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Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma

Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma. Jorge Casalderrey-Solana LBNL. Work in collaboration with Derek Teany. r ab (-b/2, b/2; - T ) A. Final distribution. Momentum Distribution. -T/2. +T/2. v. f 0 (b). f f (b). T.

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Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma

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  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. rab (-b/2, b/2; -T)A Final distribution Momentum Distribution -T/2 +T/2 v f0 (b) ff (b)

  3. T Fymvm(t1,y1)Dt1dy1 Four possible correlators Fymvm(t2,y2)Dt2dy2 Momentum Broadening From the final distribution Transverse gradient  Fluctuation of the Wilson line Dressed field strength

  4. t x Moving string is “straighten” by the coordinates c(u)=v u=1/g “World sheet black hole” at uws=1/g At v=0 both horizons coincide. Coordinate singularity!! Wilson Lines in AdS/CFT t v=0 x=vt x u=e u0=1 Horizon

  5. Kruskal Map As in usual black holes, (t, u) are only defined for u<u0 Global (Kruskal) coordinates V u=1/r2 U 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 Maldacena Son, Herzog (02)

  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 (Son+Herzog) Imposing analyticity in the WS horizon

  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 k ! Fluctuation-dissipation theorem

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

  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 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

  12. Back up Slides

  13. 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

  14. 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

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

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

  17. Changing the Contour Time

  18. 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 :

  19. 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

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