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 N=4 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. random force drag Langevin Model for Heavy Quarks Moore, Teaney 03 Heavy Quark with v<<1 neglect radiation HQ classical on medium correlations scale white noise  Mean transfer momentum Einstein relations:

4. Density Matrix of a Heavy Quark Eikonalized  E=gM >> T momentum change observation medium correlations 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 Fymvm(t1,y1)Dt1dy1 Four possible correlators Fymvm(t2,y2)Dt2dy2 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 ∞. 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 r>r0 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 Solution=Kruskal Plane The string has two endpoints, one at each boundary. Minimal surface with static world-line at each boundary V=0 U=0 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 V=0 F L R P U=0 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 x=vt at the boundary Energy and momentum flux through the string: same k ! Fluctuation-dissipation theorem Very small relaxation times (t0=1/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 v=0 when The induced metric is diagonal World sheet horizon at

16. Fluctuation Matching Along the future world sheet horizon we impose the same analyticity condition as for in the v=0 case. 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. (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 N=4 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) t L r0 (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 (L>Lc) with Q2max=6ET. At strong coupling

22. Charm Quark Flow ? (Moore, Teaney 03) (b=6.5 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. E(t1,y1)Dt1dy1 Since in k there is no time order: E(t2,y2)Dt2dy2 Force Correlators from Wilson Lines Integrating the Heavy Quark propagator: Which is obtained from small fluctuations of the Wilson line