1. Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma Jorge Casalderrey-Solana LBNL Work in collaboration with Derek Teany

2. Momentum Distribution -T/2+T/2 v f 0 (b) f f (b) Final distribution  ab (-b/2, b/2; - T ) A

3. Momentum Broadening From the final distribution Transverse gradient  Fluctuation of the Wilson line Dressed field strength F y  v  (t 1,y 1 )  t 1  y 1 F y  v  (t 2,y 2 )  t 2  y 2 Four possible correlators T

4. Wilson Lines in AdS/CFT x t x t u 0 =1 v= 0 x=vt Horizon u=  c(u)=v u=1/  Moving string is “straighten” by the coordinates “World sheet black hole” at u ws =1/  At v= 0 both horizons coincide. Coordinate singularity!!

5. Kruskal Map As in usual black holes, (t, u) are only defined for u<u 0 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 fields Son, Herzog (02) Maldacena u=1/r 2 Global (Kruskal) coordinates V U

6. String Solution in Global Coordinates v= 0 smooth crossing v ≠ 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

7. String Fluctuations The fluctuations “live” in the world-sheet In the world-sheet horizon they behave as infalling outgoing Extension to Kruskal  analyticity cond. Negative frequency modes  infalling Positive frequency modes  outgoing Imposing analyticity in the WS horizon (Son+Herzog)

8. Check: Fluctuation Dissipation theorem Slowly moving heavy quark  Brownian motion Drag and noise are not independent. Einstein relation 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

9. Consequences for Heavy Quarks Brownian motion => Diffusion equation in configuration space HQ Diffusion coefficient (Einstein): From Rapp’s talk Different number of degrees of freedom!: Putting numbers:

10. Broadening at Finite Velocity It is dependent on the energy of the quark! Diverges in the ultra-relativistic limit. But the computation is only valid for (maximum value of the electric field the brane can support) There is a new scale in the problem 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

11. Conclusions We 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

12. Back up Slides

13. Computation of (Radiative Energy Loss) (Liu, Rajagopal, Wiedemann) t L r0r0 Dipole amplitude: two parallel Wilson lines in the light cone: For small transverse distance: Order of limits: String action becomes imaginary for entropy scaling

14. Energy Dependence of (JC & X. N. Wang) From the unintegrated PDF Evolution leads to growth of the gluon density, In the DLA Saturation effects  For an infinite conformal plasma (L>L c ) with Q 2 max =6ET. At strong coupling HTL provide the initial conditions for evolution.

15. Noise from Microscopic Theory HQ momentum relaxation time: Consider times such that  microscopic force (random) charge densityelectric field

16. Heavy Quark Partition Function McLerran, Svetitsky (82) Polyakov Loop YM + Heavy Quark states YM states Integrating out the heavy quark

17. Changing the Contour Time

18.  as a Retarded Correlator  is defined as an unordered correlator: From Z HQ the only unordered correlator is Defining: In the  0 limit the contour dependence disappears :

19. 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(t 1,y 1 )  t 1  y 1 E(t 2,y 2 )  t 2  y 2