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Momentum Broadening of Heavy Probes in Strongly Couple Plasmas

Jorge Casalderrey-Solana

Lawrence Berkeley National Laboratory

Work in collaboration with Derek Teaney

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Outline

Momentum transfer in gauge theoriesLangevin model for Heavy Quarks

Momentum transfer from density matrix

Momentum transfer in terms of Wilson lines (non perturbative def.)

Computation in strongly couple N=4 SYM

String solution spanning the Kruskal plane

Small fluctuations of the string contour and retarded correlators

Comparison with other computations

Static Quark

Quark Moving at finite v

Analysis of fluctuations

World Sheet horizon

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Langevin Model for Heavy Quarks Moore, Teaney 03

TMp ~

T

1 MT

TTM

HQ classical on medium correlations scale

white noise '' tttt yy

Einstein relations:MTD 2

22TD

iDii pt

dt

dp

random force drag

Mean transfer momentum

Heavy Quark with v<<1 neglect radiation

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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

Dressed field strength

tie 1y

2y0

Fyv(t1,y1)t1y1

Fyv(t2,y2)t2y2

Four possible correlators

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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 MapAs in black holes, (t, r) –coordinates are only defined for r>r0

Proper definition of coordinate, two copies of (t, r) related by time reversal.

In the presence of black branes the space has two boundaries (L and R)

SUGRA fields on L and R boundaries are type 1 and 2 sources

Son, Herzog (02)

The presence of two boundaries leads to properly defined thermal correlators (KMS relations)

Maldacena

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Static String Solution=Kruskal PlaneThe string has two endpoints, one at each boundary.

Minimal surface with static world-line at each boundary

Transverse fluctuations of the end points can be classified in type 1 and 2, according to L, R

Fluctuation at each boundary

V=0

U=0

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Fluctuations of the Quark World LineOriginal (t,u) coordinates:

The string falls straight to the horizon

Small fluctuation problem: Solve the linearized equation of motion

At the horizon (u1/r20)

4/1);( iti ueuY

Which solution should we pick?

4/* 1);( iti ueuY

infalling

outgoing

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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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Retarded Correlators and Defining:

),(),(1

lim'

*2/10

22

uYuYu

TRG uuR

We obtain KMS relations (static HQ in equilibrium with bath)

30 Im

2lim TG

TRu

And the static momentum transfer

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Consequences for Heavy QuarksUsing the Einstein relations the diffusion coefficient:

It is not QCD but… from data:

For the Langevin process to apply

TD

2

63

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

Heavy Quark forced to move with velocity v: Wilson line x=vt at the boundary

Energy and momentum flux through the string:

same

Very small relaxation times (t0=1/D)

charm bottom

Fluctuation-dissipation theorem

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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?

2/11u

rz

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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

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Momentum BroadeningTaking derivatives of the action:

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 TTws

The mean transfer momentum isdiverges in v 1 limit

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

2TM

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Conclusions

We have provided a “non-perturbative” definition of the momentum diffusion coefficient as derivatives of a Wilson Line

This definition is suited to compute 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

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Back up Slides

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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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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

~,

cTT

Nq

k

dkq

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

ETC

Cq

A

RR

2

2

3ˆ

HTL provide the initial conditions for evolution.

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Charm Quark Flow ?(Moore, Teaney 03)

(b=6.5 fm)

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Heavy Quark Suppression (Derek Teaney, to appear)

Charm and Bottom spectrum from Cacciari et al.

Hadronization from measured fragmentation functions

Electrons from charm and bottom semileptonic decays

non photonic electros

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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

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Heavy Quark Partition FunctionMcLerran, Svetitsky (82)

Polyakov Loop

YM + Heavy Quark states YM states

Integrating out the heavy quark

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Changing the Contour Time

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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 :

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Check in Perturbation Theory

= optical theorem

= imaginary part of gluon propagator

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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