Evolution of singularities in thermalization of strongly coupled gauge theory Shu Lin RBRC J....
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Transcript of Evolution of singularities in thermalization of strongly coupled gauge theory Shu Lin RBRC J....
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
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
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.