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 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

3 Langevin Model for Heavy Quarks Moore, Teaney 03  HQ classical on medium correlations scale  white noise  Einstein relations: random force drag Mean transfer momentum Heavy Quark with v<<1  neglect radiation

4 Density Matrix of a Heavy Quark Eikonalized  E=  M >> 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 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

6 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?

7 Kruskal Map As in black holes, (t, r) –coordinates are only defined for r>r 0 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

8 Static String Solution=Kruskal Plane The 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

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/r 2 0 ) Which solution should we pick? infalling outgoing

10 Boundary Conditions for Fluctuations RL F P V=0 U=0 Son, Herzong (Unruh): Negative frequency modes Positive frequency modes  near horizon

11 Retarded Correlators and  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 x=vt at the boundary Energy and momentum flux through the string: same  Very small relaxation times (t 0 =1/  D ) charmbottom Fluctuation-dissipation theorem

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 The induced metric is diagonal Same as v=0 when World sheet horizon at

16 Fluctuation Matching The two modes are infalling and outgoing in the world sheet horizon 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. (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 The mean transfer momentum is diverges in v  1 limit 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  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.

19 Back up Slides

20 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

21 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.

22 Charm Quark Flow ? (Moore, Teaney 03) (b=6.5 fm)

23 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

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

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

26 Changing the Contour Time

27  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 :

28 Check  in Perturbation Theory = optical theorem = imaginary part of gluon propagator 

29 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