Transverse Momentum Broadening of a Fast Quark in

a N=4 Yang Mills Plasma

Jorge Casalderrey-SolanaLBNL

Work in collaboration with Derek Teany

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

Momentum BroadeningFrom the final distribution

Transverse gradient Fluctuation of the Wilson line

Dressed field strength

tie 1y

2y0

Fyv(t1,y1)t1y1

Fyv(t2,y2)t2y2

Four possible correlators

TbfbT bfp 20

22 T

Wilson Lines in AdS/CFT

x

t

x

t

u0=1

v=0 x=vt

Horizon

u=

c(u)=v u=1/

Moving string is “straighten” by the coordinates

“World sheet black hole” at uws=1/

At v=0 both horizons coincide. Coordinate singularity!!

2

tanh

2

tanh

2

tan

2

tan1ˆ2/12/11

2/12/112/12/11

2/12/11

2/1

uuuutt

uu ˆ

Kruskal MapAs in usual black holes, (t, u) are only defined for u<u0

Two copies of the (t,u) patches “time reverted”

In the presence of black branes the space has two boundaries (L and R) . There are type 1 and type 2 sources.

These correspond to the doubling of the fieldsSon, Herzog (02)

Maldacena

u=1/r2

Global (Kruskal) coordinates

*2 zteU V

U

*2 zteV

2/112/11* tanhtan

2

1uuz

String Solution in Global Coordinates

v=0 smooth crossingv≠0 logarithmic divergence in past horizon. Artifact! (probes moving from -)

String boundaries in L, R universes are type 1, 2 Wilson Lines.

Transverse fluctuations transmit from L R

2/11tanln2

uvVv

x )(uvtx

String FluctuationsThe fluctuations “live” in the world-sheet

In the world-sheet horizon they behave as

4/ˆˆˆ ˆ1)ˆ;ˆ(ˆ iti ueuY infalling

outgoing 4/ˆˆˆ* ˆ1)ˆ;ˆ(ˆ iti ueuY Extension to Kruskal analyticity cond.

Negative frequency modes infalling

Positive frequency modes outgoing

Imposing analyticity in the WS horizon

3TT

(Son+Herzog)

1ˆ u

u

cNg 2

Check: Fluctuation Dissipation theoremSlowly moving heavy quark Brownian motion

tpdt

dpiiD

i '' tttt Tyy

Drag and noise are not independent. Einstein relation

MTD 2

Direct computation of the drag force(Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser)

The string needs work over the tension to bend.

This is the momentum lost by the quark

same Fluctuation-dissipation theorem

Consequences for Heavy QuarksBrownian motion => Diffusion equation in configuration space

HQ Diffusion coefficient (Einstein):

From Rapp’s talk

Different number of degrees of freedom!:T

D2

63

6.1QCD

SYM

S

S4.2

QCD

SYM

S

S

22TD

Putting numbers:

Broadening at Finite Velocity 3TT

It is dependent on the energy of the quark!

Diverges in the ultra-relativistic limit.

But the computation is only valid for

2TM

(maximum value of the electric field the brane can support)

There is a new scale in the problem TTws It almost behaves like a temperature (KMS like relations)

Is it a (weird) blue shift?

Non trivial matching with light cone value

Liu, Rajagopal, Wiedemann

ConclusionsWe have provided a “non-perturbative” definition of the momentum broadening as derivatives of a Wilson Line

This 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 but the calculation is limited to

2TM

Back up Slides

Computation of (Radiative Energy Loss)q(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

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

From the unintegrated PDF

Evolution leads to growth of the gluon density,

In the DLA x

q

T

T

eqx1

ln22

2

~,

cT

TNq

k

dkq

2

2

22 2

Saturation effects cs LLQq ,min~ˆ 2

q

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

ETC

Cq

A

RR

2

2

3ˆ

HTL provide the initial conditions for evolution.

Noise from Microscopic Theory

HQ momentum relaxation time: DDT

Mmedium

DHQ ~

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

Changing the Contour Time

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