttUtQ

E(t1,y1)t1y1

E(t2,y2)t2y2

',,

',0',0',0,,,xxt

aa

aa xExQTxQxtExtQTxtQ

Small fluctuation of the HQ path

y

t

u0=1

v=0

Horizon

u=

AdS/CFT computationStatic Quark:straight string stretching to the horizon

We solve the small fluctuation problem

(mechanical problem)

In terms of the classical solution

3*

2/1

2

0 ),(),(1Im2lim TNguyuyu

Lcu

01414

62222

22

yuu

yuuuy uu

Broadening and Diffusion in AdS/CFT

3TNg c

In perturbation theory Ng 2~

Depends explicitly on Nc. Different from /s It is not universal!

Putting numbers

TD 2

To compare with QCD => rescale the degrees of freedom

4.2QCD

SYM

SS 6.1

QCD

SYM

SS (Liu, Rajagopal,

Wiedemann)

x

t

x=vt

c(u)=v u=1/

Probe at finite velocityTrailing string => work over tension

2

2

12 v

vTdtdp

3

21 TMTD same

Drag valid for all p !(Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser)

Broadening => repeat fluctuation problem

3TNg cT Depends on the energy of the probe!

ConclusionsWe have provided a “non-perturbative” definition of the momentum broadening as derivatives of a Wilson LineThis definition is suited to compute k in N=4 SYM by means of the AdS/CFT correspondence.

The results agree, via the Einstein relations, with the computations of the drag coefficient. This can be considered as an explicit check that AdS/CFT satisfy the fluctuation dissipation theorem.

The calculated scales as and takes much larger values than the perturbative extrapolation for QCD.

The momentum broadening at finite v diverges as .

Back up Slides

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

t

Lr0

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

2TM

LLqS eeW

ˆ41

For small transverse distance:

Order of limits: 11 v M2

String action becomes imaginary for

323.5ˆ TNgqSYM entropy scaling

fmGeVqQCD2126ˆ

Energy Dependence of (JC & X. N. Wang)

From the unintegrated PDF

Evolution leads to growth of the gluon density,

In the DLA x

q

TT

eqx1ln22

2

~,

cT

TNq

kdkq

2

2

22 2

Saturation effects cs LLQq ,min~ˆ 2

q̂

For an infinite conformal plasma (L>Lc) with Q2max=6ET.At strong coupling

ETCCq

A

RR

2

23ˆ

HTL provide the initial conditions for evolution.

Noise from Microscopic Theory

HQ momentum relaxation time: DDTM

mediumD

HQ ~1

Consider times such that mediumHQ

microscopic force (random)

charge density electric field

qE

Heavy Quark Partition FunctionMcLerran, Svetitsky (82)

Polyakov Loop

YM + Heavy Quark states YM states

Integrating out the heavy quark

Heavy Quark Partition FunctionMcLerran, Svetitsky (82)

Polyakov Loop

YM + Heavy Quark states YM states

Integrating out the heavy quark

as a Retarded Correlator

is defined as an unordered correlator:

From ZHQ the only unordered correlator is

Defining:

In the 0 limit the contour dependence disappears :

Force Correlators from Wilson Lines

Which is obtained from small fluctuations of the Wilson line

Integrating the Heavy Quark propagator:

Since in there is no time order:

E(t1,y1)t1y1

E(t2,y2)t2y2

Momentum Distribution

-T/2 +T/2

v f0 (b) ff (b)

duAi

e

mediumQ t bWC

Final distribution

Acf bWbTrbf )(

ab (

-b/2

, b/2

; -T)

A