Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma
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Transcript of Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma
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