Initial Conditions from Shock Wave Collisions in AdS5

Yuri KovchegovThe 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

proper time

rapidity

23

20 xx

30

30ln2

1

xx

xx

0x

3x

QGP

CGC

CGC (Color Glass Condensate) =classical gluon fields. The matter distribution due toclassical gluon fields is rapidity-independent.

QGP = Quark Gluon Plasma

xx

Most General Rapidity-Independent Energy-Momentum Tensor

The most general rapidity-independent energy-momentum tensor for a high energy collision of two very large nuclei is (at x3 =0)

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

3

which, due to 0 T

gives

3p

d

d

0x

1x

2x

3x

3x

2x

1x

Color Glass at Very Early Times

In CGC at very early times

z

y

x

t

T

)(000

0)(00

00)(0

000)(

3p

d

d such that, since

1,1

log~ 2 SQ

0x

1x

2x

3x

we get, at the leading log level,

Energy-momentum tensor is

(Lappi ’06Fukushima ‘07)

Color Glass at Later Times: “Free Streaming”

At late times classical CGC gives free streaming,

which is characterized by the following energy-momentum tensor:

d

d

such that

and

1

~

The total energy E~ e is conserved, as expected fornon-interacting particles.

z

y

x

t

p

pT

0000

0)(00

00)(0

000)(

0x

1x

2x

3x

SQ

1

Classical FieldsClassical Fields

CGC classical gluon field leads to energy density scaling as

1

~classical

from numerical simulations by Krasnitz, Nara, Venugopalan ‘01

Much Later Times: Bjorken Hydrodynamics

In the case of ideal hydrodynamics, the energy-momentum tensor is symmetric in all three spatial directions (isotropization):

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

p

d

d

such that

Using the ideal gas equation of state, , yieldsp3

3/4

1~

Bjorken, ‘83

The total energy E~ is not conserved, while the total entropy S is conserved.

0x

1x

2x

3x

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

z=0

Our 4dworld

5d (super) gravitylives here in the AdS space

AdS5 space – a 5-dim space with a cosmological constant = -6/L2.(L is the radius of the AdS space.)

5th dimension

222

22 2 dzdxdxdxz

Lds

2

30 xxx

AdS/CFT Correspondence (Gauge-Gravity Duality)

Large-Nc, large g2 Nc N=4 SYM theory in our 4 space-timedimensions

Weakly coupledsupergravity in 5danti-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.

Energy-momentum tensor is dual to the metric in AdS. Using Fefferman-Graham coordinates one can write the metric as

with z the 5th dimension variable and the 4d metric.

Expand near the boundary of the AdS space:

For Minkowski world and with

Holographic renormalization

22

22 ),(~ dzdxdxzxgz

Lds

),(~ zxg

),(~ zxg

de Haro, Skenderis, Solodukhin ‘00

Bjorken Hydrodynamics in AdS

AdS Dual of a Static Thermal Medium

z

z=0

Our 4dworld

AdS5 black hole metric can be written as

5th dimension

2240

4240

4

240

4

2

22 )/1(

)/1(

)/1(dzxdzzdt

zz

zz

z

Lds

black hole horizonz0

Tz 2

0 with

Black hole in AdS5 ↔ Thermal medium in N=4 SYM theory.

AdS Dual of Bjorken Hydrodynamics

z=0

R3

black hole horizonz0

Janik, Peschanski ’05: to get Bjorken hydro dual need z0 =z0

Black hole recedes into the bulk: medium in 4d expands and cools off.

Asymptotic geometry

Janik and Peschanski ’05 showed that in the rapidity-independent case the geometry of AdS space at late proper times is given by the following metric

with e0 a constant. In 4d gauge theory this gives Bjorken hydrodynamics:

with

22220

2

0

2

0

2

22

3/4

4

3/4

4

3/4

4

11

1dzdxded

e

e

z

Lds z

z

z

z

y

x

t

p

p

pT

)(000

0)(00

00)(0

000)(

3/4

1~

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

)(~ xT

Shock wave in AdS

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^-):

22242

2

2

22 )(

22 dzdxdxzxT

Ndxdx

z

Lds

C

Janik, Peschanksi ‘05

Need the metric dual to a shock wave that solves Einstein equations:

06

2

12

gL

gRR

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:

22242

22 )(2 dzdxdxzxdxdxz

Lds

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

Imagine a collision of two shock waves in AdS:

We know the metric of bothshock waves, and know thatnothing happens before the collision.

