Initial Conditions from
Shock Wave Collisions in AdS5
Yuri Kovchegov
The Ohio State University
Based on the work done with
Javier Albacete, Shu Lin, and Anastasios Taliotis,
arXiv:0805.2927 [hep-th],
arXiv:0902.3046 [hep-th],
arXiv:0911.4707 [hep-ph]
Outline




Problem of isotropization/thermalization in
heavy ion collisions
AdS/CFT techniques we use
Bjorken hydrodynamics in AdS
Colliding shock waves in AdS:



Collisions at large coupling: complete nuclear
stopping
Proton-nucleus collisions
Trapped surface and black hole production
Thermalization problem
Timeline of a Heavy Ion Collision
(particle production)
Notations
x
proper time

  x x
2
0
2
3
rapidity
1 x0  x3
  ln
2 x0  x3
CGC (Color Glass Condensate) =
classical gluon fields.
The matter distribution due to
classical gluon fields is
rapidity-independent.
QGP = Quark Gluon Plasma
x
0
x
QGP
CGC
x3

Most General Rapidity-Independent EnergyMomentum Tensor
The most general rapidity-independent energy-momentum tensor
for a high energy collision of two very large nuclei is (at x3 =0)
T
which, due to

  ( )

 0

0

 0

 T

0
0
p ( )
0
0
p ( )
0
0
0 

0 
0 

p3 ( ) 
0
x2
gives
  p3
d

d

xt 0
xx1
xy 2
xz 3
x3
x1
Color Glass at Very Early Times
In CGC at very early times  ~ log
such that, since
  p3
d

d

2
1

,
 QS  1
we get, at the leading log level,
Energy-momentum tensor is
T

(Lappi ’06
Fukushima ‘07)
0
0
0 
  ( )


 ( ) 0
0 
 0

0
0
 ( )
0 


 0

0
0


(

)


tx0
xx1
yx2
zx3
Color Glass at Later Times: “Free Streaming”
At late times  
1
QS
classical CGC gives free streaming,
which is characterized by the following energy-momentum tensor:
T

  ( )

 0

0

 0

0
p( )
0
0
0  xt 0

0
0  x1
p( ) 0  xy2

0
0  xz3
such that
0
d


d

and
 ~
1

 The total energy E~ e  is conserved, as expected for
non-interacting particles.
Classical Fields
from numerical
simulations by Krasnitz,
Nara, Venugopalan ‘01
 CGC classical gluon field leads to energy density scaling as

classical
~
1

Much Later Times: Bjorken Hydrodynamics
In the case of ideal hydrodynamics, the energy-momentum
tensor is symmetric in all three spatial directions (isotropization):
T

  ( )

 0

0

 0

0
0
p( )
0
0
p( )
0
0
0  tx0

0  xx1
0  yx2

p( )  zx3
such that
d
p

d

Using the ideal gas equation of state,   3 p , yields
~
1
 4/3
Bjorken, ‘83
 The total energy E~   is not conserved, while the
total entropy S is conserved.
The Problem

Can one show in an analytic calculation that the
energy-momentum tensor of the medium produced
in heavy ion collisions is isotropic over a
parametrically long time?

That is, can one start from a collision of two nuclei
and obtain Bjorken hydrodynamics?

Even in some idealized scenario? Like
ultrarelativistic nuclei of infinite transverse extent?

Let us proceed assuming that strong-coupling
dynamics from AdS/CFT would help accomplish this
goal.
AdS/CFT techniques
AdS/CFT Approach
z=0
Our 4d
world
5d (super) gravity
lives here in the AdS space
5th dimension
z
x0  x3
x 
2

AdS5 space – a 5-dim space
with a cosmological constant L= -6/L2.
(L is the radius of the AdS space.)

2
L
ds 2  2  2 dx  dx   dx2  dz 2
z

AdS/CFT Correspondence
(Gauge-Gravity Duality)
Large-Nc, large lg2 Nc
N=4 SYM theory in
our 4 space-time
dimensions
Weakly coupled
supergravity in 5d
anti-de Sitter space!
 Can solve Einstein equations of supergravity in 5d to
learn about energy-momentum tensor in our 4d world in
the limit of strong coupling!
 Can calculate Wilson loops by extremizing string
configurations.
 Can calculate e.v.’s of operators, correlators, etc.
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 ~


2
ds  2 g  ( x, z ) dx dx  dz
z
~
2

with z the 5th dimension variable and g  ( x, z ) the 4d metric.

