AdS/CFT “Applications” Jorge Casalderrey-Solana LBNL

Langevin Model for Heavy Quarks Heavy Quark with v<<1  neglect radiation  HQ classical on medium correlations scale  white noise  Einstein relations: random force drag

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

Force Correlators from Wilson Lines Which is obtained from small fluctuations of the Wilson line Integrating the Heavy Quark propagator: E(t 1,y 1 )  t 1  y 1 E(t 2,y 2 )  t 2  y 2  y acts as a source for the force

Adding Temperature and Heavy Quarks 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. Wilson Line The Hawking temperature of the brane becomes the temperature of the FT

Fluctuations of the Quark World Line Prescription for retarded correlators: Solve the linearized equations of motion Impose infalling boundary conditions at the horizon The retarded correlator is given by the boundary action

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 Drag force valid for ultra relativistic particle! Add an external electric field to valance the drag v Einstein relations

Consequences for Heavy Quarks Heavy probe on plasma => Brownian Motion (Langevin dynamics) HQ Diffusion coefficient (Einstein): From Rapp’s talk Different number of degrees of freedom: But, for the Langevin process to apply

Broadening of a Fast Probe Fluctuations of the “bending” string: World sheet horizon at Complication: Similar 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) v

Back Up

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

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

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.

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

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?

World Sheet Horizon We introduce The induced metric is diagonal Same as v=0 when World sheet horizon at

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)