1 AdS/CFT Calculations of Parton Energy Loss Jorge Casalderrey-Solana Lawrence Berkeley National Lab. In collaboration with D. Teaney

2 Why N = 4 Yang Mills? The QGP at T=(1-2)T c may be strongly coupled: Strong flow observed at RHIC consistent with ideal hydro. Transport requires cross sections 10 times larger than pQCD. Estimates and calculations on QGP shear viscosity yield small values. N =4 Yang Mills can be solved at strong coupling via AdS/CFT Pressure at strong coupling is ¾ P Stephan-Boltzmann. Lattice QCD  similar deviations are observed at T=(1-2)T c. Shear viscosity conjectured minimal bound. But N=4 is not QCD (scale invariant, supersymmetry…) However AdS/CFT is the only method available to address strongly coupled gauge theories Strong jet quenching (opaque medium).  s (T) is large. Non perturbative access to real time dynamics of gauge theories.

3 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)

4 Momentum Broadening Transverse momentum transferred Transverse gradient  Fluctuation of the Wilson line Four different correlators:

5 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?

6 Kruskal Map Black hole  two copies of the (boundary) field theory (Maldacena) r t r0r0 Herzog & Son: Fluctuations on (R, L) can be matched so that field correlators have the correct analytic properties (KMS relations) Each (boundary) fields are identified with type 1 and 2.

7 Fluctuations (static) of the Quark World Line Small fluctuation problem (linearized) Near the horizon (u  1/r 2 0 ) infalling outgoing Son-Herzog prescription: KMS relation (static quarks in equilibrium)

8 Consequences for Heavy Quarks Heavy probe on plasma => Brownian Motion (Langevin dynamics) HQ Diffusion coefficient (Einstein): Putting numbers: It is not QCD but… QCD at weak coupling: Different number of degrees of freedom: Liu, Rajagopal, Wiedemann

9 Drag Force (Herzog, Karch, Kovtun, Kozcaz and Yaffe ; Gubser) Direct computation: HQ forced to move with velocity v:  Wilson line x=vt at the boundary Energy and momentum flux through the string: same  Fluctuation-dissipation theorem Langevin: Einstein relation: Drag force valid for ultra relativistic particle. Add an external electric field to valance the drag v

10 Broadening of a Fast Probe Fluctuations of the “bending” string: World sheet horizon at Complications: Discontinuity in the past horizon Similar to KMS relation but: G R is infalling in the world sheet horizon The temperature of the correlator is that of the world sheet black hole (blue shift?) New Scale! Diverges in ultra relativistic limit! But the brane does not support arbitrary large electric fields (pair production)

11 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

12 Conclusions AdS/CFT provides a rigorous way to address the physics at strong coupling. The computed transport coefficient have remarkable features Large values Unusual coupling dependences Energy dependence of the momentum broadening Many other application of AdS/CFT to Heavy Ion phenomenology (fields associated to probe, hydrodynamics, production of fireball) The applicability of these results demand phenomenological work to explain them in a way which can be translated to QCD.

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

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)