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