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. Outline.

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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

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


Outline

Outline

  • 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


Stages of heavy ion collisions

Stages of heavy ion collisions

Hadronic gas

QGP fluid

Hydrodynamics

thermalization

Equilibration of matter/Glasma

0

Partonic evolution/CGC

Au

Au


Gauge gravity duality preliminary

Gauge/Gravity duality preliminary

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


Gravitational collapse model dual to thermalization

Gravitational collapse model dual to thermalization

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 beyond

Quasi-static state & beyond

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)


Toy model moving mirror in ads

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)=zs

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

I. Amado, C. Hoyos, 0807.2337

J. Erdmenger, SL, H. Ngo, 1101.5505


Singularities in the correlator

Singularities in the correlator

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:


Divergence matching method

Divergence matching method

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


Divergence matching method continued

Divergence matching method(continued)

Repeating the previous process:

with

Singular part of GR(t,t’):

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


Gravitational collapse model

Gravitational collapse model

AdS-Schwarzschild

-zs

pure AdS

Falling trajectory of the shell by Israel junction condition:

-zh


Light ray bouncing in collapse background

Light ray bouncing in collapse background

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


Boundary condition on the shell

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


Divergence matching method for shell

Divergence matching method for shell

Initial condition from WKB limit

...

Divergence matching:


Singularities in the correlator1

Singularities in the correlator

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”


Singularities in thermal correlator of 1 1d cft

Singularities in thermal correlator of 1+1D CFT

GR()

In units of 2T

Re

BTZ black hole dual to 1+1D CFT

Quasi Normal Modes

Im

Singularities at:


Singularities in thermal correlator of 3 1d cft

Singularities in thermal correlator of 3+1D CFT

GR()

Re

Quasi Normal Modes

AdS5-Schwarzschild dual to 3+1D CFT

Im

GR(t)

Imt

for ||>>T

Ret

Singularities at


Geometric optics in penrose diagram

Geometric optics in Penrose diagram

I. Amado, C. Hoyos 0807.2337


Singularities in the complex t plane

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. 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?


Evolution of qnm in gravitational collapse of btz black hole

Evolution of QNM in gravitational collapse of BTZ black hole

Ingoing wave

Outgoing wave

BTZ

Quasi static state: z=zs

z=1

pure AdS3

QNM given only by the vanishing of the denominator


Two sets of qnm

Two sets of QNM

Im

Re

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


Summary

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.


Thank you

Thank you!


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