Transcript "HIC with Perpendicular Dynamics from Evolving Geometries in AdS"

Department of Physics
HIC with Dynamics┴ from Evolving
Geometries in AdS
arXiv: 1004.3500 [hep-th],
Anastasios Taliotis
Partial Extension of arXiv:0805.2927 [hep-th],
arXiv:0902.3046 [hep-th], arXiv:0705.1234 [hep-ph]
(published in JHEP and Phys. Rev. C) [ Albacete, Kovchegov, Taliotis]
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Outline
Motivating strongly coupled dynamics in HIC
AdS/CFT: What we need for this work
State/set up the problem
Attacking the problem using AdS/CFT
Predictions/comparisons/conclusions/Summary
Future work
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Motivating strongly coupled
dynamics in HIC
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Notation/Facts
2
2


x

x
Proper time:
0
3
x0
1 x0  x3 1 x
 ln
Rapidity:   ln
2 x 0  x 3 2 x
Saturation scale Qs: The scale where
density of partons becomes high.
valid for times t >> 1/Qs
Bj QGP
Hydro
g<<1; valid up to times  ~ 1/QS.
CGC describes matter distribution due

CGC
to classical gluon fields and is rapidityindependent ( g<<1, early times).
x3
Hydro is a necessary condition for
thermalization. Bjorken Hydro describes
successfully particle spectra and spectral flow. Is
g??>>1 at late times?? Maybe; consistent with
the small MFP implied
by

 a hydro description.
D F
J
No unified framework exists that describes both
strongly & weakly coupled dynamics
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Goal: Stress-Energy (SE) Tensor
• SE of the produced medium gives useful information.
• In particular, its form (as a function of space and time
variables) allows to decide whether we could have
thermalization i.e. it provides useful criteria for the
(possible) formation of QGP.
• SE tensor will be the main object of this talk: we will
see how it can be calculated by non perturbative
methods in HIC.
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Most General Rapidity-Independent
SE Tensor
The most general rapidity-independent SE tensor for
a collision of two transversely large nuclei is (at x3 =0)
T 
  ( )

 0

0

 0

0
p( )
0
0
0
0
p( )
0
which, due to
0 x
t0

0 x
x1
0 x
y
 2
p3 ( )  x
z
3
 T

0
x1
x3
x2
  p3
d

gives
d

We will see three different regimes of p3
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Early times : τQs <<1
I.
II.
CGC
T 
Later times : τ>~1/Qs
Bjorken Hydrodynamics
CGC
0
0 t
  ( ) 0


0  x 
 0  ( ) 0

T
0
0  ( )
0 y


 0
0
0   ( )  z

 ~ log 
III. Much later times:τQs >>>1
0
0 t
  ( ) 0
  ( )



0 x
 0 p( ) 0
 0


T 
0
0 p( ) 0  y
0



 0
 0
0
0 p0( )  z


0
p( )
0
0
Isotropization
2
[Lappi ’06 Fukushima ’07: pQCD]
[Talıotıs ’10: AdS/CFT]
0
0  xt 0

0
0  x1
p( ) 0  xy2

0 p(τ)
0  xz3
[Free streaming]
~
1
thermalization

~
1
 4/3
•Classical gluon fields
•Classical gluon fields
•Hydrodynamic description
•Pert. theory applies
•Pert. Theory applies
•Does pert. Theory apply??
•Describes RHIC data well
•Energy is conserved
•Describes data successfully
(particle multiplicity dN/dn)
[Krasnitz, Nara,Venogopalan,
Lappi, Kharzeev, Levin, Nardi]
(spectra dN/d2pTdn for K, ρ,
n & elliptic flow)
[Heınz et al]
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Bjorken Hydro & strongly coupled dynamics
If
p3  0
then, as
  p3
d

d

, one gets
~
1

1 
.
Deviations  ~ 1 / 
from the energy conservation  ~ 1 / 
are due to longitudinal pressure, P3 which does work P3dV in the
longitudinal direction modifying the energy density scaling with tau.
1 
It is suggested that neither classical nor quantum gluonic or fermionic fields can
cause the transition from free streaming to Bjorken hydro within perturbation
theory. [Kovchegov’05]
On the other hand Bjorken hydro describe simulations satisfactory.
Conclude that alternative methods are needed!
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AdS/CFT: What we need for this work
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Quantifying the Conjecture
<exp z=0∫O φ0>CFT = Zs(φ|φ(z=0)= φo)
O is the CFT operator. Typically want <O1 O2…On>
φ0 =φ0 (x1,x2,… ,xd) is the source of O in the CFT picture
φ =φ (x1,x2,… ,xd ,z) is some field in string theory with B.C.
φ (z=0)= φ0
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Holographic renormalization
[Witten ‘98]
Example:
• Quantifying the Conjecture
<exp z=0∫O φ0>CFT = Zs(φ|φ(z=0)= φo)
• Know the SE Tensor of Gauge theory is given by
T

 S
1
 2
g  g

|g
  
• So gμν acts as a source => in order to calculate Tμν from
AdS/CFT must find the metric. Metric has its eq. of motion
i.e. Einsteins equations.
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Holographic renormalization
de Haro, Skenderis, Solodukhin ‘00
 Energy-momentum tensor is dual to the metric in AdS. Using
Fefferman-Graham coordinates one can write the metric as
2
L
ds 2  2  g  ( x, z ) dx  dx  dz 2   L2 d 5
z
~
with z the 5th dimension variable and g  ( x, z ) the 4d metric.

