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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