~
Expand g  ( x, z ) near the boundary of the AdS space:

For Minkowski world
and
with
Bjorken Hydrodynamics in AdS
AdS Dual of a Static Thermal Medium
Black hole in AdS5 ↔ Thermal medium in N=4 SYM theory.
z=0
Our 4d
world
5th dimension
z0
z
L2
ds  2
z
2
black hole horizon
AdS5 black hole metric can be written as
 (1  z 4 / z04 ) 2 2
2
4
4
2

dt

(
1

z
/
z
)
d
x

dz
0


4
4
(
1

z
/
z
)
0


with
z0 
2
T
AdS Dual of Bjorken Hydrodynamics
Janik, Peschanski ’05: to get Bjorken hydro dual need z0 =z0().
z=0
R3
z0
black hole horizon
Black hole recedes into the bulk: medium in 4d expands and cools off.
Asymptotic geometry

Janik and Peschanski ’05 showed that in the rapidityindependent case the geometry of AdS space at late
proper times  is given by the following metric
(
)
2
z4


1  e0  4 / 3
L
4
2
2
2
2
2
2
z
(
ds  2 
d


1

e
 d  dx )  dz 
4
0
 4/3
z
z  1  e0  4 / 3



2

(
)
with e0 a constant.
In 4d gauge theory this gives Bjorken hydrodynamics:
T

  ( )

 0

0

 0

0
0
p( )
0
0
p( )
0
0
0 t

0 x
0 y

p( )  z
with
~
1

4/3
Bjorken hydrodynamics in AdS




Looks like a proof of thermalization at large
coupling.
It almost is: however, one needs to first understand
what initial conditions lead to this Bjorken
hydrodynamics.
Is it a weakly- or strongly-coupled heavy ion
collision which leads to such asymptotics? If yes, is
the initial energy-momentum tensor similar to that
in CGC? Or does one need some pre-cooked
isotropic initial conditions to obtain Janik and
Peschanski’s late-time asymptotics?
In AdS the problem of thermalization = problem of
black hole production in the bulk
Colliding shock waves in AdS
J. Albacete, A. Taliotis, Yu.K.
arXiv:0805.2927 [hep-th], arXiv:0902.3046 [hep-th]
see also Nastase; Shuryak, Sin, Zahed; Kajantie, Louko,
Tahkokkalio; Grumiller, Romatschke.
Single Nucleus in AdS/CFT
An ultrarelativistic nucleus
is a shock wave in 4d with
the energy-momentum
tensor
T ~   ( x  )
Shock wave in AdS
Need the metric dual to a shock
wave that solves Einstein
equations:
R
1
6
 R g   2 g   0
2
L
The metric of a shock wave in
AdS corresponding to the
ultrarelativistic nucleus in 4d is
(note that T_ _ can be any
function of x^-):
L2
ds  2
z
2

2 2



4
2
2
2

2
dx
dx

T
(
x
)
z
dx

dx

dz




2
N
C

 Janik, Peschanksi ‘05
Diagrammatic interpretation
The metric of a shock wave in AdS
corresponding to the ultrarelativistic
nucleus in 4d can be represented as a
graviton exchange between the
boundary of the AdS space and the
bulk:

2
L
ds 2  2  2 dx  dx     ( x  ) z 4 dx 2  dx2  dz 2
z

cf. classical Yang-Mills field of a single ultrarelativistic nucleus
in CGC in covariant gauge: given by 1-gluon exchange
(Jalilian-Marian, Kovner, McLerran, Weigert ’96, Yu.K. ’96)
Model of heavy ion collisions in AdS



L2
ds  2
z
2
Imagine a collision of two
shock waves in AdS:
We know the metric of both
shock waves, and know that
nothing happens before the
collision.
Need to find a metric in the
forward light cone!
(cf. classical fields in CGC)
?


2 2
2 2


2
2

4
2

4
2

2
dx
dx

dx

dz

T
(
x
)
z
dx

T
(
x
)
z
dx



1 
2 


2
2
N
N
C
C


empty AdS5
1-graviton part
higher order
graviton exchanges
Heavy ion collisions in AdS
L2
ds  2
z
2


2 2
2 2


2
2

4
2

4
2

2
dx
dx

dx

dz

T
(
x
)
z
dx

T
(
x
)
z
dx



1 
2 


2
2
N
N
C
C


empty AdS5
1-graviton part
higher order
graviton exchanges
Expansion Parameter




Depends on the exact form of the energymomentum tensor of the colliding shock waves.
For T ~   ( x ) the parameter in 4d is  3 :
the expansion is good for early times  only.

For T ~ L2    ( x  ) that we will also consider
the expansion parameter in 4d is L2 2. Also valid
for early times only.
In the bulk the expansion is valid at small-z by the
same token.
What to expect

There is one important constraint of non-negativity of
energy density. It can be derived by requiring that
T t  t  0
for any time-like t.