Need to find a metric in theforward light cone! (cf. classical fields in CGC)

24

22

224

12

222

2

22 )(

2)(

22 dxzxT

NdxzxT

Ndzdxdxdx

z

Lds

CC

empty AdS5 1-graviton part higher ordergraviton exchanges

?

Heavy ion collisions in AdS

24

22

224

12

222

2

22 )(

2)(

22 dxzxT

NdxzxT

Ndzdxdxdx

z

Lds

CC

empty AdS5 1-graviton part higher ordergraviton exchanges

Expansion Parameter

Depends on the exact form of the energy-momentum tensor of the colliding shock waves.

For the parameter in 4d is :the expansion is good for early times only.

For that we will also considerthe expansion parameter in 4d is 2 2. Also valid for early times only.

In the bulk the expansion is valid at small-z by the same token.

)(~ xT

)(~ 2 xT

What to expect

There is one important constraint of non-negativity of energy density. It can be derived by requiring that

for any time-like t.

This gives (in rapidity-independent case)

along with

0 ttT

0)(

Janik, Peschanksi ‘05

Lowest Order Diagram

Simple dimensional analysis:

221~

Each graviton gives , hence get no rapidity dependence:e

tindependen Yee

Y

)(~~ 11

xT

)(~~ 22

xT Grumiller, Romatschke ‘08Albacete, Taliotis, Yu.K. ‘08

The same result comes out of detailed calculations.

Shock waves collision: problem 1

Energy density at mid-rapidity grows with time!? This violates 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)

0)('

221~

Shock waves collision: problem 2

Delta-functions are unwieldy. We will smear the shock wave:

with and . ( is the typical transverse momentum scale in the shock.)

Look at the energy-momentum tensor of a nucleus after collision:

Looks like by the light-cone time

the nucleus will run out of momentum and stop!

2224)2/,( x

aaxaxT

3/1

1~

1~

Aax

)()()( xaxa

x

3/12 Ap pAa /~ 3/1

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 rapidity-dependent 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 ithappens 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:

1p

2p

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 and are small.

2

1~p

xi

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

Not a free streaming medium. For ideal hydrodynamics expect

such that:

However, we get

Not hydrodynamics either.

2/5

)2/3(

2~

)(

1~

e

xxp

0

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

is right, we should be able to see

how it stops.

2224)2/,( x

aaxaxT

Proton Stopping

We have the original shock wave:

We have the produced stuff:

Adding them together we see thatthe 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

see also Gubser, Pufu, Yarom ’08,’09; Lin, Shuryak ’09.

Yu.K., Lin ‘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 ‘pre-horizon’, whose appearance indicates that gravitational collapse is inevitable.

Pioneered in AdS by Gubser, Pufu, Yarom ’08:

marginallytrappedsurface

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)

This is parametrically shorter than the time of shock wave stopping:

(Part of) the system thermalizes before shock waves stop!

cfmpth /07.0~

)(

1~

1~

3/13/1

3/1

1~

1~

Aax

Black Hole Production

Estimating the produced entropy by calculating the area of the trapped surface one gets the energy-scaling of particle multiplicity:

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.

3/1~~ sentropyNGubser, Pufu, Yarom, ‘08

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

Deviations from the scaling of energy density,

like are due to longitudinal pressure

, which does work in the longitudinal direction

modifying the energy density scaling with tau.

1

~

3p0,

1~

1

dVp3

Positive longitudinal pressure and isotropization

1~

3p

d

d If then, as , one gets .03 p 1

1~

↔ deviations from

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

As p goes negative at late times, this is clearly not hydrodynamics and not free streaming.

Delta-prime shocks

)(')(),(')( 222

211

xxtxxt

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…

Delta-prime shocks

Unphysical shock waves

One can show that the conclusion about nuclear stopping holds for any energy-momentum tensor of the nuclei such that

To mimic weak coupling effects in the gravity dual we propose using unphysical shock waves with not positive-definite energy-momentum tensor:

0)(,0)( 21

xTdxxTdx

0)(,0)( 21

xTdxxTdx

Unphysical shock waves

Namely we take

This gives:

Almost like CGC at early times:

Energy density is now non-negative everywhere in the forward light cone!

The system may lead to Bjorken hydro.

)(),( 222

211

xTxT

22

213 8)()()( pp

z

y

x

t

T

)(000

0)(00

00)(0

000)(

cf. Taliotis, Yu.K. ‘07

Will this lead to Bjorken hydro?

Not clear at this point. But if yes, the transition may look like this:

Janik, Peschanski‘05

(Yu.K., Taliotis ‘07)

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 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!

AGeVs /200