~
Expand g  ( x, z ) near the boundary (z=0) of the AdS space:

Using AdS/CFT can show:
2
c
2 z 0
N
 T 
lim
2
,
g  ( x , z )   
z
4
, i.e. z 4 coef .
and
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State/set up the problem
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Strategy
Initial Tµν
phenomenology
Initial Geometry
Dynamical Tµν
(our result)
Evolve
Einstein's Eq.
AdS/CFT
Dictionary
Dynamical
Geometry
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Field equations, AdS5 shockwave; ∂gMN
Tμν
 Eq. of Motion (units L=1) for gΜΝ(xM = x±, x1, x2, z) is generally given
1
R  4 g  R   52 ( J   Jg MN )
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 AdS-shockwave with bulk matter: [Janik & Peschanski ’06]
1
2
ds  2 [2dx  dx   t1 (r1 , x  ) z 4 (dx  ) 2  dx  dz 2 ]
z
2
t1   log( kr1 ) ( x )

r  (x )  (x )
1 2
2 2


r1 | r  b1 |
Then ~z4 coef. implies <Tμν (xμ)> ~ -δμ + δν + µlog(r1) δ(x+) in QFT side
Corresponding bulk tensor JMN :
J MN
 4 
 2 z  (r1 ) ( x  )
5
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Single nucleus
Single shockwave
The picture in 4d is that matter moves ultrarelativistically
along x- according to figure.
Einstein's equations are satisfied trivially except (++)
component; it satisfies a linear equation:
□(z4 t1)=J++
J 
This suggests may represent the
shockwave metric as a single vertex:
a graviton exchange between the
source J++ (the nucleus living at z=0;
the boundary of AdS) and point XM in
the bulk which gravitational field is
measured.
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4D Picture of Collision
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Superposition of two shockwaves
Non linearities of gravity

L2
ds  2
z
2

?
(1)  
(1)  
( 2)




2
2
4
2
4
2



2
dx
dx

dx

dz

t
(
r

b
)
z
dx

t
(
r

b
)
z
dx


(
x
)

(
x
)
g



1
2





Higher graviton ex.
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Flat AdS
One graviton ex.
Due to non linearities
Back-to-Back reactions for JMN
• In order to have a consistent
expansion in µ2 we must determine
( 2)
J MN
• We use geodesic analysis

• Bulk source J++ (J--) moves in the
gravitational filed of the shock t1(t2)
• Important:
is conserved iff b≠0

 J MN ~   N  ( b )
M
Self corrections to JMN
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Calculation/results
• Step 1: Choose a gauge: Fefferman-Graham coordinates
• Step 2: Linearize field eq. expanding around 1/z2 ηMN
(partial DE with w.r.t. x+,x-, z with non constant coef.).
• Step 3: Decouple the DE. In particular all components g(2)µν
obey: □g(2)µν = A(2)µν(t1(x-) ,t2 (x+) ,J) with box the d'Alembertian in AdS5.
• Step 4: Solve them imposing (BC) causality-Determine the GR
• Step 5: Determine Tμν by reading the z4 coef. of gμν
 Side Remark: Gzz encodes tracelessness of
Tµν
Gzν encode conservation of Tµν
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The Formula for Tµν
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Eccentricity-Momentum Anisotropy
Momentum Anisotropy εx= εx(x) (left) and εx= εx(1/x) (right) for

intermediate x  .
b
Agrees qualitatively with [Heinz,Kolb,
Lappi,Venugopalan,Jas,Mrowczynski]
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Conclusions
• Built perturbative expansion of dual geometry to determine Tµν ;
applies for sufficiently early times: µτ3<<1.
• Tµν evolves according to causality in an intuitive way! There is a
kinematical window where lim T ( , x1 , x2 , b) is invariant under   r .
b 0
[Gubser ‘10]
• Our exact formula (when applicable) allows as to compute Spatial
Eccentricity and Momentum Anisotropy .
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• When τ>>r1 ,r2 have ε~τ2 log2 τ-compare with ε~Q2slog2 τ
[Lappi, Fukushima]
• Despite J being localized, it still contributes to gµν and so to Tµν not only
on the light-cone but also inside.
• Impact parameter is required otherwise violate conservation of JMN and
divergences of gµν. Not a surprise for classical field theories.
• Our technique has been applied to ordinary (4d) gravity and found
MS thesis.dept.
similar behavior for gµν. Taliotis’10
of Mathematics, OSU
• A phenomenological model using the (boosted) Woods-Saxon profile:
[Gubser,Yarom,Pufu ‘08]
Note symmetry under r  
when b=0; [Gubser’10]
For τ> r1,r2
Thank you
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Supporting slides
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O(µ2) Corrections to Jµν
Remark: These corrections live on the forward light-cone as should!
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Field Equations
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