This gives (in rapidity-independent case)
 ( )  0
along with
Janik, Peschanksi ‘05
Lowest Order Diagram
~ T1 ~ 1  ( x  )
Y 
 ~ 1 2  2
The same result comes out of
detailed calculations.

~ T2 ~ 2  ( x )

Each graviton gives
Simple dimensional analysis:
e
Grumiller, Romatschke ‘08
Albacete, Taliotis, Yu.K. ‘08
, hence get no rapidity dependence:
e eY     independen t
Shock waves collision: problem 1
 ~ 1 2 


2
Energy density at mid-rapidity grows with time!?
This violates  ' ( )  0 condition. This means in
some frames energy density at some rapidity is
negative!
I do not know of a good explanation: it may be due
to some Casimir-like forces between the receding
nuclei. (see e.g. work by Kajantie, Tahkokkalio,
Louko ‘08)
Shock waves collision: problem 2

Delta-functions are unwieldy. We will smear the
shock wave:




  (x ) 
a ~ A1/ 3 / p 
with   p L A
and
. (L is the
typical transverse momentum scale in the shock.)
Look at the energy-momentum tensor of a nucleus
after collision:


a
 ( x )  (a  x )
2
1/ 3
T  ( x   a, x   a / 2) 


a
 4 2  2 x  2
1
 a L A1/ 3
the nucleus will run out of momentum and stop!
Looks like by the light-cone time
x ~
1
~
Shock waves at lowest order



We conclude that describing the whole
collision in the strong coupling framework
leads to nuclei stopping shortly after the
collision.
This would not lead to Bjorken
hydrodynamics. It is very likely to lead to
Landau-like rapidity-dependent
hydrodynamics. This is fine, as rapiditydependent hydrodynamics also describes
RHIC data rather well.
However baryon stopping data contradicts
the conclusion of nuclear stopping at RHIC.
Landau vs Bjorken
Landau hydro: results from
strong coupling dynamics (at all
times) in the collision. While
possible, contradicts baryon
stopping data at RHIC.
Bjorken hydro: describes RHIC
data well. The picture of nuclei
going through each other almost
without stopping agrees with our
perturbative/CGC understanding of
collisions. Can we show that it
happens in AA collisions?
Proton-Nucleus Collisions
pA Setup

Solving the full AA problem is hard. To gain intuition
need to start somewhere. Consider pA collisions:

1
p
p

2
pA Setup

In terms of graviton exchanges need to resum
diagrams like this:
In QCD pA with gluons cf.
A. Mueller, Yu.K., ’98;
B. Kopeliovich, A. Tarasov
and A. Schafer, ’98;
A. Dumitru, L. McLerran, ‘01.
Eikonal Approximation

Note that the nucleus is Lorentz-contracted. Hence
all
1
x ~ 
p2

i
and are small.
Physical Shocks

Summing all these graphs for the delta-function
shock waves
yields the transverse pressure:

Note the applicability region:
Physical Shocks

The full energy-momentum tensor can be easily
constructed too. In the forward light cone we get:
Physical Shocks: the Medium


Is this Bjorken hydro? Or a free-streaming medium?
Appears to be neither. At late times
p~
1
 2
(x )
x

~
e  (3 / 2)
 5/ 2

Not a free streaming medium.
For ideal hydrodynamics expect
such that:

However, we get
0
Not hydrodynamics either.
Physical Shocks: the Medium

Most likely this is an artifact of the approximation,
this is a “virtual” medium on its way to
thermalization.
Proton Stopping

What about the proton? If our
earlier conclusion about shock
wave stopping based on
T  ( x   a, x   a / 2) 

a
 4 2  2 x  2
is right, we should be able to see
how it stops.
Proton Stopping

We have the original shock wave:

We have the produced stuff:

Adding them together we see that
the shock wave is cancelled:
T++ goes to zero as x+ grows large!
Proton Stopping

We get complete proton stopping (arbitrary units):
T++
of the proton
X+
Colliding shock waves: trapped surface
analysis
Yu.K., Lin ‘09
see also Gubser, Pufu, Yarom ’08,’09; Lin, Shuryak ’09.
Trapped Surface: Shock Waves with
Sources


To determine whether the black hole is produced and to
estimate the generated entropy use the trick invented
by Penrose – find a ‘trapped surface’, which is a ‘prehorizon’, whose appearance indicates that gravitational
collapse is inevitable.
Pioneered in AdS by Gubser, Pufu, Yarom ’08:
marginally
trapped
surface
Trapped Surface: Shock Waves without
Sources

