Momentum Broadening of Heavy Probes in Strongly Couple Plasmas

1
Momentum Broadening of Heavy Probes in Strongly
Couple Plasmas

• Jorge Casalderrey-Solana
• Lawrence Berkeley National Laboratory

Work in collaboration with Derek Teaney
2
Outline
Momentum transfer in gauge theories
Langevin model for Heavy Quarks
Momentum transfer from density matrix
Momentum transfer in terms of Wilson lines (non
perturbative def.)
Computation in strongly couple N4 SYM
Static Quark
String solution spanning the Kruskal plane
Small fluctuations of the string contour and
retarded correlators
Comparison with other computations
Quark Moving at finite v
Analysis of fluctuations
World Sheet horizon
3
Langevin Model for Heavy Quarks
Moore, Teaney 03
Heavy Quark with vltlt1? neglect radiation
? HQ classical on medium correlations scale
? white noise ?
Mean transfer momentum
Einstein relations
4
Density Matrix of a Heavy Quark
Eikonalized ? EgM gtgt T
momentum change
medium correlations
observation
Fixed gauge field ? propagation color rotation
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)
5
Momentum Broadening
Transverse momentum transferred
Transverse gradient ? Fluctuation of the Wilson
line
Dressed field strength
6
Wilson Line From Classical Strings
Heavy Quark ? Move one brane to 8.
The dynamics are described by a classical
string between black and boundary barnes
Nambu-Goto action ? minimal surface with boundary
the quark world-line.
If the branes are not extremal they are black
branes? horizon
Which String Configuration corresponds to the 1-2
Wilson Line?
7
Kruskal Map
As in black holes, (t, r) coordinates are only
defined for rgtr0
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
Maldacena
Son, Herzog (02)
The presence of two boundaries leads to properly
defined thermal correlators (KMS relations)
8
Static String SolutionKruskal Plane
The string has two endpoints, one at each
boundary.
Minimal surface with static world-line at each
boundary
V0
U0
Transverse fluctuations of the end points can be
classified in type 1 and 2, according to L, R
? Fluctuation at each boundary
9
Fluctuations of the Quark World Line
Original (t,u) coordinates
The string falls straight to the horizon
Small fluctuation problem Solve the
linearized equation of motion
At the horizon (u?1/r20)
infalling
outgoing
Which solution should we pick?
10
Boundary Conditions for Fluctuations
Son, Herzong (Unruh)
Negative frequency modes
near horizon
?
Positive frequency modes
11
Retarded Correlators and k
Defining
We obtain KMS relations (static HQ in equilibrium
with bath)
And the static momentum transfer
12
Consequences for Heavy Quarks
Using the Einstein relations the diffusion
coefficient
It is not QCD but from data
For the Langevin process to apply
13
Drag Force
(Herzog, Karch, Kovtun, Kozcaz and Yaffe
Gubser)
Heavy Quark forced to move with velocity v
? Wilson line xvt at the boundary
Energy and momentum flux through the string
same k !
Fluctuation-dissipation theorem
Very small relaxation times (t01/hD)
charm
bottom
14
Fluctuations of moving string
String solution at finite v
discontinous across the past horizon (artifact)
Small transverse fluctuations in (t,u) coordinates
Both solutions are infalling at the AdS horizon
Which solution should we pick?
15
World Sheet Horizon
We introduce
Same as v0 when
The induced metric is diagonal
World sheet horizon at
16
Fluctuation Matching
The two modes are infalling and outgoing in the
world sheet horizon
Close to V0 both behave as v0 case ? same
analyticity continuation
The fluctuations are smooth along the future
(AdS) horizon.
(prescription to go around the
pole)
17
Momentum Broadening
Taking 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
diverges in v ? 1 limit
The mean transfer momentum is
But the brane does not support arbitrary large
electric fields (pair production)
18
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 k in N4 SYM
by means of the AdS/CFT correspondence.
The calculated k scales as and takes much
larger values than the perturbative extrapolation
for QCD.
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 momentum broadening k at finite v diverges as
but the calculation is limited to
.
19
Back up Slides
20
Computation of (Radiative Energy Loss)
(Liu, Rajagopal, Wiedemann)
Dipole amplitude two parallel Wilson lines
in the light cone
Order of limits
String action becomes imaginary for
For small transverse distance
entropy scaling
21
Energy Dependence of
(JC X. N. Wang)
From the unintegrated PDF
Evolution leads to growth of the gluon density,
In the DLA
HTL provide the initial conditions for evolution.
Saturation effects ?
For an infinite conformal plasma (LgtLc) with
Q2max6ET.
At strong coupling
22
Charm Quark Flow ?
(Moore, Teaney 03)
(b6.5 fm)
23
Heavy Quark Suppression
(Derek Teaney, to appear)
non photonic electros
Charm and Bottom spectrum from Cacciari et al.
Hadronization from measured fragmentation
functions
Electrons from charm and bottom semileptonic
decays
24
Noise from Microscopic Theory
HQ momentum relaxation time
Consider times such that

microscopic force (random)
?
charge density
electric field
25
Heavy Quark Partition Function
McLerran, Svetitsky (82)
YM states
YM Heavy Quark states
Integrating out the heavy quark
Polyakov Loop
26
Changing the Contour Time
27
k as a Retarded Correlator
k is defined as an unordered correlator
From ZHQ the only unordered correlator is
Defining
In the w?0 limit the contour dependence
disappears
28
Check k in Perturbation Theory

optical theorem
imaginary part of gluon propagator
?
29
Force Correlators from Wilson Lines
Integrating the Heavy Quark propagator
Which is obtained from small fluctuations of the
Wilson line