Evolution of singularities in thermalization of strongly coupled gauge theory

Shu LinRBRC

J. Erdmenger, SL: 1205.6873J. Erdmenger, C. Hoyos, SL: 1112.1963J. Erdmenger, SL, H. Ngo: 1101.5505SL, E. Shuryak: 0808.0910

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

Stages of heavy ion collisions

0

Au Au

QGP fluid

Partonic evolution/CGC

Equilibration of matter/Glasma

Hydrodynamics

Hadronic gas

thermalization

Gauge/Gravity duality preliminaryLarge Nc , strong coupling limit of N=4 SYM

string theory in AdS background

4

4

2222

22

2222

22

1

))(/)((

)(

hz

zf

zfdzxddtzfz

Lds

dzxddtz

Lds

N=4 SYM at temperature(plasma) hz

T1

N=4 SYM at zero temperature(vacuum)

bulk fieldA

g

boundary operatorTrF2+

JT

Pure AdS

AdS-Schwarzshild

Gravitational collapse model dual to thermalization

shell falling

boundary z=0

“horizon”: z=zh

AdS-Schwarzschild

pure AdS

z=

SL, E. Shuryak0808.0910 [hep-th]

No spatial gradient, similar to quantum quench.

Quasi-static state & beyond

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

O(t,x)O(t’,0) = O(t-t’)O(x)

),(),,(

)]0,0(),,'([)'(),( 3

kGzk

OxttOttxedtdkG

R

ikxtiR

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

O(t,x)O(t’,0) O(t-t’)O(x)

),',(),,',(

)]0,'(),,([)'(),',( 3

kttGzktt

tOxtOttxedkttG

R

ikxR

shell

AdS-Schwarzschild

pure AdS

Toy model: Moving Mirror in AdSMirror at z=f(t).Dirichlet boundary condition on the mirror

)',(...)'(),',( 4 ttGzttzttG RR

)]0,'(),,([)'()',( 3 tOxtOttxdttGR zero momentum sector

Two sovable examples:

standing mirror f(t)=zs

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

I. Amado, C. Hoyos, 0807.2337J. Erdmenger, SL, H. Ngo, 1101.5505

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: )1/1ln(

2~)0,(

~)0,(

00

22/5

2

uu

nitzt

z

nezt

nnn

snn

tin

n

n

Divergence matching method

J. Erdmenger, C. Hoyos, SL 1112.1963

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!

Initial condition:

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

matching near t0

matching near 1t

...

natural splitting between positive/negative frequency contributions

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

AdS-Schwarzschild

pure AdS

Falling trajectory of the shell by Israel junction condition:

-zs

-zh

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

z=0

z=zh

z=zsOnly finite bouncing is possible:The warping factor freeze both the shell and the light ray near horizon

t’

t’

1/zs

Boundary condition on the shell: scalar fieldn: normal vector on the shellQuantities with index f: above the shellQuantities without: below the shell

To study retarded correlator, use infalling wave below the shell:

)(~

)(~)2(2/

2/

)1(2/

2/

zHze

zHze

ddti

ddti

positive frequencynegative frequency

Boundary condition on the shell involves both time and radial derivaives and scalar itself

Divergence matching method for shell

Initial condition from WKB limit

...

Divergence matching:

Singularities in the correlator

For d=4, c=5/2

Results tested against quasi-static state

nt as )(' sn zTt

“thermalization time”T=0.35GeVzs=1/1.5GeV tth=0.02fm/c

Singularities in thermal correlator of 1+1D CFT

BTZ black hole dual to 1+1D CFT

GR()Re

Im

Quasi Normal Modes

Singularities at:

In units of 2T

2 t

Singularities in thermal correlator of 3+1D CFT

AdS5-Schwarzschild dual to 3+1D CFT

GR()Re

Im

Quasi Normal Modes

GR(t)

Ret

Imt

for ||>>T

Singularities at

00~2,2 xmtmxt

2 t

Geometric optics in Penrose diagram

I. Amado, C. Hoyos 0807.2337

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.

Initial condition from WKB limit

Recall

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

BTZ

pure AdS3

Quasi static state: z=zsz=1

Ingoing wave Outgoing wave

QNM given only by the vanishing of the denominator

Two sets of QNM

Set 1:

Asymptotically Normal Modes

Agrees with results from divergence matching

Set 2: i-(2n-1) and i2n-1as 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

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!