1. Evolution of singularities in thermalization of strongly coupled gauge theory Shu Lin RBRC J. Erdmenger, SL: 1205.6873 J. Erdmenger, C. Hoyos, SL: 1112.1963 J. Erdmenger, SL, H. Ngo: 1101.5505 SL, E. Shuryak: 0808.0910

2. Outline Hope: to understand thermalization with gauge/gravity duality Toy model and divergence matching method Application of the divergence matching method to gravitational collapse model Evolution of singularities of unequal time correlator and the dual evolution of QNM

3. Stages of heavy ion collisions 0 Au QGP fluid Partonic evolution/CGC Equilibration of matter/Glasma Hydrodynamics Hadronic gas thermalization

4. Gauge/Gravity duality preliminary Large N c, strong coupling limit of N=4 SYM string theory in AdS background N=4 SYM at temperature (plasma) N=4 SYM at zero temperature(vacuum) bulk field  A  g  boundary operator TrF 2 +  J  T  Pure AdS AdS- Schwarzshild

5. Gravitational collapse model dual to thermalization shell falling boundary z=0 “horizon”: z=z h AdS-Schwarzschild pure AdS z=  SL, E. Shuryak 0808.0910 [hep-th] No spatial gradient, similar to quantum quench.

6. Quasi-static state & beyond quasi-static state: shell at z=z s <z h  O(t,x)O(t’,0)  =  O(t-t’)O(x)  Beyond quasi-static: falling shell z=z s (t)  O(t,x)O(t’,0)    O(t-t’)O(x)  shell AdS-Schwarzschild pure AdS

7. Toy model: Moving Mirror in AdS Mirror at z=f(t). Dirichlet boundary condition on the mirror zero momentum sector Two sovable examples: standing mirror f(t)=z s scaling mirror f(t)=t/u 0 with u0>1 I. Amado, C. Hoyos, 0807.2337 J. Erdmenger, SL, H. Ngo, 1101.5505

8. Singularities in the correlator In high frequency(WKB) limit, singularities of G R (t,t’) occur at, consistent with a geometric optics picture in the bulk. Bulk-cone singularities conjecture: Hubeny, Liu and Rangamani hep-th/0610041 Singularities in time contains information on the “spectrum” of the particular operator O: Standing mirror: Scaling mirror:

9. Divergence matching method J. Erdmenger, C. Hoyos, SL 1112.1963 G R (t,t’,z) singular near the segments (-,0), (+,1), (-,1) etc Matching along the mirror trajectory and on the boundary allows us to determine the singualr part of G R (t,t’,z) without solving PDE! Initial condition: for our world d=4, c=5/2 matching near t 0 matching near... natural splitting between positive/negative frequency contributions

10. Divergence matching method(continued) Repeating the previous process: with Singular part of G R (t,t’): for our world d=4, c=5/2

11. Gravitational collapse model AdS-Schwarzschild pure AdS Falling trajectory of the shell by Israel junction condition: -z s -z h

12. Light ray bouncing in collapse background Expectation from geometric optics picture suggests singularities of G R (t,t’) when the light ray starting off at t’ returns to the boundary z=0 z=z h z=z s Only finite bouncing is possible: The warping factor freeze both the shell and the light ray near horizon t’ 1/z s

13. Boundary condition on the shell  : scalar field n: normal vector on the shell Quantities with index f: above the shell Quantities without: below the shell To study retarded correlator, use infalling wave below the shell: positive frequency negative frequency Boundary condition on the shell involves both time and radial derivaives and scalar itself

14. Divergence matching method for shell Initial condition from WKB limit... Divergence matching:

15. Singularities in the correlator For d=4, c=5/2 Results tested against quasi-static state as “thermalization time” T=0.35GeV z s =1/1.5GeV  t th =0.02fm/c

16. Singularities in thermal correlator of 1+1D CFT BTZ black hole dual to 1+1D CFT GR()GR() Re  Im  Quasi Normal Modes Singularities at: In units of 2  T

17. Singularities in thermal correlator of 3+1D CFT AdS 5 -Schwarzschild dual to 3+1D CFT GR()GR() Re  Im  Quasi Normal Modes GR(t)GR(t) Ret Imt for |  |>>T Singularities at

18. Geometric optics in Penrose diagram I. Amado, C. Hoyos 0807.2337

19. Singularities in the complex t plane? We have seen the disappearance of singularities on the real t axis as we probe later stage of a thermalizing state. G R (t,t’) What about singularities in the complex t plane? Do they emerge as the field thermalizes and eventually reduce to the singularities pattern in the thermal correlator? The singularities on the real t axis we obtained come from real frequency contributions, i.e. Normal Modes, while singularities of thermal correlator come from QNM contribution. Initial condition from WKB limit Recall Essentially a real frequency WKB. Can complex WKB give us singularities in the complex plane?

20. Evolution of QNM in gravitational collapse of BTZ black hole BTZ pure AdS 3 Quasi static state: z=z s z=1 Ingoing waveOutgoing wave QNM given only by the vanishing of the denominator

21. Two sets of QNM Set 1: Asymptotically Normal Modes  Agrees with results from divergence matching Set 2: i  -(2n-1) and i  2n-1 as opposed to i  =-2n for retarded correlator and i  =2n for advanced correlator The QNM evolution does not seem to reduce to the pattern of the thermal state Re  Im 

22. Summary Starting with toy models, we have developed a divergence matching method for obtaining the singular part of unequal time correlator. Applying the method to gravitational collapse model, we obtain the evolution of singularities of correlator in thermalizing state. Motivated by the emergence of singularities in complex plane from contribution of QNM, we explored the evolution of QNM in quasi static state, but failed to reduce to themal QNM.

23. Thank you!