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Evolution of singularities in thermalization of strongly coupled gauge theory

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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

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

Hadronic gas

QGP fluid

Hydrodynamics

thermalization

Equilibration of matter/Glasma

0

Partonic evolution/CGC

Au

Au

Large Nc , strong coupling limit of N=4 SYM

string theory in AdS background

N=4 SYM at zero temperature(vacuum)

Pure AdS

AdS-Schwarzshild

N=4 SYM at temperature

(plasma)

bulk field

A

g

boundary operator

TrF2+

J

T

boundary z=0

AdS-Schwarzschild

shell falling

“horizon”: z=zh

pure AdS

z=

No spatial gradient, similar to quantum quench.

SL, E. Shuryak

0808.0910 [hep-th]

quasi-static state: shell at z=zs<zh

O(t,x)O(t’,0) = O(t-t’)O(x)

AdS-Schwarzschild

shell

pure AdS

Beyond quasi-static: falling shell z=zs(t)

O(t,x)O(t’,0) O(t-t’)O(x)

Mirror at z=f(t).

Dirichlet boundary condition on the mirror

zero momentum sector

Two sovable examples:

standing mirror f(t)=zs

scaling mirror f(t)=t/u0 withu0>1

I. Amado, C. Hoyos, 0807.2337

J. Erdmenger, SL, H. Ngo, 1101.5505

In high frequency(WKB) limit, singularities of GR(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:

GR(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 GR(t,t’,z) without solving PDE!

natural splitting between positive/negative frequency contributions

Initial condition:

matching near t0

matching near

...

J. Erdmenger, C. Hoyos, SL 1112.1963

for our world d=4, c=5/2

Repeating the previous process:

with

Singular part of GR(t,t’):

for our world d=4, c=5/2

AdS-Schwarzschild

-zs

pure AdS

Falling trajectory of the shell by Israel junction condition:

-zh

Expectation from geometric optics picture suggests singularities of GR(t,t’) when the light ray starting off at t’ returns to the boundary

t’

z=0

Only finite bouncing is possible:

The warping factor freeze both the shell and the light ray near horizon

z=zs

t’

z=zh

1/zs

: 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

Initial condition from WKB limit

...

Divergence matching:

Results tested against quasi-static state

For d=4, c=5/2

as

T=0.35GeV

zs=1/1.5GeV

tth=0.02fm/c

“thermalization time”

GR()

In units of 2T

Re

BTZ black hole dual to 1+1D CFT

Quasi Normal Modes

Im

Singularities at:

GR()

Re

Quasi Normal Modes

AdS5-Schwarzschild dual to 3+1D CFT

Im

GR(t)

Imt

for ||>>T

Ret

Singularities at

I. Amado, C. Hoyos 0807.2337

We have seen the disappearance of singularities on the real t axis as we probe later stage of a thermalizing state. GR(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.

Recall

Initial condition from WKB limit

Essentially a real frequency WKB. Can complex WKB give us singularities in the complex plane?

Ingoing wave

Outgoing wave

BTZ

Quasi static state: z=zs

z=1

pure AdS3

QNM given only by the vanishing of the denominator

Im

Re

Set 1:

Asymptotically Normal Modes

Agrees with results from divergence matching

Set 2:

i-(2n-1) and i2n-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

- 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.