Sources in the bulk are sometimes hard to interpret
in gauge theory. However, if one gets rid of sources
by sending them off to IR the trapped surface
remains:
Yu.K., Shu Lin, ‘09
Black Hole Production

Using trapped surface analysis one can estimate the
thermalization time (Yu.K., Lin ’09; see also
Grumiller, Romatschke ’08)
 th ~


1
 1/ 3
~
1
~ 0.07 fm / c
 1/ 3
(p )
This is parametrically shorter than the time of shock
wave stopping:
1
1

x ~
~
 a L A1/ 3
(Part of) the system thermalizes before shock waves
stop!
Black Hole Production

Estimating the produced entropy by calculating the
area of the trapped surface one gets the energyscaling of particle multiplicity:
1/ 3
N ~ entropy ~ s
Gubser, Pufu,
Yarom, ‘08
where s is the cms energy.


The power of 1/3 is not too far from the
phenomenologically preferred 0.288 (HERA) and 0.2
(RHIC).
However, one has to understand dN/d in AdS and
the amount of baryon stopping to make a more
comprehensive comparison.
Black Hole Production




It appears that the black hole is at z= ∞ with a
horizon at finite z, independent of transverse
coordinates, similar to Janik and Peschanski case.
In our case we have rapidity-dependence.
We conclude that thermalization does happen in
heavy ion collisions at strong coupling.
We expect that it happens before the shock waves
stop.
Conclusions




We have constructed graviton expansion for the
collision of two shock waves in AdS, with the goal of
obtaining energy-momentum tensor of the produced
strongly-coupled matter in the gauge theory.
We have solved the pA scattering problem in AdS in
the eikonal approximation.
Shock waves stop and probably lead to Landau-like
rapidity-dependent hydrodynamics.
We performed a trapped-surface analysis showing
that thermalization does happen in heavy ion
collisions at strong coupling, and is much quicker
than shock wave stopping.
Backup Slides
Rapidity-Independent Energy-Momentum
Tensor
If
p3  0
then, as
Deviations from the
like
p3
~
1
 1 
,
  p3
d

d

 ~
0
, which does work
1

, one gets
~
1

1 
scaling of energy density,
are due to longitudinal pressure
p3 dV
in the longitudinal direction
modifying the energy density scaling with tau.
 Positive longitudinal
pressure and isotropization
↔ deviations from
 ~
1

.
Delta-prime shocks

For delta-prime shock waves the result is surprising.
The all-order eikonal answer for pA is given by
LO+NLO terms:
+

That is, graviton exchange series terminates at NLO.
Delta-prime shocks

The answer for transverse pressure is
with the shock waves
t1 ( x  )  L21  ' ( x  ),

t2 ( x  )  L22  ' ( x  )
As p goes negative at late times, this is clearly not
hydrodynamics and not free streaming.
Delta-prime shocks


Note that the energy momentum tensor becomes
rapidity-dependent:
Thus we conclude that initially the matter
distribution is rapidity-dependent. Hence at late
times it will be rapidity-dependent too (causality).
Can one get Bjorken hydro still? Probably not…
Unphysical shock waves

One can show that the conclusion about nuclear
stopping holds for any energy-momentum tensor of
the nuclei such that






dx
T
(
x
)  0,
1






dx
T
(
x
) 0
2



To mimic weak coupling effects in the gravity dual
we propose using unphysical shock waves with not
positive-definite energy-momentum tensor:






dx
T
(
x
)  0,
1 



dx
T
(
x
) 0
2 

Unphysical shock waves


Namely we take T1  L21    ( x  ), T2  L22    ( x  )
 ( )  p( )   p3 ( )  8L L
2
1
This gives:
2
2
cf. Taliotis, Yu.K. ‘07

Almost like CGC at early times:
T



0
0
0 
  ( )


 ( ) 0
0 
 0

0
0
 ( )
0 


 0
0
0
  ( ) 

t
x
y
z
Energy density is now non-negative everywhere in
the forward light cone!
The system may lead to Bjorken hydro.
Will this lead to Bjorken hydro?

Not clear at this point. But if yes, the transition may
look like this:
(Yu.K., Taliotis ‘07)
Janik,
Peschanski
‘05
cf. Beuf et al ’09,
Chesler & Yaffe ‘09
Isotropization time

One can estimate this isotropization time from
AdS/CFT (Yu.K, Taliotis ‘07) obtaining
where e0 is the coefficient in Bjorken energy-scaling:

For central Au+Au collisions at RHIC at s  200 GeV / A
hydrodynamics requires =15 GeV/fm3 at =0.6 fm/c
(Heinz, Kolb ‘03), giving 0=38 fm-8/3. This leads to
in good agreement with hydrodynamics!